Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

The following is a conversation with The following is a conversation with Terrence Tao. Widely considered to be Terrence Tao. Widely considered to be Terrence Tao. Widely considered to be one of the greatest mathematicians in one of the greatest mathematicians in one of the greatest mathematicians in history. Often referred to as the Mozart history. Often referred to as the Mozart history. Often referred to as the Mozart of math, he won the Fields Medal and the of math, he won the Fields Medal and the of math, he won the Fields Medal and the Breakthrough Prize in mathematics and Breakthrough Prize in mathematics and Breakthrough Prize in mathematics and has contributed groundbreaking work to a has contributed groundbreaking work to a has contributed groundbreaking work to a truly astonishing range of fields in truly astonishing range of fields in truly astonishing range of fields in mathematics and physics. mathematics and physics. mathematics and physics. This was a huge honor for me for many This was a huge honor for me for many This was a huge honor for me for many reasons, including the humility and reasons, including the humility and reasons, including the humility and kindness that Terry showed to me kindness that Terry showed to me kindness that Terry showed to me throughout all our interactions. It throughout all our interactions. It throughout all our interactions. It means the world. This is the Lex means the world. This is the Lex means the world. This is the Lex Freedman podcast. To support it, please Freedman podcast. To support it, please Freedman podcast. To support it, please check out our sponsors in the check out our sponsors in the check out our sponsors in the description or at description or at description or at lexfreedman.com/sponsors. lexfreedman.com/sponsors. lexfreedman.com/sponsors. And now, dear friends, here's Terren And now, dear friends, here's Terren And now, dear friends, here's Terren Tao. Tao. Tao. What was the first really difficult What was the first really difficult What was the first really difficult research level math problem that you research level math problem that you research level math problem that you encountered? One that gave you pause encountered? One that gave you pause encountered? One that gave you pause maybe. Well, I mean in your maybe. Well, I mean in your maybe. Well, I mean in your undergraduate um education, you learn undergraduate um education, you learn undergraduate um education, you learn about the really hard impossible about the really hard impossible about the really hard impossible problems like the reman hypothesis, the problems like the reman hypothesis, the problems like the reman hypothesis, the twin primes conjecture. You can make twin primes conjecture. You can make twin primes conjecture. You can make problems arbitrarily difficult. That's problems arbitrarily difficult. That's problems arbitrarily difficult. That's not really a problem. In fact, there's not really a problem. In fact, there's not really a problem. In fact, there's even problems that we know to be even problems that we know to be even problems that we know to be unsolvable. What's really interesting unsolvable. What's really interesting unsolvable. What's really interesting are the problems just at the on the are the problems just at the on the are the problems just at the on the boundary between what we can do boundary between what we can do boundary between what we can do relatively easily and what are hopeless.

relatively easily and what are hopeless. relatively easily and what are hopeless. Um but what are problems where like Um but what are problems where like Um but what are problems where like existing techniques can do like 90% of existing techniques can do like 90% of existing techniques can do like 90% of the job and then you just need that the job and then you just need that the job and then you just need that remaining 10%. Um I think as a PhD remaining 10%. Um I think as a PhD remaining 10%. Um I think as a PhD student the CA problem certainly caught student the CA problem certainly caught student the CA problem certainly caught my eye and it just got solved actually. my eye and it just got solved actually. my eye and it just got solved actually. It's a problem I've worked on a lot in It's a problem I've worked on a lot in It's a problem I've worked on a lot in my early research. Historically it came my early research. Historically it came my early research. Historically it came from a little puzzle by the Japanese from a little puzzle by the Japanese from a little puzzle by the Japanese mathematician Soji Kaya uh in like 1918 mathematician Soji Kaya uh in like 1918 mathematician Soji Kaya uh in like 1918 or so. Um, so the puzzle is that you you or so. Um, so the puzzle is that you you or so. Um, so the puzzle is that you you you have um a needle um in on the plane. you have um a needle um in on the plane. you have um a needle um in on the plane. Um think like like a like driving like Um think like like a like driving like Um think like like a like driving like on on on a road something and you you on on on a road something and you you on on on a road something and you you want it to execute a U-turn. You want to want it to execute a U-turn. You want to want it to execute a U-turn. You want to turn the needle around. Um but you want turn the needle around. Um but you want turn the needle around. Um but you want to do it in as little space as possible. to do it in as little space as possible. to do it in as little space as possible. So you want to use as little area in So you want to use as little area in So you want to use as little area in order to turn it around. So um but the order to turn it around. So um but the order to turn it around. So um but the needle is infinitely maneuverable. needle is infinitely maneuverable. needle is infinitely maneuverable. So you can imagine just spinning it So you can imagine just spinning it So you can imagine just spinning it around its um as a unit needle. You can around its um as a unit needle. You can around its um as a unit needle. You can spin it around its center. Um, and I spin it around its center. Um, and I spin it around its center. Um, and I think, um, that gives you a disc of of think, um, that gives you a disc of of think, um, that gives you a disc of of area, I think pi over four. Um, or you area, I think pi over four. Um, or you area, I think pi over four. Um, or you can do a three-point U-turn, which is can do a three-point U-turn, which is can do a three-point U-turn, which is what they we teach people in in the what they we teach people in in the what they we teach people in in the driving schools to do. Uh, and that driving schools to do. Uh, and that driving schools to do. Uh, and that actually takes area pi over 8. So, it's actually takes area pi over 8. So, it's actually takes area pi over 8. So, it's it's a little bit more efficient than um it's a little bit more efficient than um it's a little bit more efficient than um a rotation. And so, for a while, people a rotation. And so, for a while, people a rotation. And so, for a while, people thought that was the most efficient uh thought that was the most efficient uh thought that was the most efficient uh way to turn things around. But, way to turn things around. But, way to turn things around. But, Mazikovich uh showed that in fact, you Mazikovich uh showed that in fact, you Mazikovich uh showed that in fact, you could actually uh turn the needle around could actually uh turn the needle around could actually uh turn the needle around using as little area as you wanted. So using as little area as you wanted. So using as little area as you wanted. So 0001 there was some really fancy multi- 0001 there was some really fancy multi- 0001 there was some really fancy multi- um u back and forth U-turn thing that um u back and forth U-turn thing that um u back and forth U-turn thing that you could you could do that that you you could you could do that that you you could you could do that that you could turn a needle around and in so

could turn a needle around and in so could turn a needle around and in so doing it would pass through every doing it would pass through every doing it would pass through every intermediate direction. Is this in the intermediate direction. Is this in the intermediate direction. Is this in the two dimensional plane? This is in the two dimensional plane? This is in the two dimensional plane? This is in the two dimensional plane. Yeah. So we two dimensional plane. Yeah. So we two dimensional plane. Yeah. So we understand everything in two dimensions. understand everything in two dimensions. understand everything in two dimensions. So the next question is what happens in So the next question is what happens in So the next question is what happens in three dimensions. So suppose like the three dimensions. So suppose like the three dimensions. So suppose like the Hubble space telescope is tube in space Hubble space telescope is tube in space Hubble space telescope is tube in space and you want to observe every single and you want to observe every single and you want to observe every single star in the universe. So you want to star in the universe. So you want to star in the universe. So you want to rotate the telescope to reach every rotate the telescope to reach every rotate the telescope to reach every single direction. And here's unrealistic single direction. And here's unrealistic single direction. And here's unrealistic part. Suppose that space is at a part. Suppose that space is at a part. Suppose that space is at a premium, which it totally is not. Uh you premium, which it totally is not. Uh you premium, which it totally is not. Uh you want to occupy as little volume as want to occupy as little volume as want to occupy as little volume as possible in order to rotate your your possible in order to rotate your your possible in order to rotate your your needle around in order to see every needle around in order to see every needle around in order to see every single star in the sky. Um how small a single star in the sky. Um how small a single star in the sky. Um how small a volume do you need to do that? And so volume do you need to do that? And so volume do you need to do that? And so you can modify basic construction. And you can modify basic construction. And you can modify basic construction. And so if your telescope has zero thickness, so if your telescope has zero thickness, so if your telescope has zero thickness, then you can use as little volume as you then you can use as little volume as you then you can use as little volume as you need. That's a simple modification of need. That's a simple modification of need. That's a simple modification of the two dimensional construction. But the two dimensional construction. But the two dimensional construction. But the question is that if your telescope the question is that if your telescope the question is that if your telescope is not zero thickness but but just very is not zero thickness but but just very is not zero thickness but but just very very thin some thickness delta what is very thin some thickness delta what is very thin some thickness delta what is the minimum volume needed to be able to the minimum volume needed to be able to the minimum volume needed to be able to see every single direction as a function see every single direction as a function see every single direction as a function of delta. So as delta gets smaller as of delta. So as delta gets smaller as of delta. So as delta gets smaller as you need gets thinner the volume should you need gets thinner the volume should you need gets thinner the volume should go down but but how fast does it go go down but but how fast does it go go down but but how fast does it go down? Um and the conjecture was that it down? Um and the conjecture was that it down? Um and the conjecture was that it goes down very very slowly um like goes down very very slowly um like goes down very very slowly um like logarithmically um uh roughly speaking logarithmically um uh roughly speaking logarithmically um uh roughly speaking and that was proved after a lot of work.

and that was proved after a lot of work. and that was proved after a lot of work. So this seems like a puzzle. Why is it So this seems like a puzzle. Why is it So this seems like a puzzle. Why is it interesting? So it turns out to be interesting? So it turns out to be interesting? So it turns out to be surprisingly connected to a lot of surprisingly connected to a lot of surprisingly connected to a lot of problems in partial differential problems in partial differential problems in partial differential equations, in number theory, in equations, in number theory, in equations, in number theory, in geometry, comics. For example, in in geometry, comics. For example, in in geometry, comics. For example, in in wave propagation, you splash some some wave propagation, you splash some some wave propagation, you splash some some water around um you create water waves water around um you create water waves water around um you create water waves and they they travel in various and they they travel in various and they they travel in various directions. Um but waves exhibit both directions. Um but waves exhibit both directions. Um but waves exhibit both both particle and wave type behavior. So both particle and wave type behavior. So both particle and wave type behavior. So you can have what's called a wave you can have what's called a wave you can have what's called a wave packet, which is like a a very localized packet, which is like a a very localized packet, which is like a a very localized wave that is localized in space and wave that is localized in space and wave that is localized in space and moving a certain direction in time. And moving a certain direction in time. And moving a certain direction in time. And so if you plot it in both space and so if you plot it in both space and so if you plot it in both space and time, it occupies a region which looks time, it occupies a region which looks time, it occupies a region which looks like a tube. And so what can happen is like a tube. And so what can happen is like a tube. And so what can happen is that you can have a wave which initially that you can have a wave which initially that you can have a wave which initially is very dispersed but it all comes it is very dispersed but it all comes it is very dispersed but it all comes it all focuses at a single point later in all focuses at a single point later in all focuses at a single point later in time. Like you can imagine dropping a time. Like you can imagine dropping a time. Like you can imagine dropping a pebble into a pond and ripples spread pebble into a pond and ripples spread pebble into a pond and ripples spread out. But then if you time reverse that out. But then if you time reverse that out. But then if you time reverse that that um that scenario and the equations that um that scenario and the equations that um that scenario and the equations of wave motion are time reversible. You of wave motion are time reversible. You of wave motion are time reversible. You can imagine ripples that are converging can imagine ripples that are converging can imagine ripples that are converging um to a single point and then a big um to a single point and then a big um to a single point and then a big splash occurs um maybe even a splash occurs um maybe even a splash occurs um maybe even a singularity. singularity. singularity. Um and so it's possible to do that. Uh Um and so it's possible to do that. Uh Um and so it's possible to do that. Uh and geometrically what's going on is and geometrically what's going on is and geometrically what's going on is that there's always s of light rays. Um that there's always s of light rays. Um that there's always s of light rays. Um so like if if if this wave represents so like if if if this wave represents so like if if if this wave represents light for example um you can imagine light for example um you can imagine light for example um you can imagine this wave as a superp position of this wave as a superp position of this wave as a superp position of photons um all traveling at the speed of photons um all traveling at the speed of photons um all traveling at the speed of light. They all travel on these light light. They all travel on these light light. They all travel on these light rays and they're all focusing at this rays and they're all focusing at this rays and they're all focusing at this one point. So you can have a very one point. So you can have a very one point. So you can have a very dispersed wave focus into a very dispersed wave focus into a very dispersed wave focus into a very concentrated wave at one point in space concentrated wave at one point in space concentrated wave at one point in space and time, but then it defocuses again and time, but then it defocuses again and time, but then it defocuses again and it separates. But potentially if the and it separates. But potentially if the and it separates. But potentially if the conjecture had a negative solution. So conjecture had a negative solution. So conjecture had a negative solution. So what that meant is that there's there's

what that meant is that there's there's what that meant is that there's there's a very efficient way to pack um tubes a very efficient way to pack um tubes a very efficient way to pack um tubes pointing different directions into a pointing different directions into a pointing different directions into a very very narrow region of of of very very very narrow region of of of very very very narrow region of of of very narrow volume. Then you would also be narrow volume. Then you would also be narrow volume. Then you would also be able to create waves that start out some able to create waves that start out some able to create waves that start out some there'll be some arrangement of waves there'll be some arrangement of waves there'll be some arrangement of waves that start out very very dispersed but that start out very very dispersed but that start out very very dispersed but they would concentrate not just at a they would concentrate not just at a they would concentrate not just at a single point but um um there'll be a single point but um um there'll be a single point but um um there'll be a large um there'll be a lot of large um there'll be a lot of large um there'll be a lot of concentrations in space and time and uh concentrations in space and time and uh concentrations in space and time and uh um and you could create what's called a um and you could create what's called a um and you could create what's called a blowup where these waves their amplitude blowup where these waves their amplitude blowup where these waves their amplitude becomes so great that the laws of becomes so great that the laws of becomes so great that the laws of physics that they're governed by are no physics that they're governed by are no physics that they're governed by are no longer wave equations but something more longer wave equations but something more longer wave equations but something more complicated and nonlinear. Um and so in complicated and nonlinear. Um and so in complicated and nonlinear. Um and so in mathematical physics we care a lot about mathematical physics we care a lot about mathematical physics we care a lot about whether certain equations in in wave whether certain equations in in wave whether certain equations in in wave equations are stable or not whether they equations are stable or not whether they equations are stable or not whether they can create um these singularities. can create um these singularities. can create um these singularities. There's a famous unsolved problem called There's a famous unsolved problem called There's a famous unsolved problem called the Navia Stokes regularity problem. So the Navia Stokes regularity problem. So the Navia Stokes regularity problem. So the Navia Stokes equations equations the Navia Stokes equations equations the Navia Stokes equations equations that govern the fluid flow for that govern the fluid flow for that govern the fluid flow for incompressible fluids like water. The incompressible fluids like water. The incompressible fluids like water. The question asks if you start with a smooth question asks if you start with a smooth question asks if you start with a smooth velocity field of water can it ever velocity field of water can it ever velocity field of water can it ever concentrate so much that like the concentrate so much that like the concentrate so much that like the velocity becomes infinite at some point velocity becomes infinite at some point velocity becomes infinite at some point that's called a singularity. We don't that's called a singularity. We don't that's called a singularity. We don't see that um in real life. You know, if see that um in real life. You know, if see that um in real life. You know, if you splash around water on the bathtub, you splash around water on the bathtub, you splash around water on the bathtub, it won't explode on you. Um or or have it won't explode on you. Um or or have it won't explode on you. Um or or have have water leaving at the speed of have water leaving at the speed of have water leaving at the speed of light, I think. But potentially, it is light, I think. But potentially, it is light, I think. But potentially, it is possible. Um and in fact, in recent possible. Um and in fact, in recent possible. Um and in fact, in recent years, the the consensus has has drifted years, the the consensus has has drifted years, the the consensus has has drifted towards the uh the belief that uh that towards the uh the belief that uh that towards the uh the belief that uh that in fact for certain very special initial in fact for certain very special initial in fact for certain very special initial configurations of of say water that configurations of of say water that configurations of of say water that singularities can form. But people have

singularities can form. But people have singularities can form. But people have not yet been able to uh to actually not yet been able to uh to actually not yet been able to uh to actually establish this. The clay foundation has establish this. The clay foundation has establish this. The clay foundation has these seven millennium prize problems these seven millennium prize problems these seven millennium prize problems has a million dollar prize for solving has a million dollar prize for solving has a million dollar prize for solving one of these problems that this is one one of these problems that this is one one of these problems that this is one of them. Of these seven only one of them of them. Of these seven only one of them of them. Of these seven only one of them has been solved the point conjecture by has been solved the point conjecture by has been solved the point conjecture by Pelman. So the Ka conjecture is not Pelman. So the Ka conjecture is not Pelman. So the Ka conjecture is not directly directly related to the Navis directly directly related to the Navis directly directly related to the Navis Stokes problem but understanding it Stokes problem but understanding it Stokes problem but understanding it would help us understand some aspects of would help us understand some aspects of would help us understand some aspects of things like wave concentration which things like wave concentration which things like wave concentration which would indirectly probably help us would indirectly probably help us would indirectly probably help us understand the Navis problem better. Can understand the Navis problem better. Can understand the Navis problem better. Can you speak to the neighbors? So the you speak to the neighbors? So the you speak to the neighbors? So the existence and smoothness like you said existence and smoothness like you said existence and smoothness like you said millennial prize problem right you've millennial prize problem right you've millennial prize problem right you've made a lot of progress on this one in made a lot of progress on this one in made a lot of progress on this one in 2016 you published a paper finite time 2016 you published a paper finite time 2016 you published a paper finite time blow up for an averaged threedimensional blow up for an averaged threedimensional blow up for an averaged threedimensional navia stoke equation right navia stoke equation right navia stoke equation right so we're trying to figure out if this so we're trying to figure out if this so we're trying to figure out if this thing usually doesn't blow up right but thing usually doesn't blow up right but thing usually doesn't blow up right but can we say for sure it never blows up can we say for sure it never blows up can we say for sure it never blows up right yeah so yeah that is literally the right yeah so yeah that is literally the right yeah so yeah that is literally the the million- dollar question yeah so the million- dollar question yeah so the million- dollar question yeah so this is what distinguishes this is what distinguishes this is what distinguishes mathematicians from pretty much mathematicians from pretty much mathematicians from pretty much everybody else like it everybody else like it everybody else like it If something holds 99.99% of the time, If something holds 99.99% of the time, If something holds 99.99% of the time, um that's good enough for most, you um that's good enough for most, you um that's good enough for most, you know, uh for for most things, but know, uh for for most things, but know, uh for for most things, but mathematicians are one of the few people mathematicians are one of the few people mathematicians are one of the few people who really care about whether every like who really care about whether every like who really care about whether every like 100% really 100% of all um situations 100% really 100% of all um situations 100% really 100% of all um situations are covered by by um yeah, so most fluid are covered by by um yeah, so most fluid are covered by by um yeah, so most fluid most of the time um water that does not most of the time um water that does not most of the time um water that does not blow up. But could you design a very blow up. But could you design a very blow up. But could you design a very special initial state that does this?

special initial state that does this? special initial state that does this? And maybe we should say that this is a And maybe we should say that this is a And maybe we should say that this is a this is a set of equations that govern this is a set of equations that govern this is a set of equations that govern in the field of fluid dynamics. Trying in the field of fluid dynamics. Trying in the field of fluid dynamics. Trying to understand how fluid behaves and it's to understand how fluid behaves and it's to understand how fluid behaves and it's actually turns out to be a really comp actually turns out to be a really comp actually turns out to be a really comp you know fluid is yeah extremely you know fluid is yeah extremely you know fluid is yeah extremely complicated thing to try to model. Yeah. complicated thing to try to model. Yeah. complicated thing to try to model. Yeah. So it has practical importance. So this So it has practical importance. So this So it has practical importance. So this clay price problem concerns what's clay price problem concerns what's clay price problem concerns what's called the incompressible navio stokes called the incompressible navio stokes called the incompressible navio stokes which governs things like water. There's which governs things like water. There's which governs things like water. There's something called the compressible navio something called the compressible navio something called the compressible navio stokes which governs things like air. stokes which governs things like air. stokes which governs things like air. And that's particularly important for And that's particularly important for And that's particularly important for weather prediction. Weather prediction weather prediction. Weather prediction weather prediction. Weather prediction it does a lot of computational fluid it does a lot of computational fluid it does a lot of computational fluid dynamics. A lot of it is actually just dynamics. A lot of it is actually just dynamics. A lot of it is actually just trying to solve the ny stokes equations trying to solve the ny stokes equations trying to solve the ny stokes equations as best they can. Um also gathering a as best they can. Um also gathering a as best they can. Um also gathering a lot of data so that they can get they lot of data so that they can get they lot of data so that they can get they can in initialize the equation. There's can in initialize the equation. There's can in initialize the equation. There's a lot of moving parts. So it's very a lot of moving parts. So it's very a lot of moving parts. So it's very important practically. Why is it important practically. Why is it important practically. Why is it difficult to prove general things difficult to prove general things difficult to prove general things about the set of equations like it not about the set of equations like it not about the set of equations like it not not blowing up? Short answer is not blowing up? Short answer is not blowing up? Short answer is Maxwell's demon. Um so exos demon is a Maxwell's demon. Um so exos demon is a Maxwell's demon. Um so exos demon is a concept in thermodynamics like if you concept in thermodynamics like if you concept in thermodynamics like if you have a box of two gases and oxygen and have a box of two gases and oxygen and have a box of two gases and oxygen and hydrogen uh and maybe you start with all hydrogen uh and maybe you start with all hydrogen uh and maybe you start with all the oxygen one side and nitrogen the the oxygen one side and nitrogen the the oxygen one side and nitrogen the other side but there's no barrier other side but there's no barrier other side but there's no barrier between them right then they will mix um between them right then they will mix um between them right then they will mix um and they should stay mixed right there and they should stay mixed right there and they should stay mixed right there there's no reason why they should unmix there's no reason why they should unmix there's no reason why they should unmix but in principle because of all the but in principle because of all the but in principle because of all the collisions between them there could be collisions between them there could be collisions between them there could be some sort of weird conspiracy that that some sort of weird conspiracy that that some sort of weird conspiracy that that um like maybe there's a microscopic um like maybe there's a microscopic um like maybe there's a microscopic demon called Maxwell's demon that will demon called Maxwell's demon that will demon called Maxwell's demon that will um every time a oxygen and nitrogen atom um every time a oxygen and nitrogen atom um every time a oxygen and nitrogen atom collide they will bounce off in such a collide they will bounce off in such a collide they will bounce off in such a way that the oxygen sort of drifts onto way that the oxygen sort of drifts onto way that the oxygen sort of drifts onto one side and then goes to the other and one side and then goes to the other and one side and then goes to the other and uh you could have an extremely uh you could have an extremely uh you could have an extremely improbable configuration emerge. Uh improbable configuration emerge. Uh improbable configuration emerge. Uh which we never see. Um and and we which we never see. Um and and we which we never see. Um and and we statistically it's extremely unlikely

statistically it's extremely unlikely statistically it's extremely unlikely but mathematically it's possible that but mathematically it's possible that but mathematically it's possible that this can happen and we can't rule it this can happen and we can't rule it this can happen and we can't rule it out. Um and this is a situation that out. Um and this is a situation that out. Um and this is a situation that shows up a lot in mathematics. Um a shows up a lot in mathematics. Um a shows up a lot in mathematics. Um a basic example is the digits of pi basic example is the digits of pi basic example is the digits of pi 3.14159 and so forth. The digits look 3.14159 and so forth. The digits look 3.14159 and so forth. The digits look like they have no pattern and we believe like they have no pattern and we believe like they have no pattern and we believe they have no pattern. On the long term, they have no pattern. On the long term, they have no pattern. On the long term, you should see as many ones and twos and you should see as many ones and twos and you should see as many ones and twos and threes as fours and fives and sixes. threes as fours and fives and sixes. threes as fours and fives and sixes. There should be no preference in the There should be no preference in the There should be no preference in the digits of pi to favor let's say 7 over digits of pi to favor let's say 7 over digits of pi to favor let's say 7 over 8. Um, but maybe there's some demon in 8. Um, but maybe there's some demon in 8. Um, but maybe there's some demon in the digits of pi that that like every the digits of pi that that like every the digits of pi that that like every time you compute more digits, it sort of time you compute more digits, it sort of time you compute more digits, it sort of biases one digit to another. Um, and biases one digit to another. Um, and biases one digit to another. Um, and this is a conspiracy that should not this is a conspiracy that should not this is a conspiracy that should not happen. There's no reason it should happen. There's no reason it should happen. There's no reason it should happen, but um there's there's there's happen, but um there's there's there's happen, but um there's there's there's no way to prove it. no way to prove it. no way to prove it. uh with our current technology. Okay. So uh with our current technology. Okay. So uh with our current technology. Okay. So getting back to Nabia Stokes, a fluid getting back to Nabia Stokes, a fluid getting back to Nabia Stokes, a fluid has a certain amount of energy and has a certain amount of energy and has a certain amount of energy and because a fluid is in motion, the energy because a fluid is in motion, the energy because a fluid is in motion, the energy gets transported around and water is gets transported around and water is gets transported around and water is also viscous. So if the energy is spread also viscous. So if the energy is spread also viscous. So if the energy is spread out over many different locations, the out over many different locations, the out over many different locations, the natural viscosity of the fluid will just natural viscosity of the fluid will just natural viscosity of the fluid will just damp out the energy and will it will go damp out the energy and will it will go damp out the energy and will it will go to zero. Um and this is what happens um to zero. Um and this is what happens um to zero. Um and this is what happens um in um uh when we actually experiment in um uh when we actually experiment in um uh when we actually experiment with water like you splash around there.

with water like you splash around there. with water like you splash around there. there's some turbulence and waves and so there's some turbulence and waves and so there's some turbulence and waves and so forth. But eventually it it settles down forth. But eventually it it settles down forth. But eventually it it settles down and and and the the lower the amplitude, and and and the the lower the amplitude, and and and the the lower the amplitude, the smaller the velocity, the the more the smaller the velocity, the the more the smaller the velocity, the the more calm it gets. Um but potentially there calm it gets. Um but potentially there calm it gets. Um but potentially there is some sort of a demon that keeps is some sort of a demon that keeps is some sort of a demon that keeps pushing the uh the energy of the fluid pushing the uh the energy of the fluid pushing the uh the energy of the fluid into a smaller and smaller scale and it into a smaller and smaller scale and it into a smaller and smaller scale and it will move faster and faster and at will move faster and faster and at will move faster and faster and at faster speeds the effective viscosity is faster speeds the effective viscosity is faster speeds the effective viscosity is relatively less. And so it could happen relatively less. And so it could happen relatively less. And so it could happen that that it it creates a some sort of that that it it creates a some sort of that that it it creates a some sort of um um what's called a self similar um um what's called a self similar um um what's called a self similar blowup scenario where you know um the blowup scenario where you know um the blowup scenario where you know um the energy of fluid starts off at some um energy of fluid starts off at some um energy of fluid starts off at some um large scale and then it all sort of um large scale and then it all sort of um large scale and then it all sort of um transfers it energy into a smaller um transfers it energy into a smaller um transfers it energy into a smaller um region of of of the fluid which then at region of of of the fluid which then at region of of of the fluid which then at a much faster rate um moves into um an a much faster rate um moves into um an a much faster rate um moves into um an even smaller region and so forth. Um and even smaller region and so forth. Um and even smaller region and so forth. Um and and each time it does this uh it takes and each time it does this uh it takes and each time it does this uh it takes maybe half as as long as as the previous maybe half as as long as as the previous maybe half as as long as as the previous one and then you you could you could one and then you you could you could one and then you you could you could actually uh converge to all the energy actually uh converge to all the energy actually uh converge to all the energy concentrating in one point in a finite concentrating in one point in a finite concentrating in one point in a finite amount of time. Um and that that's uh amount of time. Um and that that's uh amount of time. Um and that that's uh that scenario is called finite blow up.

that scenario is called finite blow up. that scenario is called finite blow up. Um so in practice this doesn't happen. Um so in practice this doesn't happen. Um so in practice this doesn't happen. Um so water is what's called turbulent. Um so water is what's called turbulent. Um so water is what's called turbulent. Um so it is true that um if you have a Um so it is true that um if you have a Um so it is true that um if you have a big eddy of water it will tend to break big eddy of water it will tend to break big eddy of water it will tend to break up into smaller eddies but it won't up into smaller eddies but it won't up into smaller eddies but it won't transfer all the the energy from one big transfer all the the energy from one big transfer all the the energy from one big eddy into one smaller eddy. It will eddy into one smaller eddy. It will eddy into one smaller eddy. It will transfer into maybe three or four and transfer into maybe three or four and transfer into maybe three or four and then those must split up into maybe then those must split up into maybe then those must split up into maybe three or four small edies of their own three or four small edies of their own three or four small edies of their own and so the energy gets dispersed to the and so the energy gets dispersed to the and so the energy gets dispersed to the point where the viscosity can can then point where the viscosity can can then point where the viscosity can can then keep that thing under control. Um but if keep that thing under control. Um but if keep that thing under control. Um but if it can somehow um concentrate um all the it can somehow um concentrate um all the it can somehow um concentrate um all the energy keep it all together um and do it energy keep it all together um and do it energy keep it all together um and do it fast enough that the viscous effects fast enough that the viscous effects fast enough that the viscous effects don't have enough time to calm don't have enough time to calm don't have enough time to calm everything down then this blob can everything down then this blob can everything down then this blob can occur. So there were papers who had occur. So there were papers who had occur. So there were papers who had claimed that oh you just need to take claimed that oh you just need to take claimed that oh you just need to take into account conservation energy and into account conservation energy and into account conservation energy and just carefully use the viscosity and you just carefully use the viscosity and you just carefully use the viscosity and you can keep everything under control for can keep everything under control for can keep everything under control for not just Navia Stokes but for many many not just Navia Stokes but for many many not just Navia Stokes but for many many types of equations like this and so in types of equations like this and so in types of equations like this and so in the past there have been many attempts the past there have been many attempts the past there have been many attempts to try to obtain what's called global to try to obtain what's called global to try to obtain what's called global regularity for Navio Stokes which is the regularity for Navio Stokes which is the regularity for Navio Stokes which is the opposite of final time blow up that opposite of final time blow up that opposite of final time blow up that velocity say smooth and it all failed velocity say smooth and it all failed velocity say smooth and it all failed there was always some sign error or some there was always some sign error or some there was always some sign error or some subtle mistake and and it couldn't be subtle mistake and and it couldn't be subtle mistake and and it couldn't be salvaged. Um so what I was interested in salvaged. Um so what I was interested in salvaged. Um so what I was interested in doing was trying to explain why we were doing was trying to explain why we were doing was trying to explain why we were not able to disprove um planet time blow not able to disprove um planet time blow not able to disprove um planet time blow up. I couldn't do it for the actual up. I couldn't do it for the actual up. I couldn't do it for the actual equations of fluids which were too equations of fluids which were too equations of fluids which were too complicated. But if I could average the complicated. But if I could average the complicated. But if I could average the equations of motion of naval basically equations of motion of naval basically equations of motion of naval basically if if um if I could turn off certain if if um if I could turn off certain if if um if I could turn off certain types of of ways in which water types of of ways in which water types of of ways in which water interacts and only keep the ones that I interacts and only keep the ones that I interacts and only keep the ones that I want. Um, so in particular, um, if, um, want. Um, so in particular, um, if, um, want. Um, so in particular, um, if, um, if there's a fluid and it could transfer

if there's a fluid and it could transfer if there's a fluid and it could transfer energy from a large Eddie into this energy from a large Eddie into this energy from a large Eddie into this small Eddie or this other small Eddie, I small Eddie or this other small Eddie, I small Eddie or this other small Eddie, I would turn off the energy channel that would turn off the energy channel that would turn off the energy channel that would transfer energy to this this one would transfer energy to this this one would transfer energy to this this one and and direct it only into um, this and and direct it only into um, this and and direct it only into um, this smaller Eddie while still preserving the smaller Eddie while still preserving the smaller Eddie while still preserving the law of conservation of energy. So you're law of conservation of energy. So you're law of conservation of energy. So you're trying to make it blow up. Yeah. Yeah. trying to make it blow up. Yeah. Yeah. trying to make it blow up. Yeah. Yeah. So I I I basically engineer um, a blow So I I I basically engineer um, a blow So I I I basically engineer um, a blow up by changing the laws of physics, up by changing the laws of physics, up by changing the laws of physics, which is one thing that mathematicians which is one thing that mathematicians which is one thing that mathematicians are allowed to do. We can change the are allowed to do. We can change the are allowed to do. We can change the equation. How does that help you get equation. How does that help you get equation. How does that help you get closer to the proof of something? Right? closer to the proof of something? Right? closer to the proof of something? Right? So, it provides what's called an So, it provides what's called an So, it provides what's called an obstruction in mathematics. Um, so, so obstruction in mathematics. Um, so, so obstruction in mathematics. Um, so, so what I did was that uh basically if I what I did was that uh basically if I what I did was that uh basically if I turned off the um certain parts of the turned off the um certain parts of the turned off the um certain parts of the equation, so which usually when you turn equation, so which usually when you turn equation, so which usually when you turn off certain interactions make it less off certain interactions make it less off certain interactions make it less nonlinear, it makes it more regular and nonlinear, it makes it more regular and nonlinear, it makes it more regular and less likely to blow up. But I found that less likely to blow up. But I found that less likely to blow up. But I found that by turning off a very well-designed set by turning off a very well-designed set by turning off a very well-designed set of of of of interactions, I could force of of of of interactions, I could force of of of of interactions, I could force all the energy to blow in finite time. all the energy to blow in finite time. all the energy to blow in finite time. So what that means is that if you wanted So what that means is that if you wanted So what that means is that if you wanted to prove um global regularity for Navia to prove um global regularity for Navia to prove um global regularity for Navia Stokes um for the actual equation you Stokes um for the actual equation you Stokes um for the actual equation you had you must use some feature of the had you must use some feature of the had you must use some feature of the true equation which which my artificial true equation which which my artificial true equation which which my artificial equation um does not satisfy. So it it equation um does not satisfy. So it it equation um does not satisfy. So it it rules out certain um certain approaches.

rules out certain um certain approaches. rules out certain um certain approaches. So um the thing about math is is it's So um the thing about math is is it's So um the thing about math is is it's not just about finding you know taking a not just about finding you know taking a not just about finding you know taking a technique that is going to work and technique that is going to work and technique that is going to work and applying it but you you need to not take applying it but you you need to not take applying it but you you need to not take the techniques that don't work. Um and the techniques that don't work. Um and the techniques that don't work. Um and for the problems that are really hard, for the problems that are really hard, for the problems that are really hard, often there are dozens of ways that you often there are dozens of ways that you often there are dozens of ways that you might think might apply to solve the might think might apply to solve the might think might apply to solve the problem. But uh it's only after a lot of problem. But uh it's only after a lot of problem. But uh it's only after a lot of experience that you realize there's no experience that you realize there's no experience that you realize there's no way that these methods are going to way that these methods are going to way that these methods are going to work. So having these counter examples work. So having these counter examples work. So having these counter examples for nearby problems um kind of rules out for nearby problems um kind of rules out for nearby problems um kind of rules out um uh it saves you a lot of time because um uh it saves you a lot of time because um uh it saves you a lot of time because you you're not wasting um energy on on you you're not wasting um energy on on you you're not wasting um energy on on things that you now know cannot possibly things that you now know cannot possibly things that you now know cannot possibly ever work. How deeply connected is it to ever work. How deeply connected is it to ever work. How deeply connected is it to that specific problem of fluid dynamics that specific problem of fluid dynamics that specific problem of fluid dynamics or just some more general intuition you or just some more general intuition you or just some more general intuition you build up about mathematics? Right. Yeah. build up about mathematics? Right. Yeah. build up about mathematics? Right. Yeah. So the key phenomenon that uh my my So the key phenomenon that uh my my So the key phenomenon that uh my my technique exploits is what's called technique exploits is what's called technique exploits is what's called superc criticality. So in partial superc criticality. So in partial superc criticality. So in partial differential equations often these differential equations often these differential equations often these equations are like a tugof-war between equations are like a tugof-war between equations are like a tugof-war between different forces. So in Navia Stokes different forces. So in Navia Stokes different forces. So in Navia Stokes there's the dissipation um force coming there's the dissipation um force coming there's the dissipation um force coming from viscosity and it's very well from viscosity and it's very well from viscosity and it's very well understood. It's linear. It calms things understood. It's linear. It calms things understood. It's linear. It calms things down. If if viscosity was all there was, down. If if viscosity was all there was, down. If if viscosity was all there was, then then nothing bad would ever happen.

then then nothing bad would ever happen. then then nothing bad would ever happen. Um but there's also transport um that Um but there's also transport um that Um but there's also transport um that that energy from in one location of that energy from in one location of that energy from in one location of space can get transported because the space can get transported because the space can get transported because the fluid is in motion to to other fluid is in motion to to other fluid is in motion to to other locations. Um and that's a nonlinear locations. Um and that's a nonlinear locations. Um and that's a nonlinear effect and that causes all the all the effect and that causes all the all the effect and that causes all the all the problems. Um so there are these two problems. Um so there are these two problems. Um so there are these two competing terms in the Davis equation competing terms in the Davis equation competing terms in the Davis equation the dissipation term and the transport the dissipation term and the transport the dissipation term and the transport term. If the dissipation term dominates, term. If the dissipation term dominates, term. If the dissipation term dominates, if it's if it's large, then basically if it's if it's large, then basically if it's if it's large, then basically you get regularity. And if um if the you get regularity. And if um if the you get regularity. And if um if the transport term dominates, then uh then transport term dominates, then uh then transport term dominates, then uh then we don't know what's going on. It's a we don't know what's going on. It's a we don't know what's going on. It's a very nonlinear situation. It's very nonlinear situation. It's very nonlinear situation. It's unpredictable. It's turbulent. So unpredictable. It's turbulent. So unpredictable. It's turbulent. So sometimes these forces are in balance at sometimes these forces are in balance at sometimes these forces are in balance at small scales, but not in balance at small scales, but not in balance at small scales, but not in balance at large scales or or vice versa. Um so large scales or or vice versa. Um so large scales or or vice versa. Um so Navis Stokes is what's called Navis Stokes is what's called Navis Stokes is what's called supercritical. So at at smaller and supercritical. So at at smaller and supercritical. So at at smaller and smaller scales, the transport terms are smaller scales, the transport terms are smaller scales, the transport terms are much stronger than the viscosity terms. much stronger than the viscosity terms. much stronger than the viscosity terms. So the viscosity are the things that So the viscosity are the things that So the viscosity are the things that calm things down. Um and so this is um calm things down. Um and so this is um calm things down. Um and so this is um um this is why the problem is hard in um this is why the problem is hard in um this is why the problem is hard in two dimensions. So the Soviet two dimensions. So the Soviet two dimensions. So the Soviet mathematician ladish skaya she in the mathematician ladish skaya she in the mathematician ladish skaya she in the 60s shows in two dimensions there is no 60s shows in two dimensions there is no 60s shows in two dimensions there is no blow up and in two dimensions the nav blow up and in two dimensions the nav blow up and in two dimensions the nav equations is what's called critical the equations is what's called critical the equations is what's called critical the effect of transport and the effect of effect of transport and the effect of effect of transport and the effect of viscosity about the same strength even viscosity about the same strength even viscosity about the same strength even at very very small scales and we have a at very very small scales and we have a at very very small scales and we have a lot of technology to handle critical and lot of technology to handle critical and lot of technology to handle critical and also subcritical equations and proof um also subcritical equations and proof um also subcritical equations and proof um regularity but for superc critical regularity but for superc critical regularity but for superc critical equations it was not clear what was equations it was not clear what was equations it was not clear what was going on going on going on and I did a lot of work and then there's and I did a lot of work and then there's and I did a lot of work and then there's been a lot of follow-up showing that for been a lot of follow-up showing that for been a lot of follow-up showing that for many other types of superc critical many other types of superc critical many other types of superc critical equations you create all kinds of blow equations you create all kinds of blow equations you create all kinds of blow up examples. Once the nonlinear effects up examples. Once the nonlinear effects up examples. Once the nonlinear effects dominate the linear effects at small dominate the linear effects at small dominate the linear effects at small scales, you can have all kinds of bad

scales, you can have all kinds of bad scales, you can have all kinds of bad things happen. So this is sort of one of things happen. So this is sort of one of things happen. So this is sort of one of the main insights of this this line of the main insights of this this line of the main insights of this this line of work is that superc criticality versus work is that superc criticality versus work is that superc criticality versus criticality and subcriticality. This criticality and subcriticality. This criticality and subcriticality. This this makes a big difference. I mean this makes a big difference. I mean this makes a big difference. I mean that's a key qualitative feature that that's a key qualitative feature that that's a key qualitative feature that distinguishes some equations for being distinguishes some equations for being distinguishes some equations for being sort of nice and predictable and you sort of nice and predictable and you sort of nice and predictable and you know like like planetary motion and I know like like planetary motion and I know like like planetary motion and I mean there are certain equations that mean there are certain equations that mean there are certain equations that that you can predict for millions of that you can predict for millions of that you can predict for millions of years and or thousands at least. Again, years and or thousands at least. Again, years and or thousands at least. Again, it's not really a problem, but but it's not really a problem, but but it's not really a problem, but but there's a reason why we can't predict there's a reason why we can't predict there's a reason why we can't predict the weather past 2 weeks into the future the weather past 2 weeks into the future the weather past 2 weeks into the future because it's a super critical equation. because it's a super critical equation. because it's a super critical equation. Lots of really strange things are going Lots of really strange things are going Lots of really strange things are going on at very fine scales. So, whenever on at very fine scales. So, whenever on at very fine scales. So, whenever there is some huge source of there is some huge source of there is some huge source of nonlinearity, nonlinearity, nonlinearity, yeah, that can create a huge problem for yeah, that can create a huge problem for yeah, that can create a huge problem for predicting what's going to happen. Yeah. predicting what's going to happen. Yeah. predicting what's going to happen. Yeah. And if the nonlinearity is somehow more And if the nonlinearity is somehow more And if the nonlinearity is somehow more and more featured and interesting at at and more featured and interesting at at and more featured and interesting at at small scales. Um I mean there's there's small scales. Um I mean there's there's small scales. Um I mean there's there's many equations that are nonlinear but um many equations that are nonlinear but um many equations that are nonlinear but um in in many equations you can approximate in in many equations you can approximate in in many equations you can approximate things by the bulk. Um so for example things by the bulk. Um so for example things by the bulk. Um so for example planetary motion you know if you want to planetary motion you know if you want to planetary motion you know if you want to understand the orbit of the moon or Mars understand the orbit of the moon or Mars understand the orbit of the moon or Mars or something you don't really need the or something you don't really need the or something you don't really need the micro structure of like the seismology micro structure of like the seismology micro structure of like the seismology of the moon or or like exactly how the of the moon or or like exactly how the of the moon or or like exactly how the mass is distributed. um you just mass is distributed. um you just mass is distributed. um you just basically you can almost approximate basically you can almost approximate basically you can almost approximate these planets by point masses and just these planets by point masses and just these planets by point masses and just the aggregate behavior is important um the aggregate behavior is important um the aggregate behavior is important um but if you want to model a fluid um like but if you want to model a fluid um like but if you want to model a fluid um like like the weather you can't just say in like the weather you can't just say in like the weather you can't just say in Los Angeles the temperature is this the Los Angeles the temperature is this the Los Angeles the temperature is this the wind speed is this for super critical wind speed is this for super critical wind speed is this for super critical equations the finance confirmation is is equations the finance confirmation is is equations the finance confirmation is is really important if we can just linger really important if we can just linger really important if we can just linger on the narto's uh equations a little bit on the narto's uh equations a little bit on the narto's uh equations a little bit so you've suggested maybe you can so you've suggested maybe you can so you've suggested maybe you can describe it that one of the ways to uh

describe it that one of the ways to uh describe it that one of the ways to uh solve it or to negatively resolve it solve it or to negatively resolve it solve it or to negatively resolve it would be to would be to would be to sort of to construct a liquid a kind of sort of to construct a liquid a kind of sort of to construct a liquid a kind of liquid computer, right? And then show liquid computer, right? And then show liquid computer, right? And then show that the halting problem from that the halting problem from that the halting problem from computation theory has consequences for computation theory has consequences for computation theory has consequences for fluid dynamics. So uh show it in that fluid dynamics. So uh show it in that fluid dynamics. So uh show it in that way. Can you describe this this Yeah. So way. Can you describe this this Yeah. So way. Can you describe this this Yeah. So this came out of of this work of this came out of of this work of this came out of of this work of constructing this this this average constructing this this this average constructing this this this average equation that that blew up. Um so one um equation that that blew up. Um so one um equation that that blew up. Um so one um as as part of how I had to do this. So as as part of how I had to do this. So as as part of how I had to do this. So there this naive way to do it. You you there this naive way to do it. You you there this naive way to do it. You you just keep pushing um um every time you just keep pushing um um every time you just keep pushing um um every time you you get energy at one scale you you push you get energy at one scale you you push you get energy at one scale you you push it immediately to the next scale as as it immediately to the next scale as as it immediately to the next scale as as fast as possible. This is sort of the fast as possible. This is sort of the fast as possible. This is sort of the naive way to to to to force blow up. Um naive way to to to to force blow up. Um naive way to to to to force blow up. Um it turns out in five and high dimensions it turns out in five and high dimensions it turns out in five and high dimensions this works. Um but in three dimensions this works. Um but in three dimensions this works. Um but in three dimensions there was this funny phenomenon that I there was this funny phenomenon that I there was this funny phenomenon that I discovered that if you if you keep if if discovered that if you if you keep if if discovered that if you if you keep if if you change the laws of physics you just you change the laws of physics you just you change the laws of physics you just always keep trying to push um the energy always keep trying to push um the energy always keep trying to push um the energy into smaller smaller scales. Um what into smaller smaller scales. Um what into smaller smaller scales. Um what happens is that the energy starts happens is that the energy starts happens is that the energy starts getting spread out into multi many getting spread out into multi many getting spread out into multi many scales at once. Um so that you you have scales at once. Um so that you you have scales at once. Um so that you you have energy at one scale you're pushing it energy at one scale you're pushing it energy at one scale you're pushing it into the next scale and then um as soon into the next scale and then um as soon into the next scale and then um as soon as it enters that scale you also push it as it enters that scale you also push it as it enters that scale you also push it to the next scale but there's still some to the next scale but there's still some to the next scale but there's still some energy left over from the previous energy left over from the previous energy left over from the previous scale. um you're trying to do everything scale. um you're trying to do everything scale. um you're trying to do everything at once. Um and this spreads out the at once. Um and this spreads out the at once. Um and this spreads out the energy too much. Um and then it turns energy too much. Um and then it turns energy too much. Um and then it turns out that that um it makes it vulnerable out that that um it makes it vulnerable out that that um it makes it vulnerable for viscosity to come in and actually for viscosity to come in and actually for viscosity to come in and actually just damp out everything. So um so it just damp out everything. So um so it just damp out everything. So um so it turns out this this direct bush doesn't turns out this this direct bush doesn't turns out this this direct bush doesn't doesn't actually work. There was a doesn't actually work. There was a doesn't actually work. There was a separate paper by some other authors separate paper by some other authors separate paper by some other authors that actually showed this um in three that actually showed this um in three that actually showed this um in three dimensions. Um so what I needed was to

dimensions. Um so what I needed was to dimensions. Um so what I needed was to program a delay. Um so kind of like air program a delay. Um so kind of like air program a delay. Um so kind of like air locks. So um I needed an equation which locks. So um I needed an equation which locks. So um I needed an equation which would start with a fluid doing something would start with a fluid doing something would start with a fluid doing something at one scale. It would push this energy at one scale. It would push this energy at one scale. It would push this energy into the next scale but it would stay into the next scale but it would stay into the next scale but it would stay there until all the energy from the from there until all the energy from the from there until all the energy from the from the larger scale got transferred and the larger scale got transferred and the larger scale got transferred and only after you pushed all the energy in only after you pushed all the energy in only after you pushed all the energy in then you sort of open the next gate and then you sort of open the next gate and then you sort of open the next gate and and then you you push that in as well. and then you you push that in as well. and then you you push that in as well. So um by doing that it kind of the So um by doing that it kind of the So um by doing that it kind of the energy inches forward scale by scale in energy inches forward scale by scale in energy inches forward scale by scale in such a way that it's always um localized such a way that it's always um localized such a way that it's always um localized at one scale at a time. Um and then it at one scale at a time. Um and then it at one scale at a time. Um and then it can resist the effects of viscosity can resist the effects of viscosity can resist the effects of viscosity because it's not dispersed. Um so in because it's not dispersed. Um so in because it's not dispersed. Um so in order to make that happen um yeah I had order to make that happen um yeah I had order to make that happen um yeah I had to construct a rather complicated to construct a rather complicated to construct a rather complicated nonlinearity. Um and it was basically nonlinearity. Um and it was basically nonlinearity. Um and it was basically like um you know like was constructed like um you know like was constructed like um you know like was constructed like electronic circuit. So I I actually like electronic circuit. So I I actually like electronic circuit. So I I actually thank my wife for this because she was thank my wife for this because she was thank my wife for this because she was trained as a electrical engineer. Um and trained as a electrical engineer. Um and trained as a electrical engineer. Um and um you know he talked about um uh you um you know he talked about um uh you um you know he talked about um uh you know he had to design circuits and so know he had to design circuits and so know he had to design circuits and so forth. And you know if if you want a forth. And you know if if you want a forth. And you know if if you want a circuit that does a certain thing like circuit that does a certain thing like circuit that does a certain thing like maybe have a light that that flashes on maybe have a light that that flashes on maybe have a light that that flashes on and then turns off and then on and then and then turns off and then on and then and then turns off and then on and then off. You can build it from from more off. You can build it from from more off. You can build it from from more primitive components you know capacitors primitive components you know capacitors primitive components you know capacitors and resistors and so forth and you have and resistors and so forth and you have and resistors and so forth and you have to build a diagram and you um and these to build a diagram and you um and these to build a diagram and you um and these diagrams you can you can sort of follow diagrams you can you can sort of follow diagrams you can you can sort of follow your eyeballs and say oh yeah the the your eyeballs and say oh yeah the the your eyeballs and say oh yeah the the current will build up here and then it current will build up here and then it current will build up here and then it will stop and then it will do that. So I will stop and then it will do that. So I will stop and then it will do that. So I knew how to build the analog of basic knew how to build the analog of basic knew how to build the analog of basic electronic components, you know, like electronic components, you know, like electronic components, you know, like resistors and capacitors and so forth.

resistors and capacitors and so forth. resistors and capacitors and so forth. And and I would I would stack them And and I would I would stack them And and I would I would stack them together um in in such a way that that I together um in in such a way that that I together um in in such a way that that I would create something that would open would create something that would open would create something that would open one gate and then there'll be a clock one gate and then there'll be a clock one gate and then there'll be a clock that would and then once the clock hits that would and then once the clock hits that would and then once the clock hits a certain threshold it would close it a certain threshold it would close it a certain threshold it would close it kind of a rude Goldberg type machine but kind of a rude Goldberg type machine but kind of a rude Goldberg type machine but described mathematically and this ended described mathematically and this ended described mathematically and this ended up working. So what I realized is that up working. So what I realized is that up working. So what I realized is that if you could pull the same thing off for if you could pull the same thing off for if you could pull the same thing off for the actual equations. So if the the actual equations. So if the the actual equations. So if the equations of water support a computation equations of water support a computation equations of water support a computation so um like if you can imagine kind of a so um like if you can imagine kind of a so um like if you can imagine kind of a steampunk but really water punk uh type steampunk but really water punk uh type steampunk but really water punk uh type of thing where um you know so modern of thing where um you know so modern of thing where um you know so modern computers are electronic you know they computers are electronic you know they computers are electronic you know they they they're powered by by electrons they they're powered by by electrons they they're powered by by electrons passing through very tiny wires and passing through very tiny wires and passing through very tiny wires and interacting with other electrons and so interacting with other electrons and so interacting with other electrons and so forth. But instead of electrons, you can forth. But instead of electrons, you can forth. But instead of electrons, you can imagine these pulses of of water moving imagine these pulses of of water moving imagine these pulses of of water moving at certain velocity and maybe it's at certain velocity and maybe it's at certain velocity and maybe it's they're two different configurations they're two different configurations they're two different configurations corresponding to a bit being up or down. corresponding to a bit being up or down. corresponding to a bit being up or down. Probably if you had two of these moving Probably if you had two of these moving Probably if you had two of these moving bodies of water collide, it would come bodies of water collide, it would come bodies of water collide, it would come out with some new configuration which is out with some new configuration which is out with some new configuration which is which would be something like an ANDgate which would be something like an ANDgate which would be something like an ANDgate or orgate. you know that if the the the or orgate. you know that if the the the or orgate. you know that if the the the output would depend in a very output would depend in a very output would depend in a very predictable way on on the inputs and predictable way on on the inputs and predictable way on on the inputs and like you could chain these together and like you could chain these together and like you could chain these together and maybe create a touring machine and and maybe create a touring machine and and maybe create a touring machine and and then you could you have computers which then you could you have computers which then you could you have computers which are made completely out of water um and are made completely out of water um and are made completely out of water um and if you have computers then maybe you can if you have computers then maybe you can if you have computers then maybe you can do robotics so I you know hydraulics and do robotics so I you know hydraulics and do robotics so I you know hydraulics and so forth um and so you could create some so forth um and so you could create some so forth um and so you could create some machine which is basically a fluid machine which is basically a fluid machine which is basically a fluid analog what's called a vonomian machine analog what's called a vonomian machine analog what's called a vonomian machine so vonomian proposed if you want to so vonomian proposed if you want to so vonomian proposed if you want to colonize Mars. The sheer cost of colonize Mars. The sheer cost of colonize Mars. The sheer cost of transporting people machines to Mars is

transporting people machines to Mars is transporting people machines to Mars is just ridiculous. But if you could just ridiculous. But if you could just ridiculous. But if you could transport one machine to Mars and this transport one machine to Mars and this transport one machine to Mars and this machine had the ability to mine the machine had the ability to mine the machine had the ability to mine the planet, create some more materials to planet, create some more materials to planet, create some more materials to smelt them and build more copies of the smelt them and build more copies of the smelt them and build more copies of the same machine. Um, then you could same machine. Um, then you could same machine. Um, then you could colonize a whole planet um over time. colonize a whole planet um over time. colonize a whole planet um over time. Um, so uh if you could build a fluid Um, so uh if you could build a fluid Um, so uh if you could build a fluid machine, which uh yeah, so it's it's machine, which uh yeah, so it's it's machine, which uh yeah, so it's it's it's a it's a robot. Okay. And what it it's a it's a robot. Okay. And what it it's a it's a robot. Okay. And what it would do it its purpose in life, it's would do it its purpose in life, it's would do it its purpose in life, it's programmed so that it would create a programmed so that it would create a programmed so that it would create a smaller version of itself in some sort smaller version of itself in some sort smaller version of itself in some sort of cold state. It wouldn't start just of cold state. It wouldn't start just of cold state. It wouldn't start just yet. Once it's ready, the big robot yet. Once it's ready, the big robot yet. Once it's ready, the big robot configuration water would transfer all configuration water would transfer all configuration water would transfer all his energy into the smaller his energy into the smaller his energy into the smaller configuration and then power down. Okay? configuration and then power down. Okay? configuration and then power down. Okay? And then like I clean itself up. And And then like I clean itself up. And And then like I clean itself up. And then what's left is this newest state then what's left is this newest state then what's left is this newest state which would then turn on and do the same which would then turn on and do the same which would then turn on and do the same thing but smaller and faster. And then thing but smaller and faster. And then thing but smaller and faster. And then the equation has a certain scaling the equation has a certain scaling the equation has a certain scaling symmetry. Once you do that, it can just symmetry. Once you do that, it can just symmetry. Once you do that, it can just keep iterating. So this in principle keep iterating. So this in principle keep iterating. So this in principle would create a blow up uh for the actual would create a blow up uh for the actual would create a blow up uh for the actual Navia Stokes and this is what I managed Navia Stokes and this is what I managed Navia Stokes and this is what I managed to accomplish for this average Navia to accomplish for this average Navia to accomplish for this average Navia Stokes. So it provided the sort of road Stokes. So it provided the sort of road Stokes. So it provided the sort of road map to solve the problem. Now this is uh map to solve the problem. Now this is uh map to solve the problem. Now this is uh a pipe dream because uh there are so a pipe dream because uh there are so a pipe dream because uh there are so many things that are missing for this to many things that are missing for this to many things that are missing for this to actually be a reality. Um so um I I I actually be a reality. Um so um I I I actually be a reality. Um so um I I I can't create these basic logic gates. Um can't create these basic logic gates. Um can't create these basic logic gates. Um I I don't I don't have these in these I I don't I don't have these in these I I don't I don't have these in these special configurations of water. Um, I special configurations of water. Um, I special configurations of water. Um, I mean there's candidates there things mean there's candidates there things mean there's candidates there things called vortex rings that might possibly called vortex rings that might possibly called vortex rings that might possibly work but um um but also you know analog work but um um but also you know analog work but um um but also you know analog computing is really nasty um compared to computing is really nasty um compared to computing is really nasty um compared to digital computing. I mean because digital computing. I mean because digital computing. I mean because there's always errors um you you have to

there's always errors um you you have to there's always errors um you you have to you have to do a lot of error correction you have to do a lot of error correction you have to do a lot of error correction along the way. I don't know how to along the way. I don't know how to along the way. I don't know how to completely power down the big machine so completely power down the big machine so completely power down the big machine so that it doesn't interfere with the the that it doesn't interfere with the the that it doesn't interfere with the the running of the smaller machine but running of the smaller machine but running of the smaller machine but everything in principle can happen like everything in principle can happen like everything in principle can happen like it doesn't contradict any of the laws of it doesn't contradict any of the laws of it doesn't contradict any of the laws of physics. Um so it's sort of evidence physics. Um so it's sort of evidence physics. Um so it's sort of evidence that this thing is possible. Um there that this thing is possible. Um there that this thing is possible. Um there are other groups who are now pursuing are other groups who are now pursuing are other groups who are now pursuing ways to make navis blow up which are ways to make navis blow up which are ways to make navis blow up which are nowhere near as ridiculously complicated nowhere near as ridiculously complicated nowhere near as ridiculously complicated as this. Um um they they actually are as this. Um um they they actually are as this. Um um they they actually are pursuing much closer to the the direct pursuing much closer to the the direct pursuing much closer to the the direct self similar model which can it doesn't self similar model which can it doesn't self similar model which can it doesn't quite work as is but there could be some quite work as is but there could be some quite work as is but there could be some simpler scheme than what I just simpler scheme than what I just simpler scheme than what I just described to make this work. There is a described to make this work. There is a described to make this work. There is a real leap of genius here to go from real leap of genius here to go from real leap of genius here to go from Navia Stokes to this touring machine. So Navia Stokes to this touring machine. So Navia Stokes to this touring machine. So it goes from what the self similar blob it goes from what the self similar blob it goes from what the self similar blob scenario that you're trying to get the scenario that you're trying to get the scenario that you're trying to get the smaller and smaller blob to now having a smaller and smaller blob to now having a smaller and smaller blob to now having a liquid toying machine gets smaller and liquid toying machine gets smaller and liquid toying machine gets smaller and smaller and smaller and somehow seeing smaller and smaller and somehow seeing smaller and smaller and somehow seeing how that how that how that could be used could be used could be used to say something about a blowup. I mean to say something about a blowup. I mean to say something about a blowup. I mean that's a big leap. So there's precedent.

that's a big leap. So there's precedent. that's a big leap. So there's precedent. I mean um so the the thing about I mean um so the the thing about I mean um so the the thing about mathematics is that it's really good at mathematics is that it's really good at mathematics is that it's really good at um spotting connections between what you um spotting connections between what you um spotting connections between what you think of what you might think of as think of what you might think of as think of what you might think of as completely different um problems. Um but completely different um problems. Um but completely different um problems. Um but if if the mathematical form is the same if if the mathematical form is the same if if the mathematical form is the same you you can you you can you can draw a you you can you you can you can draw a you you can you you can you can draw a connection um so um there's a lot of connection um so um there's a lot of connection um so um there's a lot of work previously on what called cellular work previously on what called cellular work previously on what called cellular automator um the most famous of which is automator um the most famous of which is automator um the most famous of which is Conway's game of life. there's this Conway's game of life. there's this Conway's game of life. there's this infinite discrete grid and at any given infinite discrete grid and at any given infinite discrete grid and at any given time the grid is either occupied by a time the grid is either occupied by a time the grid is either occupied by a cell or it's empty and there's a very cell or it's empty and there's a very cell or it's empty and there's a very simple rule that uh tells you how these simple rule that uh tells you how these simple rule that uh tells you how these cells evolve. So sometimes cells live cells evolve. So sometimes cells live cells evolve. So sometimes cells live and sometimes they die. Um and this um and sometimes they die. Um and this um and sometimes they die. Um and this um you know um when I was a a student it you know um when I was a a student it you know um when I was a a student it was a very popular screen saver to was a very popular screen saver to was a very popular screen saver to actually just have these these actually just have these these actually just have these these animations going and and they look very animations going and and they look very animations going and and they look very chaotic. In fact they look a little bit chaotic. In fact they look a little bit chaotic. In fact they look a little bit like turbulent float sometimes. But at like turbulent float sometimes. But at like turbulent float sometimes. But at some point people discovered more and some point people discovered more and some point people discovered more and more interesting structures within this more interesting structures within this more interesting structures within this game of life. Um so for example they game of life. Um so for example they game of life. Um so for example they discovered this thing called a glider. discovered this thing called a glider. discovered this thing called a glider. So a glider is a very tiny configuration So a glider is a very tiny configuration So a glider is a very tiny configuration of like four or five cells which evolves of like four or five cells which evolves of like four or five cells which evolves and it just moves at a certain direction and it just moves at a certain direction and it just moves at a certain direction and that's like this this vortex rings and that's like this this vortex rings and that's like this this vortex rings this um yeah so this is an analogy the this um yeah so this is an analogy the this um yeah so this is an analogy the game of life is kind of like a discrete game of life is kind of like a discrete game of life is kind of like a discrete equation and and um the flu navis is a equation and and um the flu navis is a equation and and um the flu navis is a continuous equation but mathematically continuous equation but mathematically continuous equation but mathematically they have some similar features um and they have some similar features um and they have some similar features um and um so over time people discovered more um so over time people discovered more um so over time people discovered more and more interesting things you could and more interesting things you could and more interesting things you could build within the game of life. The game build within the game of life. The game build within the game of life. The game life is a very simple system. It only life is a very simple system. It only life is a very simple system. It only has like three or four rules um to to do has like three or four rules um to to do has like three or four rules um to to do it, but but you can design all kinds of it, but but you can design all kinds of it, but but you can design all kinds of interesting configurations inside it. Um interesting configurations inside it. Um interesting configurations inside it. Um there's something called a glider gun there's something called a glider gun there's something called a glider gun that does nothing to spit out gliders that does nothing to spit out gliders that does nothing to spit out gliders one at a one one at a time. Um and then

one at a one one at a time. Um and then one at a one one at a time. Um and then after a lot of effort, people managed to after a lot of effort, people managed to after a lot of effort, people managed to to create um and gates and or gates for to create um and gates and or gates for to create um and gates and or gates for gliders. Like there's this massive gliders. Like there's this massive gliders. Like there's this massive ridiculous structure which if you if a ridiculous structure which if you if a ridiculous structure which if you if a if you have a stream of gliders um if you have a stream of gliders um if you have a stream of gliders um coming in here and a stream of gliders coming in here and a stream of gliders coming in here and a stream of gliders coming in here then you may produce a coming in here then you may produce a coming in here then you may produce a stream of gliders coming out. If so stream of gliders coming out. If so stream of gliders coming out. If so maybe if both of of the um streams um maybe if both of of the um streams um maybe if both of of the um streams um have gliders then there'll be an output have gliders then there'll be an output have gliders then there'll be an output stream but if only one of them does then stream but if only one of them does then stream but if only one of them does then nothing comes out. Mhm. So they could nothing comes out. Mhm. So they could nothing comes out. Mhm. So they could build something like that. And once you build something like that. And once you build something like that. And once you could build and um these basic gates could build and um these basic gates could build and um these basic gates then just from software engineering you then just from software engineering you then just from software engineering you can build almost anything. Um you can can build almost anything. Um you can can build almost anything. Um you can build a touring machine. I mean it's build a touring machine. I mean it's build a touring machine. I mean it's like an enormous steampunk type things. like an enormous steampunk type things. like an enormous steampunk type things. They look ridiculous. But then people They look ridiculous. But then people They look ridiculous. But then people also generated self-replicating objects also generated self-replicating objects also generated self-replicating objects in the game of life. A massive machine a in the game of life. A massive machine a in the game of life. A massive machine a bon machine which over a huge period of bon machine which over a huge period of bon machine which over a huge period of time and it always look like glider guns time and it always look like glider guns time and it always look like glider guns inside doing these very steampunk inside doing these very steampunk inside doing these very steampunk calculations. it would create another calculations. it would create another calculations. it would create another version of itself which could replicate. version of itself which could replicate. version of itself which could replicate. It's so incredible. A lot of this was It's so incredible. A lot of this was It's so incredible. A lot of this was like community crowdsourced by like like community crowdsourced by like like community crowdsourced by like amateur mathematicians actually. Um so I amateur mathematicians actually. Um so I amateur mathematicians actually. Um so I knew about that that that work and so knew about that that that work and so knew about that that that work and so that is part of what inspired me to that is part of what inspired me to that is part of what inspired me to propose the same thing with Navia propose the same thing with Navia propose the same thing with Navia Stokes. Um which is a much as I said Stokes. Um which is a much as I said Stokes. Um which is a much as I said analog is much worse than digital like analog is much worse than digital like analog is much worse than digital like it's going to be um you can't just it's going to be um you can't just it's going to be um you can't just directly take the constructions in the directly take the constructions in the directly take the constructions in the game of life and plunk them in. But game of life and plunk them in. But game of life and plunk them in. But again it just it shows it's possible.

again it just it shows it's possible. again it just it shows it's possible. You know, there's a kind of emergence You know, there's a kind of emergence You know, there's a kind of emergence that happens with these cellular automa. that happens with these cellular automa. that happens with these cellular automa. Local rules. Local rules. Local rules. Maybe it's similar to fluids. I don't Maybe it's similar to fluids. I don't Maybe it's similar to fluids. I don't know. But local rules operating at scale know. But local rules operating at scale know. But local rules operating at scale can create these incredibly complex can create these incredibly complex can create these incredibly complex dynamic structures. Do you think any of dynamic structures. Do you think any of dynamic structures. Do you think any of that is amendable to mathematical that is amendable to mathematical that is amendable to mathematical analysis? analysis? analysis? Do we have the tools to say something Do we have the tools to say something Do we have the tools to say something profound about that? The thing is you profound about that? The thing is you profound about that? The thing is you can get this emerg in very complicated can get this emerg in very complicated can get this emerg in very complicated structures but only with very carefully structures but only with very carefully structures but only with very carefully prepared initial conditions. Yeah. So so prepared initial conditions. Yeah. So so prepared initial conditions. Yeah. So so these these these glider guns and and these these these glider guns and and these these these glider guns and and gates and and so forth machines if you gates and and so forth machines if you gates and and so forth machines if you just plunk down randomly some cells and just plunk down randomly some cells and just plunk down randomly some cells and you and you will not see any of these. you and you will not see any of these. you and you will not see any of these. Um and that's the analogous situation Um and that's the analogous situation Um and that's the analogous situation with Navia Stokes again you know that with Navia Stokes again you know that with Navia Stokes again you know that that with with typical initial that with with typical initial that with with typical initial conditions you you will not have any of conditions you you will not have any of conditions you you will not have any of this weird computation going on. Um but this weird computation going on. Um but this weird computation going on. Um but basically through engineering you know basically through engineering you know basically through engineering you know by by by specially designing things in a by by by specially designing things in a by by by specially designing things in a very special way you can make clever very special way you can make clever very special way you can make clever constructions. I wonder if it's possible constructions. I wonder if it's possible constructions. I wonder if it's possible to prove the sort of the negative of to prove the sort of the negative of to prove the sort of the negative of like basically prove that only through like basically prove that only through like basically prove that only through engineering can you ever create engineering can you ever create engineering can you ever create something interesting. This this is a something interesting. This this is a something interesting. This this is a recurring challenge in mathematics that recurring challenge in mathematics that recurring challenge in mathematics that um I call it the dichotomy between um I call it the dichotomy between um I call it the dichotomy between structure and randomness. That most structure and randomness. That most structure and randomness. That most objects that you can generate in objects that you can generate in objects that you can generate in mathematics are random. They look like mathematics are random. They look like mathematics are random. They look like rand like the digits of pi. Well, we rand like the digits of pi. Well, we rand like the digits of pi. Well, we believe is a good example. Um, but believe is a good example. Um, but believe is a good example. Um, but there's a very small number of things there's a very small number of things there's a very small number of things that have patterns. Um, but um, now you that have patterns. Um, but um, now you that have patterns. Um, but um, now you can prove something has a pattern by

can prove something has a pattern by can prove something has a pattern by just constructing, you know, like if just constructing, you know, like if just constructing, you know, like if something has a simple pattern and you something has a simple pattern and you something has a simple pattern and you have a proof that it it does something have a proof that it it does something have a proof that it it does something like repeat itself every so often. You like repeat itself every so often. You like repeat itself every so often. You can do that. But um, and you you can can do that. But um, and you you can can do that. But um, and you you can prove that that for example, you can you prove that that for example, you can you prove that that for example, you can you can prove that most sequences of of can prove that most sequences of of can prove that most sequences of of digits have no pattern. Um, so like if digits have no pattern. Um, so like if digits have no pattern. Um, so like if you just pick digits randomly, there's you just pick digits randomly, there's you just pick digits randomly, there's something called low large numbers. It something called low large numbers. It something called low large numbers. It tells you you're going to get as many tells you you're going to get as many tells you you're going to get as many ones as as twos in the long run. Um but ones as as twos in the long run. Um but ones as as twos in the long run. Um but um we have a lot fewer tools to to to if um we have a lot fewer tools to to to if um we have a lot fewer tools to to to if I give you a specific pattern like the I give you a specific pattern like the I give you a specific pattern like the digits of pi how can I show that this digits of pi how can I show that this digits of pi how can I show that this doesn't have some weird pattern to it. doesn't have some weird pattern to it. doesn't have some weird pattern to it. Some other work that I spend a lot of Some other work that I spend a lot of Some other work that I spend a lot of time on is to prove what are called time on is to prove what are called time on is to prove what are called structure theorems or inverse theorems structure theorems or inverse theorems structure theorems or inverse theorems that give tests for when something is is that give tests for when something is is that give tests for when something is is very structured. So some functions are very structured. So some functions are very structured. So some functions are what's called additive like if you have what's called additive like if you have what's called additive like if you have a function that maps natural numbers a function that maps natural numbers a function that maps natural numbers with natural numbers. So maybe um you with natural numbers. So maybe um you with natural numbers. So maybe um you know two maps to four three maps to six know two maps to four three maps to six know two maps to four three maps to six and so forth. um some functions what's and so forth. um some functions what's and so forth. um some functions what's called additive which means that if you called additive which means that if you called additive which means that if you add if you add two inputs together the add if you add two inputs together the add if you add two inputs together the output gets gets added as well uh for output gets gets added as well uh for output gets gets added as well uh for example multiplying by a constant if you example multiplying by a constant if you example multiplying by a constant if you multiply a number by 10 um if you if you multiply a number by 10 um if you if you multiply a number by 10 um if you if you multiply a plus b by 10 that's the same multiply a plus b by 10 that's the same multiply a plus b by 10 that's the same as multiplying a by 10 and b by 10 and as multiplying a by 10 and b by 10 and as multiplying a by 10 and b by 10 and then adding them together so some um then adding them together so some um then adding them together so some um functions are additive some are kind of functions are additive some are kind of functions are additive some are kind of additive but not completely additive um additive but not completely additive um additive but not completely additive um so for example if I take a number n I so for example if I take a number n I so for example if I take a number n I multiply by the square root of two and I multiply by the square root of two and I multiply by the square root of two and I take the integer part of that So 10 by take the integer part of that So 10 by take the integer part of that So 10 by square of two is like 14 point square of two is like 14 point square of two is like 14 point something. So 10 up to 14. Um 20 up to something. So 10 up to 14. Um 20 up to something. So 10 up to 14. Um 20 up to 28. Um so in that case additively is 28. Um so in that case additively is 28. Um so in that case additively is true then. So 10 + 10 is 20 and 14 + 14 true then. So 10 + 10 is 20 and 14 + 14 true then. So 10 + 10 is 20 and 14 + 14 is 28. But because of this rounding

is 28. But because of this rounding is 28. But because of this rounding sometimes there's roundoff errors and sometimes there's roundoff errors and sometimes there's roundoff errors and and sometimes when you um add a plus b and sometimes when you um add a plus b and sometimes when you um add a plus b this function doesn't quite give you the this function doesn't quite give you the this function doesn't quite give you the sum of of the two individual outputs but sum of of the two individual outputs but sum of of the two individual outputs but the sum plus minus one. Um so it's the sum plus minus one. Um so it's the sum plus minus one. Um so it's almost additive but not quite additive. almost additive but not quite additive. almost additive but not quite additive. Um so there's a lot of useful results in Um so there's a lot of useful results in Um so there's a lot of useful results in mathematics and I've worked a lot on mathematics and I've worked a lot on mathematics and I've worked a lot on developing things like this to the developing things like this to the developing things like this to the effect that if if a function exhibits effect that if if a function exhibits effect that if if a function exhibits some structure like this then um it's some structure like this then um it's some structure like this then um it's basically there's a reason for why it's basically there's a reason for why it's basically there's a reason for why it's true and the reason is because there's true and the reason is because there's true and the reason is because there's there's some other nearby function which there's some other nearby function which there's some other nearby function which is actually um completely structured is actually um completely structured is actually um completely structured which is explaining this sort of partial which is explaining this sort of partial which is explaining this sort of partial pattern that you have. Um and so if you pattern that you have. Um and so if you pattern that you have. Um and so if you have these so inverse theorems it um it have these so inverse theorems it um it have these so inverse theorems it um it creates this sort of dichotomy that they creates this sort of dichotomy that they creates this sort of dichotomy that they either the objects that you study are either the objects that you study are either the objects that you study are either have no structure at all or they either have no structure at all or they either have no structure at all or they are somehow related to something that is are somehow related to something that is are somehow related to something that is structured. Um and in either way in structured. Um and in either way in structured. Um and in either way in either um in either case you can make either um in either case you can make either um in either case you can make progress. Um a good example of this is progress. Um a good example of this is progress. Um a good example of this is that there's this old theorem in that there's this old theorem in that there's this old theorem in mathematics called sim theorem proven in mathematics called sim theorem proven in mathematics called sim theorem proven in the 1970s. It concerns trying to find a the 1970s. It concerns trying to find a the 1970s. It concerns trying to find a certain type of pattern in a set of certain type of pattern in a set of certain type of pattern in a set of numbers. the patterns that have make numbers. the patterns that have make numbers. the patterns that have make progression things like 3 five and seven progression things like 3 five and seven progression things like 3 five and seven or or or 10 15 and 20 andreli or or or 10 15 and 20 andreli or or or 10 15 and 20 andreli proved that um any set of of numbers proved that um any set of of numbers proved that um any set of of numbers that are sufficiently big um what's that are sufficiently big um what's that are sufficiently big um what's called positive density has um called positive density has um called positive density has um arithmetic progressions in it of of any arithmetic progressions in it of of any arithmetic progressions in it of of any length you wish um so for example um the length you wish um so for example um the length you wish um so for example um the odd numbers have a set of density 1/2 um odd numbers have a set of density 1/2 um odd numbers have a set of density 1/2 um and they contain arithmetic progressions and they contain arithmetic progressions and they contain arithmetic progressions of any length um so in that case it's of any length um so in that case it's of any length um so in that case it's obvious because the the odd numbers are obvious because the the odd numbers are obvious because the the odd numbers are really really structured I can just take

really really structured I can just take really really structured I can just take 11 13 15 17 I just I can I can easily 11 13 15 17 I just I can I can easily 11 13 15 17 I just I can I can easily find arithmetic progressions in in in find arithmetic progressions in in in find arithmetic progressions in in in that set. Um but um zerminism also that set. Um but um zerminism also that set. Um but um zerminism also applies to random sets. If I take the applies to random sets. If I take the applies to random sets. If I take the set of odd numbers and I flip a coin um set of odd numbers and I flip a coin um set of odd numbers and I flip a coin um and for each number and I only keep the and for each number and I only keep the and for each number and I only keep the numbers which for which I got a heads numbers which for which I got a heads numbers which for which I got a heads okay so I just flip coins. I just okay so I just flip coins. I just okay so I just flip coins. I just randomly take out half the numbers I randomly take out half the numbers I randomly take out half the numbers I keep one half. So that's a set that has keep one half. So that's a set that has keep one half. So that's a set that has no no patterns at all. But just from no no patterns at all. But just from no no patterns at all. But just from random fluctuations, you will still get random fluctuations, you will still get random fluctuations, you will still get a lot of um um of arithmetic a lot of um um of arithmetic a lot of um um of arithmetic progressions in that set. Can you prove progressions in that set. Can you prove progressions in that set. Can you prove that that that there's arithmetic progressions of there's arithmetic progressions of there's arithmetic progressions of arbitrary length within a random? Yes. arbitrary length within a random? Yes. arbitrary length within a random? Yes. Um have you heard of the infinite monkey Um have you heard of the infinite monkey Um have you heard of the infinite monkey theorem? Usually mathematicians give theorem? Usually mathematicians give theorem? Usually mathematicians give boring names to theorists, but boring names to theorists, but boring names to theorists, but occasionally they they give colorful occasionally they they give colorful occasionally they they give colorful names. Yes. The popular version of the names. Yes. The popular version of the names. Yes. The popular version of the infinite monkey theorem is that if you infinite monkey theorem is that if you infinite monkey theorem is that if you have an infinite number of monkeys in a have an infinite number of monkeys in a have an infinite number of monkeys in a room with each with a typewriter they room with each with a typewriter they room with each with a typewriter they type out uh text randomly almost surely type out uh text randomly almost surely type out uh text randomly almost surely one of them is going to generate the one of them is going to generate the one of them is going to generate the entire screw of Hamlet or any other entire screw of Hamlet or any other entire screw of Hamlet or any other finite string of text. Uh it will just finite string of text. Uh it will just finite string of text. Uh it will just take some time quite a lot of time take some time quite a lot of time take some time quite a lot of time actually but if you have an infinite actually but if you have an infinite actually but if you have an infinite number then it happens. Um so um number then it happens. Um so um number then it happens. Um so um basically the the if you take an basically the the if you take an basically the the if you take an infinite string of of digits or whatever infinite string of of digits or whatever infinite string of of digits or whatever um eventually any finite pattern you um eventually any finite pattern you um eventually any finite pattern you wish will emerge. Um it may take a long wish will emerge. Um it may take a long wish will emerge. Um it may take a long time but it will eventually happen. Um time but it will eventually happen. Um time but it will eventually happen. Um in particular arithmetic progressions of in particular arithmetic progressions of in particular arithmetic progressions of any length will eventually happen. Okay.

any length will eventually happen. Okay. any length will eventually happen. Okay. But you need that but you need an But you need that but you need an But you need that but you need an extremely long random sequence for this extremely long random sequence for this extremely long random sequence for this to happen. I suppose that's intuitive. to happen. I suppose that's intuitive. to happen. I suppose that's intuitive. It's just infinity. Yeah. Infinity It's just infinity. Yeah. Infinity It's just infinity. Yeah. Infinity absorbs a lot of sins. Yeah. How are we absorbs a lot of sins. Yeah. How are we absorbs a lot of sins. Yeah. How are we humans supposed to deal with infinity? humans supposed to deal with infinity? humans supposed to deal with infinity? Well, you can think of infinity as as as Well, you can think of infinity as as as Well, you can think of infinity as as as just an abstraction of um a finite just an abstraction of um a finite just an abstraction of um a finite number for which you you do not have a number for which you you do not have a number for which you you do not have a bound for um that uh you know I mean so bound for um that uh you know I mean so bound for um that uh you know I mean so nothing in real life is truly infinite. nothing in real life is truly infinite. nothing in real life is truly infinite. Um but you know you can um you know you Um but you know you can um you know you Um but you know you can um you know you can ask yourself questions like you know can ask yourself questions like you know can ask yourself questions like you know what if I had as much money as I wanted what if I had as much money as I wanted what if I had as much money as I wanted you know or what if I could go as fast you know or what if I could go as fast you know or what if I could go as fast as I wanted and a way in which as I wanted and a way in which as I wanted and a way in which mathematicians formalize that is mathematicians formalize that is mathematicians formalize that is mathematics has found a formalism to mathematics has found a formalism to mathematics has found a formalism to idealize instead of something being idealize instead of something being idealize instead of something being extremely large or extremely small to extremely large or extremely small to extremely large or extremely small to actually be exactly infinite or zero. Um actually be exactly infinite or zero. Um actually be exactly infinite or zero. Um and often the the mathematics becomes a and often the the mathematics becomes a and often the the mathematics becomes a lot cleaner when you do that. I mean in lot cleaner when you do that. I mean in lot cleaner when you do that. I mean in physics we we joke about uh assuming physics we we joke about uh assuming physics we we joke about uh assuming spherical cows. um you know like real spherical cows. um you know like real spherical cows. um you know like real world problems have got all kinds of world problems have got all kinds of world problems have got all kinds of real world effects but you can idealize real world effects but you can idealize real world effects but you can idealize send certain things to infinity send send certain things to infinity send send certain things to infinity send certain things to zero um and um and the certain things to zero um and um and the certain things to zero um and um and the mathematics becomes a lot simpler to mathematics becomes a lot simpler to mathematics becomes a lot simpler to work with there. I wonder how often work with there. I wonder how often work with there. I wonder how often using infinity using infinity using infinity uh forces us to deviate from um the uh forces us to deviate from um the uh forces us to deviate from um the physics of reality. Yeah. So there's a physics of reality. Yeah. So there's a physics of reality. Yeah. So there's a lot of pitfalls. Um so you know we we lot of pitfalls. Um so you know we we lot of pitfalls. Um so you know we we spend a lot of time in undergraduate spend a lot of time in undergraduate spend a lot of time in undergraduate math classes teaching analysis. Um and math classes teaching analysis. Um and math classes teaching analysis. Um and analysis is often about how to take analysis is often about how to take analysis is often about how to take limits and and and and whether you you limits and and and and whether you you limits and and and and whether you you know so for example a plus b is always b know so for example a plus b is always b know so for example a plus b is always b plus a. Um so when you have a finite

plus a. Um so when you have a finite plus a. Um so when you have a finite number of terms you add them you can number of terms you add them you can number of terms you add them you can swap them and there there's no problem. swap them and there there's no problem. swap them and there there's no problem. But when you have infinite number of But when you have infinite number of But when you have infinite number of terms there these sort of shell games terms there these sort of shell games terms there these sort of shell games you can play where you can have a series you can play where you can have a series you can play where you can have a series which converges to one value but you which converges to one value but you which converges to one value but you rearrange it and it suddenly converges rearrange it and it suddenly converges rearrange it and it suddenly converges to another value. And so you can make to another value. And so you can make to another value. And so you can make mistakes. You have to know what you're mistakes. You have to know what you're mistakes. You have to know what you're doing when you allow infinity. Um you doing when you allow infinity. Um you doing when you allow infinity. Um you have to introduce these epsilons and have to introduce these epsilons and have to introduce these epsilons and deltas and and this there's a certain deltas and and this there's a certain deltas and and this there's a certain type of way of reasoning that helps you type of way of reasoning that helps you type of way of reasoning that helps you avoid mistakes. Um avoid mistakes. Um avoid mistakes. Um in more recent years um people have in more recent years um people have in more recent years um people have started taking results that are true in started taking results that are true in started taking results that are true in infinite limits and what's called infinite limits and what's called infinite limits and what's called finetizing them. Um so you know that finetizing them. Um so you know that finetizing them. Um so you know that something's true eventually but um you something's true eventually but um you something's true eventually but um you don't know when. Now give me a rate. don't know when. Now give me a rate. don't know when. Now give me a rate. Okay. Okay, so it's such a if I have Okay. Okay, so it's such a if I have Okay. Okay, so it's such a if I have don't have an infinite number of monkeys don't have an infinite number of monkeys don't have an infinite number of monkeys but but a large finite number of but but a large finite number of but but a large finite number of monkeys, how long do I have to wait for monkeys, how long do I have to wait for monkeys, how long do I have to wait for H to come out? Um and that's a more H to come out? Um and that's a more H to come out? Um and that's a more quantitative question. Um and this is quantitative question. Um and this is quantitative question. Um and this is something that you can you can um attack something that you can you can um attack something that you can you can um attack by purely finite methods and you can use by purely finite methods and you can use by purely finite methods and you can use your finite intuition. Um and in this your finite intuition. Um and in this your finite intuition. Um and in this case it turns out to be exponential in case it turns out to be exponential in case it turns out to be exponential in the length of the text that you're the length of the text that you're the length of the text that you're you're trying to generate. Um so um and you're trying to generate. Um so um and you're trying to generate. Um so um and so this is why you never see the monkeys so this is why you never see the monkeys so this is why you never see the monkeys create Hamilton. you can maybe see them create Hamilton. you can maybe see them create Hamilton. you can maybe see them create a four-letter word, but nothing create a four-letter word, but nothing create a four-letter word, but nothing that big. And so I personally find once that big. And so I personally find once that big. And so I personally find once you finitize an infinite statement, it's you finitize an infinite statement, it's you finitize an infinite statement, it's it does become much more intuitive and it does become much more intuitive and it does become much more intuitive and it's no longer so so weird. Um so even it's no longer so so weird. Um so even it's no longer so so weird. Um so even if you're working with infinity, it's if you're working with infinity, it's if you're working with infinity, it's good to finitize so that you can have good to finitize so that you can have good to finitize so that you can have some intuition. Yeah. The downside is some intuition. Yeah. The downside is some intuition. Yeah. The downside is that the finite groups are just much that the finite groups are just much that the finite groups are just much much messier and and uh yeah. So so the

much messier and and uh yeah. So so the much messier and and uh yeah. So so the infinite ones are found first usually infinite ones are found first usually infinite ones are found first usually like decades earlier and then later on like decades earlier and then later on like decades earlier and then later on people finize them. So since we people finize them. So since we people finize them. So since we mentioned a lot of math and a lot of mentioned a lot of math and a lot of mentioned a lot of math and a lot of physics uh what is the difference physics uh what is the difference physics uh what is the difference between mathematics and physics as between mathematics and physics as between mathematics and physics as disciplines as ways of understanding of disciplines as ways of understanding of disciplines as ways of understanding of seeing the world maybe we can throw in seeing the world maybe we can throw in seeing the world maybe we can throw in engineering in there you mentioned your engineering in there you mentioned your engineering in there you mentioned your wife is an engineer give it new wife is an engineer give it new wife is an engineer give it new perspective on circuits right so this perspective on circuits right so this perspective on circuits right so this different way of looking at the world different way of looking at the world different way of looking at the world given that you've done mathematical given that you've done mathematical given that you've done mathematical physics so you you've you've worn all physics so you you've you've worn all physics so you you've you've worn all the hats right so I think science in the hats right so I think science in the hats right so I think science in general is interaction between three general is interaction between three general is interaction between three things um there's the real world um things um there's the real world um things um there's the real world um there's is what we observe of the there's is what we observe of the there's is what we observe of the reward, our observations and then our reward, our observations and then our reward, our observations and then our mental models as to how we think the mental models as to how we think the mental models as to how we think the world works. Um so um we can't directly world works. Um so um we can't directly world works. Um so um we can't directly access reality. Okay. Uh all we have are access reality. Okay. Uh all we have are access reality. Okay. Uh all we have are the observations which are incomplete the observations which are incomplete the observations which are incomplete and they they have errors. Um and um and they they have errors. Um and um and they they have errors. Um and um there are many many cases where we would there are many many cases where we would there are many many cases where we would um uh we want to know for example what um uh we want to know for example what um uh we want to know for example what is the weather like tomorrow and we is the weather like tomorrow and we is the weather like tomorrow and we don't yet have the observation we'd like don't yet have the observation we'd like don't yet have the observation we'd like to a prediction. Um and then we have to a prediction. Um and then we have to a prediction. Um and then we have these simplified models sometimes making these simplified models sometimes making these simplified models sometimes making unrealistic assumptions you know unrealistic assumptions you know unrealistic assumptions you know spherical cow type things. Those are the spherical cow type things. Those are the spherical cow type things. Those are the mathematical models. Mathematics is mathematical models. Mathematics is mathematical models. Mathematics is concerned with the models. Science concerned with the models. Science concerned with the models. Science collects the observations and it collects the observations and it collects the observations and it proposes the models that might explain proposes the models that might explain proposes the models that might explain these observations. What mathematics these observations. What mathematics these observations. What mathematics does we we stay within the model and we does we we stay within the model and we does we we stay within the model and we ask what are the consequences of that ask what are the consequences of that ask what are the consequences of that model? what observations would what model? what observations would what model? what observations would what predictions would the model make of the predictions would the model make of the predictions would the model make of the of future observations um or past of future observations um or past of future observations um or past observations does it fit observed data

observations does it fit observed data observations does it fit observed data um so there's definitely a symbiosis um um so there's definitely a symbiosis um um so there's definitely a symbiosis um it's ma I guess mathematics is is it's ma I guess mathematics is is it's ma I guess mathematics is is unusual among other disciplines is that unusual among other disciplines is that unusual among other disciplines is that we start from hypothesis like the axims we start from hypothesis like the axims we start from hypothesis like the axims of a model and ask what conclusions come of a model and ask what conclusions come of a model and ask what conclusions come up from that that model um in almost any up from that that model um in almost any up from that that model um in almost any other discipline uh you start with the other discipline uh you start with the other discipline uh you start with the conclusions you know I want to do this I conclusions you know I want to do this I conclusions you know I want to do this I want to build a bridge, you know, I I want to build a bridge, you know, I I want to build a bridge, you know, I I want to to make money. I want to do want to to make money. I want to do want to to make money. I want to do this. Okay. And then you you you find this. Okay. And then you you you find this. Okay. And then you you you find the path to get there. Um the path to get there. Um the path to get there. Um a lot there there's a lot less sort of a lot there there's a lot less sort of a lot there there's a lot less sort of speculation about suppose I did this, speculation about suppose I did this, speculation about suppose I did this, what would happen? Um you know, planning what would happen? Um you know, planning what would happen? Um you know, planning and and and modeling um uh speculative and and and modeling um uh speculative and and and modeling um uh speculative fiction maybe is one other place. Uh but fiction maybe is one other place. Uh but fiction maybe is one other place. Uh but uh that's about it actually. Most of uh that's about it actually. Most of uh that's about it actually. Most of things we do in life is conclusions things we do in life is conclusions things we do in life is conclusions driven including physics and science. driven including physics and science. driven including physics and science. You I mean they want to know you know You I mean they want to know you know You I mean they want to know you know where is this asteroid going to go? What where is this asteroid going to go? What where is this asteroid going to go? What was what what is the weather going to be was what what is the weather going to be was what what is the weather going to be tomorrow? Um but um Bathe also has this tomorrow? Um but um Bathe also has this tomorrow? Um but um Bathe also has this other direction of of going from the uh other direction of of going from the uh other direction of of going from the uh the axioms. What do you think there is the axioms. What do you think there is the axioms. What do you think there is this tension in physics between theory this tension in physics between theory this tension in physics between theory and experiment? Mhm. What do you think and experiment? Mhm. What do you think and experiment? Mhm. What do you think is the more powerful way of discovering is the more powerful way of discovering is the more powerful way of discovering truly novel ideas about reality? Well, truly novel ideas about reality? Well, truly novel ideas about reality? Well, you need both top down and bottom up. Um you need both top down and bottom up. Um you need both top down and bottom up. Um yeah, it's it's a real interaction yeah, it's it's a real interaction yeah, it's it's a real interaction between all these things. So over time between all these things. So over time between all these things. So over time the observations and the theory and the the observations and the theory and the the observations and the theory and the modeling should both get closer to modeling should both get closer to modeling should both get closer to reality. But initially and it is I mean reality. But initially and it is I mean reality. But initially and it is I mean this is um this is always the case. You this is um this is always the case. You this is um this is always the case. You know they're always far apart to begin know they're always far apart to begin know they're always far apart to begin with. Um but you need one to figure out with. Um but you need one to figure out with. Um but you need one to figure out where to push the other you know. So um

where to push the other you know. So um where to push the other you know. So um if your model is predicting anomalies um if your model is predicting anomalies um if your model is predicting anomalies um that are not picked up by experiment that are not picked up by experiment that are not picked up by experiment that tells experimenters where to look that tells experimenters where to look that tells experimenters where to look you know um to to to to find more data you know um to to to to find more data you know um to to to to find more data to refine the models. Um yeah so it it to refine the models. Um yeah so it it to refine the models. Um yeah so it it it goes it goes back and forth. Um it goes it goes back and forth. Um it goes it goes back and forth. Um within mathematics itself there's within mathematics itself there's within mathematics itself there's there's also a theory and experimental there's also a theory and experimental there's also a theory and experimental component. It's just that until very component. It's just that until very component. It's just that until very recently theory has dominated almost recently theory has dominated almost recently theory has dominated almost completely like 99% of mathematics is completely like 99% of mathematics is completely like 99% of mathematics is theoretical mathematics and there's a theoretical mathematics and there's a theoretical mathematics and there's a very tiny amount of experimental very tiny amount of experimental very tiny amount of experimental mathematics. Um I mean people do do it mathematics. Um I mean people do do it mathematics. Um I mean people do do it you know like if they want to study you know like if they want to study you know like if they want to study prime numbers or whatever they can just prime numbers or whatever they can just prime numbers or whatever they can just generate large data sets and with a so generate large data sets and with a so generate large data sets and with a so once we had computers um we be to do it once we had computers um we be to do it once we had computers um we be to do it a little bit. Um although even before a little bit. Um although even before a little bit. Um although even before well like Gaus for example he discovered well like Gaus for example he discovered well like Gaus for example he discovered he conjectured the most basic theorem in he conjectured the most basic theorem in he conjectured the most basic theorem in in number theory to call the prime in number theory to call the prime in number theory to call the prime number theorem which predicts how many number theorem which predicts how many number theorem which predicts how many primes that up to a million up to a primes that up to a million up to a primes that up to a million up to a trillion. It's not an obvious question trillion. It's not an obvious question trillion. It's not an obvious question and basically what he did was that he and basically what he did was that he and basically what he did was that he computed I mean mostly um by himself but computed I mean mostly um by himself but computed I mean mostly um by himself but also hired human computers um people who also hired human computers um people who also hired human computers um people who whose professional job it was to do whose professional job it was to do whose professional job it was to do arithmetic um to compute the first arithmetic um to compute the first arithmetic um to compute the first 100,000 tribes or something and made 100,000 tribes or something and made 100,000 tribes or something and made tables and made a prediction um that was tables and made a prediction um that was tables and made a prediction um that was an early example of experimental an early example of experimental an early example of experimental mathematics mathematics mathematics um but until very recently it was not um um but until very recently it was not um um but until very recently it was not um yeah I mean theoretical mathematics was yeah I mean theoretical mathematics was yeah I mean theoretical mathematics was just much more successful I mean because just much more successful I mean because just much more successful I mean because doing complicated mathematical doing complicated mathematical doing complicated mathematical computations is uh was just not not computations is uh was just not not computations is uh was just not not feasible until very recently. Uh and feasible until very recently. Uh and feasible until very recently. Uh and even nowadays, you know, even though we even nowadays, you know, even though we even nowadays, you know, even though we have powerful computers, only some have powerful computers, only some have powerful computers, only some mathematical things can be um explored

mathematical things can be um explored mathematical things can be um explored numerically. There's something called numerically. There's something called numerically. There's something called the comatorial explosion. If you want us the comatorial explosion. If you want us the comatorial explosion. If you want us to study, for example, Zodius the you to study, for example, Zodius the you to study, for example, Zodius the you want to study all possible subsets of want to study all possible subsets of want to study all possible subsets of the numbers 1 to a,000. There's only the numbers 1 to a,000. There's only the numbers 1 to a,000. There's only 1,000 numbers. How bad could it be? It 1,000 numbers. How bad could it be? It 1,000 numbers. How bad could it be? It turns out the number of different turns out the number of different turns out the number of different subsets of of 1 to a,000 is 2 to the^ subsets of of 1 to a,000 is 2 to the^ subsets of of 1 to a,000 is 2 to the^ 1,000 which is way bigger than than that 1,000 which is way bigger than than that 1,000 which is way bigger than than that any computer can currently can can in any computer can currently can can in any computer can currently can can in fact anybody ever will ever um fact anybody ever will ever um fact anybody ever will ever um enumerate. Um so you have you have to be enumerate. Um so you have you have to be enumerate. Um so you have you have to be um there are certain math problems that um there are certain math problems that um there are certain math problems that very quickly become just intractable to very quickly become just intractable to very quickly become just intractable to attack by direct brute force attack by direct brute force attack by direct brute force computation. Uh chess is another um computation. Uh chess is another um computation. Uh chess is another um famous example. The number of chess famous example. The number of chess famous example. The number of chess positions uh we can't get a computer to positions uh we can't get a computer to positions uh we can't get a computer to fully explore. fully explore. fully explore. But now we have AI um um we have tools But now we have AI um um we have tools But now we have AI um um we have tools to explore this space not with 100% to explore this space not with 100% to explore this space not with 100% guarantees of success but with guarantees of success but with guarantees of success but with experiment you know so like um we can experiment you know so like um we can experiment you know so like um we can empirically solve chess now for example empirically solve chess now for example empirically solve chess now for example we have we have very very good AIs that we have we have very very good AIs that we have we have very very good AIs that that can you know they don't explore that can you know they don't explore that can you know they don't explore every single position in in the game every single position in in the game every single position in in the game tree but they have found some very good tree but they have found some very good tree but they have found some very good approximation um and people are using approximation um and people are using approximation um and people are using actually these chess engines to make uh actually these chess engines to make uh actually these chess engines to make uh to do experimental chess um that they're to do experimental chess um that they're to do experimental chess um that they're revisiting old chess theories about, oh, revisiting old chess theories about, oh, revisiting old chess theories about, oh, you know, when you this type of opening, you know, when you this type of opening, you know, when you this type of opening, you know, this is a good, this is a good you know, this is a good, this is a good you know, this is a good, this is a good type of move, this is not, and they can type of move, this is not, and they can type of move, this is not, and they can use these chess engines to actually use these chess engines to actually use these chess engines to actually refine in some case overturn um um refine in some case overturn um um refine in some case overturn um um conventional wisdom about chess. And I conventional wisdom about chess. And I conventional wisdom about chess. And I do hope that uh that mathematics will do hope that uh that mathematics will do hope that uh that mathematics will will have a larger experimental will have a larger experimental will have a larger experimental component in the future perhaps powered component in the future perhaps powered component in the future perhaps powered by AI. We'll of course talk about that by AI. We'll of course talk about that by AI. We'll of course talk about that but in the case of chess and there's a

but in the case of chess and there's a but in the case of chess and there's a similar thing in mathematics that I similar thing in mathematics that I similar thing in mathematics that I don't believe it's providing a kind of don't believe it's providing a kind of don't believe it's providing a kind of formal explanation of the different formal explanation of the different formal explanation of the different positions. It's just saying which positions. It's just saying which positions. It's just saying which position is better or not that you can position is better or not that you can position is better or not that you can intuit it as a human being and then from intuit it as a human being and then from intuit it as a human being and then from that we humans can construct a theory of that we humans can construct a theory of that we humans can construct a theory of the matter. You've mentioned the Plato's the matter. You've mentioned the Plato's the matter. You've mentioned the Plato's cave allegory. Mhm. So in case people cave allegory. Mhm. So in case people cave allegory. Mhm. So in case people don't know, it's where people are don't know, it's where people are don't know, it's where people are observing shadows of reality, not observing shadows of reality, not observing shadows of reality, not reality itself, and they believe what reality itself, and they believe what reality itself, and they believe what they're observing to be reality. Is that they're observing to be reality. Is that they're observing to be reality. Is that in some sense what mathematicians and in some sense what mathematicians and in some sense what mathematicians and maybe all humans are doing is um looking maybe all humans are doing is um looking maybe all humans are doing is um looking at shadows at shadows at shadows of reality? Is it possible for us to of reality? Is it possible for us to of reality? Is it possible for us to truly access truly access truly access reality? Well, there these three reality? Well, there these three reality? Well, there these three onlogical things. there's actual onlogical things. there's actual onlogical things. there's actual reality, there's our observations and reality, there's our observations and reality, there's our observations and our our models. Um, and technically they our our models. Um, and technically they our our models. Um, and technically they are distinct and I think they will are distinct and I think they will are distinct and I think they will always be distinct. Um, but they can get always be distinct. Um, but they can get always be distinct. Um, but they can get closer um over time. Um, you know, so um closer um over time. Um, you know, so um closer um over time. Um, you know, so um and the process of getting closer often and the process of getting closer often and the process of getting closer often means that you you have to discard your means that you you have to discard your means that you you have to discard your initial intuitions. Um so um like initial intuitions. Um so um like initial intuitions. Um so um like astronomy provides great examples you astronomy provides great examples you astronomy provides great examples you know like you know like you an initial know like you know like you an initial know like you know like you an initial model of the world is is flat because it model of the world is is flat because it model of the world is is flat because it looks flat you know and um and that it's looks flat you know and um and that it's looks flat you know and um and that it's and it's big you know and the rest of and it's big you know and the rest of and it's big you know and the rest of the universe the skies is not you know the universe the skies is not you know the universe the skies is not you know like the sun for example looks really like the sun for example looks really like the sun for example looks really tiny um and so you start off with a

tiny um and so you start off with a tiny um and so you start off with a model which is actually really far from model which is actually really far from model which is actually really far from reality um but it fits kind of the reality um but it fits kind of the reality um but it fits kind of the observations that you have um you know observations that you have um you know observations that you have um you know so you know so things look good you know so you know so things look good you know so you know so things look good you know but but over time as you make more and but but over time as you make more and but but over time as you make more and more observations bring it closer to to more observations bring it closer to to more observations bring it closer to to reality Okay. Um the model gets dragged reality Okay. Um the model gets dragged reality Okay. Um the model gets dragged along with it and so over time we had to along with it and so over time we had to along with it and so over time we had to realize that the earth was round that it realize that the earth was round that it realize that the earth was round that it spins. It goes around the solar system. spins. It goes around the solar system. spins. It goes around the solar system. Solar system goes around the galaxy and Solar system goes around the galaxy and Solar system goes around the galaxy and so on and so forth. And the guys so on and so forth. And the guys so on and so forth. And the guys universe is expanding the expansion universe is expanding the expansion universe is expanding the expansion itself expanding accelerating and in itself expanding accelerating and in itself expanding accelerating and in fact very recently in this year. So this fact very recently in this year. So this fact very recently in this year. So this uh even the acceleration of the universe uh even the acceleration of the universe uh even the acceleration of the universe itself is this evidence that this itself is this evidence that this itself is this evidence that this non-constant and uh the explanation non-constant and uh the explanation non-constant and uh the explanation behind why that is it's catching up. Um behind why that is it's catching up. Um behind why that is it's catching up. Um it's catching up. I mean it's still you it's catching up. I mean it's still you it's catching up. I mean it's still you know the dark matter or dark energy this know the dark matter or dark energy this know the dark matter or dark energy this this kind of thing. We have we have a this kind of thing. We have we have a this kind of thing. We have we have a model that sort of explains that fits model that sort of explains that fits model that sort of explains that fits the data really well. It just has a few the data really well. It just has a few the data really well. It just has a few parameters that um you have to specify. parameters that um you have to specify. parameters that um you have to specify. Um but so you know people say that's Um but so you know people say that's Um but so you know people say that's fudge factors you know with with enough fudge factors you know with with enough fudge factors you know with with enough fudge factors you can explain anything. fudge factors you can explain anything. fudge factors you can explain anything. Um but uh the mathematical point of the Um but uh the mathematical point of the Um but uh the mathematical point of the model is that um you want to have fewer model is that um you want to have fewer model is that um you want to have fewer parameters in your model than data parameters in your model than data parameters in your model than data points in your observational set. So if points in your observational set. So if points in your observational set. So if you have a model with 10 parameters that you have a model with 10 parameters that you have a model with 10 parameters that explains 10 10 observations that is a explains 10 10 observations that is a explains 10 10 observations that is a completely useless model. It's what's completely useless model. It's what's completely useless model. It's what's called overfitted. But like if you have called overfitted. But like if you have called overfitted. But like if you have a model with you know two parameters and a model with you know two parameters and a model with you know two parameters and it explains a trillion observations it explains a trillion observations it explains a trillion observations which is basically uh so yeah the the which is basically uh so yeah the the which is basically uh so yeah the the the dark matter model I think has like the dark matter model I think has like the dark matter model I think has like 14 parameters and it explains pabytes of 14 parameters and it explains pabytes of 14 parameters and it explains pabytes of data um that that that the astronomers data um that that that the astronomers data um that that that the astronomers have. Um you can think of of a theory have. Um you can think of of a theory have. Um you can think of of a theory like one way to think about um physical

like one way to think about um physical like one way to think about um physical math theory theory is it's a compression math theory theory is it's a compression math theory theory is it's a compression of of the universe um and data of of the universe um and data of of the universe um and data compression. So you know you have these compression. So you know you have these compression. So you know you have these pabytes of observations you'd like to pabytes of observations you'd like to pabytes of observations you'd like to compress it to a model which you can compress it to a model which you can compress it to a model which you can describe in five pages and specify a describe in five pages and specify a describe in five pages and specify a certain number of parameters and if it certain number of parameters and if it certain number of parameters and if it can fit to reasonable accuracy you know can fit to reasonable accuracy you know can fit to reasonable accuracy you know almost all of your observations. I mean almost all of your observations. I mean almost all of your observations. I mean the more compression that you make the the more compression that you make the the more compression that you make the better your theory. In fact, one of the better your theory. In fact, one of the better your theory. In fact, one of the great surprises of our universe and of great surprises of our universe and of great surprises of our universe and of everything in it is that it's everything in it is that it's everything in it is that it's compressible at all. It's the compressible at all. It's the compressible at all. It's the unreasonable effectiveness of unreasonable effectiveness of unreasonable effectiveness of mathematics. Yeah, Einstein had a quote mathematics. Yeah, Einstein had a quote mathematics. Yeah, Einstein had a quote like that. The the most incomprehensible like that. The the most incomprehensible like that. The the most incomprehensible thing about the universe is that it is thing about the universe is that it is thing about the universe is that it is comprehensible, right? And not just comprehensible, right? And not just comprehensible, right? And not just comprehensible. You can do an equation comprehensible. You can do an equation comprehensible. You can do an equation like E= MC². There is actually a some like E= MC². There is actually a some like E= MC². There is actually a some mathematical possible explanation for mathematical possible explanation for mathematical possible explanation for that. Um, so there's this phenomenon in that. Um, so there's this phenomenon in that. Um, so there's this phenomenon in mathematics called universality. So many mathematics called universality. So many mathematics called universality. So many complex systems at the macro scale are complex systems at the macro scale are complex systems at the macro scale are coming out of lots of tiny interactions coming out of lots of tiny interactions coming out of lots of tiny interactions at the macro scale and normally because at the macro scale and normally because at the macro scale and normally because of the common form of explosion you of the common form of explosion you of the common form of explosion you would think that uh the macros scale would think that uh the macros scale would think that uh the macros scale equations must be like infinitely equations must be like infinitely equations must be like infinitely exponentially more complicated than than exponentially more complicated than than exponentially more complicated than than the uh the microscale ones and they are the uh the microscale ones and they are the uh the microscale ones and they are if you want to solve them completely if you want to solve them completely if you want to solve them completely exactly like if you want to model um all exactly like if you want to model um all exactly like if you want to model um all the atoms in a box of of air that's like the atoms in a box of of air that's like the atoms in a box of of air that's like Avagadro's number is humongous right Avagadro's number is humongous right Avagadro's number is humongous right there's a huge number of particles if there's a huge number of particles if there's a huge number of particles if you actually have to track each one you actually have to track each one you actually have to track each one it'll be ridiculous. this but certain it'll be ridiculous. this but certain it'll be ridiculous. this but certain laws emerge at the microscopic scale laws emerge at the microscopic scale laws emerge at the microscopic scale that almost don't depend on what's going that almost don't depend on what's going that almost don't depend on what's going on at the micros scale or only depend on on at the micros scale or only depend on on at the micros scale or only depend on a very small number of parameters. So if a very small number of parameters. So if a very small number of parameters. So if you want to model a gas um of you know you want to model a gas um of you know you want to model a gas um of you know quintilion particles in a box you just

quintilion particles in a box you just quintilion particles in a box you just need to know it temperature and pressure need to know it temperature and pressure need to know it temperature and pressure and volume and a few parameters like and volume and a few parameters like and volume and a few parameters like five or six and it models almost five or six and it models almost five or six and it models almost everything you need need to know about everything you need need to know about everything you need need to know about these 10 to 23 or whatever particles. Um these 10 to 23 or whatever particles. Um these 10 to 23 or whatever particles. Um so we we have um we we don't understand so we we have um we we don't understand so we we have um we we don't understand universality anywhere near as we would universality anywhere near as we would universality anywhere near as we would like mathematically but there are much like mathematically but there are much like mathematically but there are much simpler toy models where we do um have a simpler toy models where we do um have a simpler toy models where we do um have a good understanding of why univers good understanding of why univers good understanding of why univers universality occurs. Um um most basic universality occurs. Um um most basic universality occurs. Um um most basic one is is the central limit theorem that one is is the central limit theorem that one is is the central limit theorem that explains why the bell curve shows up explains why the bell curve shows up explains why the bell curve shows up everywhere in nature that so many things everywhere in nature that so many things everywhere in nature that so many things are distributed by what's called a are distributed by what's called a are distributed by what's called a Gaussian distribution famous bell curve. Gaussian distribution famous bell curve. Gaussian distribution famous bell curve. There's now even a meme with this curve There's now even a meme with this curve There's now even a meme with this curve and even the meme applies broadly and even the meme applies broadly and even the meme applies broadly universality to the meme. Yeah. Yes, you universality to the meme. Yeah. Yes, you universality to the meme. Yeah. Yes, you can go meta if you like. But there are can go meta if you like. But there are can go meta if you like. But there are many many processes for example you can many many processes for example you can many many processes for example you can take lots and lots of independent um take lots and lots of independent um take lots and lots of independent um random variables and average them random variables and average them random variables and average them together um uh in in various ways. you together um uh in in various ways. you together um uh in in various ways. you take a simple average or more take a simple average or more take a simple average or more complicated average and we can prove in complicated average and we can prove in complicated average and we can prove in various cases that that these these bell various cases that that these these bell various cases that that these these bell curves these gaussians emerge and it is curves these gaussians emerge and it is curves these gaussians emerge and it is a satisfying satisfying explanation. Um a satisfying satisfying explanation. Um a satisfying satisfying explanation. Um sometimes they don't. Um so so if you sometimes they don't. Um so so if you sometimes they don't. Um so so if you have many different inputs and they're have many different inputs and they're have many different inputs and they're all correlated in some systemic way then all correlated in some systemic way then all correlated in some systemic way then you can get something very far from a you can get something very far from a you can get something very far from a bow curve show up. Uh and this is also bow curve show up. Uh and this is also bow curve show up. Uh and this is also important to know when this system important to know when this system important to know when this system fails. So universality is not a 100% fails. So universality is not a 100% fails. So universality is not a 100% reliable thing to rely on that um um the reliable thing to rely on that um um the reliable thing to rely on that um um the global financial crisis was a a famous global financial crisis was a a famous global financial crisis was a a famous example of this. Uh people thought that example of this. Uh people thought that example of this. Uh people thought that uh um mortgage defaults um had this sort uh um mortgage defaults um had this sort uh um mortgage defaults um had this sort of um Gaussian type behavior that that

of um Gaussian type behavior that that of um Gaussian type behavior that that if you if you ask if a population of of if you if you ask if a population of of if you if you ask if a population of of of uh you know 100,000 Americans with of uh you know 100,000 Americans with of uh you know 100,000 Americans with mortgages ask what what proportion of mortgages ask what what proportion of mortgages ask what what proportion of them would default on the mortgages. Um them would default on the mortgages. Um them would default on the mortgages. Um if everything was decorated it would be if everything was decorated it would be if everything was decorated it would be an asset bell curve and and like you can an asset bell curve and and like you can an asset bell curve and and like you can you can manage risk with options and you can manage risk with options and you can manage risk with options and derivatives and so forth and um and it derivatives and so forth and um and it derivatives and so forth and um and it there's a very beautiful theory um but there's a very beautiful theory um but there's a very beautiful theory um but if there are systemic shocks in the if there are systemic shocks in the if there are systemic shocks in the economy uh that can push everybody to economy uh that can push everybody to economy uh that can push everybody to default at the same time that's very default at the same time that's very default at the same time that's very non-gian behavior um and uh this wasn't non-gian behavior um and uh this wasn't non-gian behavior um and uh this wasn't fully accounted for in 2008 fully accounted for in 2008 fully accounted for in 2008 now I think there's some more awareness now I think there's some more awareness now I think there's some more awareness that this is systemic risk is actually a that this is systemic risk is actually a that this is systemic risk is actually a much bigger issue and uh just because much bigger issue and uh just because much bigger issue and uh just because the model is pretty uh and nice uh it the model is pretty uh and nice uh it the model is pretty uh and nice uh it may not match reality. Right. So, so the may not match reality. Right. So, so the may not match reality. Right. So, so the mathematics of working out what models mathematics of working out what models mathematics of working out what models do is really important. Um, but um also do is really important. Um, but um also do is really important. Um, but um also the science of validating when the the science of validating when the the science of validating when the models fit reality and when they don't. models fit reality and when they don't. models fit reality and when they don't. Um, I mean that you need both. Um, and Um, I mean that you need both. Um, and Um, I mean that you need both. Um, and but mathematics can help because it it but mathematics can help because it it but mathematics can help because it it can for example these central limit can for example these central limit can for example these central limit theorems it tells you that if you have theorems it tells you that if you have theorems it tells you that if you have certain aums like like non-correlation certain aums like like non-correlation certain aums like like non-correlation that if all the inputs were not that if all the inputs were not that if all the inputs were not correlated to each other um then you correlated to each other um then you correlated to each other um then you have this kind of behavior things are have this kind of behavior things are have this kind of behavior things are fine. it it tells you where to look for fine. it it tells you where to look for fine. it it tells you where to look for weaknesses in the model. So if you have weaknesses in the model. So if you have weaknesses in the model. So if you have a mathematical understanding of central a mathematical understanding of central a mathematical understanding of central limit theorem and someone proposes use limit theorem and someone proposes use limit theorem and someone proposes use these Gaussian copy or whatever to to these Gaussian copy or whatever to to these Gaussian copy or whatever to to model um default risk um if you're model um default risk um if you're model um default risk um if you're mathematically um trained you would say mathematically um trained you would say mathematically um trained you would say okay but what if this systemic okay but what if this systemic okay but what if this systemic correlation between all your inputs and

correlation between all your inputs and correlation between all your inputs and so then then you can ask the economists so then then you can ask the economists so then then you can ask the economists you know how how how much of a risk is you know how how how much of a risk is you know how how how much of a risk is that um and then you can you can you can that um and then you can you can you can that um and then you can you can you can go look for that. So there's always this go look for that. So there's always this go look for that. So there's always this this this synergy between science and this this synergy between science and this this synergy between science and and mathematics. A little bit on the and mathematics. A little bit on the and mathematics. A little bit on the topic of universality. Mhm. topic of universality. Mhm. topic of universality. Mhm. You're known and celebrated for working You're known and celebrated for working You're known and celebrated for working across an incredible breadth of across an incredible breadth of across an incredible breadth of mathematics reminiscent of Hilbert a mathematics reminiscent of Hilbert a mathematics reminiscent of Hilbert a century ago. In fact, the great Fields century ago. In fact, the great Fields century ago. In fact, the great Fields Medal winning mathematician Tim Gow has Medal winning mathematician Tim Gow has Medal winning mathematician Tim Gow has said that you are the closest thing we said that you are the closest thing we said that you are the closest thing we get to Hilbert. get to Hilbert. get to Hilbert. He's a colleague of yours. Oh yeah. Good He's a colleague of yours. Oh yeah. Good He's a colleague of yours. Oh yeah. Good friend. But anyway, so you are known for friend. But anyway, so you are known for friend. But anyway, so you are known for this ability to go both deep and broad this ability to go both deep and broad this ability to go both deep and broad in mathematics. So you're the perfect in mathematics. So you're the perfect in mathematics. So you're the perfect person to ask, do you think there are person to ask, do you think there are person to ask, do you think there are threads that connect all the disparate threads that connect all the disparate threads that connect all the disparate areas of mathematics? Is there a kind of areas of mathematics? Is there a kind of areas of mathematics? Is there a kind of deep underlying structure deep underlying structure deep underlying structure uh to all of mathematics? There's uh to all of mathematics? There's uh to all of mathematics? There's certainly a lot of connecting threads. certainly a lot of connecting threads. certainly a lot of connecting threads. Um and a lot of the progress of Um and a lot of the progress of Um and a lot of the progress of mathematics has can be represented by mathematics has can be represented by mathematics has can be represented by taking by stories of two fields of taking by stories of two fields of taking by stories of two fields of mathematics that were previously not mathematics that were previously not mathematics that were previously not connected and finding connections. Um an connected and finding connections. Um an connected and finding connections. Um an ancient example is um geometry and ancient example is um geometry and ancient example is um geometry and number theory you know. So so in the number theory you know. So so in the number theory you know. So so in the times of the ancient Greeks these were times of the ancient Greeks these were times of the ancient Greeks these were considered different subjects. Um I mean considered different subjects. Um I mean considered different subjects. Um I mean mathematicians worked on both. You know mathematicians worked on both. You know mathematicians worked on both. You know you could work both on on geometry most you could work both on on geometry most you could work both on on geometry most famously but also on numbers. Um but famously but also on numbers. Um but famously but also on numbers. Um but they were not really considered related.

they were not really considered related. they were not really considered related. Um I mean a little bit like you know you Um I mean a little bit like you know you Um I mean a little bit like you know you could say that that this length was five could say that that this length was five could say that that this length was five times this length because you could take times this length because you could take times this length because you could take five copies of this length and so forth. five copies of this length and so forth. five copies of this length and so forth. But it wasn't until Deart who really But it wasn't until Deart who really But it wasn't until Deart who really realized that who developed analytic realized that who developed analytic realized that who developed analytic geometry that you can you can geometry that you can you can geometry that you can you can parameterize the plane a geometric parameterize the plane a geometric parameterize the plane a geometric object by um by two real numbers. Every object by um by two real numbers. Every object by um by two real numbers. Every point can be and so geometric problems point can be and so geometric problems point can be and so geometric problems can be turned into into problems about can be turned into into problems about can be turned into into problems about numbers. Um and the the today this feels numbers. Um and the the today this feels numbers. Um and the the today this feels almost trivial like like there's there's almost trivial like like there's there's almost trivial like like there's there's there's no content to this like of there's no content to this like of there's no content to this like of course uh you you know um a plane is xx course uh you you know um a plane is xx course uh you you know um a plane is xx and y and because that's what we teach and y and because that's what we teach and y and because that's what we teach and it's internalized. Um but it was an and it's internalized. Um but it was an and it's internalized. Um but it was an important development that these these important development that these these important development that these these two fields were unified. Um and this two fields were unified. Um and this two fields were unified. Um and this process has just gone on throughout process has just gone on throughout process has just gone on throughout mathematics over and over again. algebra mathematics over and over again. algebra mathematics over and over again. algebra and geometry were separated and now we and geometry were separated and now we and geometry were separated and now we have a student algebraic geometry that have a student algebraic geometry that have a student algebraic geometry that connects them and over and over again connects them and over and over again connects them and over and over again and that's certainly the type of and that's certainly the type of and that's certainly the type of mathematics that that I enjoy the most. mathematics that that I enjoy the most. mathematics that that I enjoy the most. So I think there's sort of different So I think there's sort of different So I think there's sort of different styles to being a mathematician. I think styles to being a mathematician. I think styles to being a mathematician. I think hedgehogs and fox a fox knows many hedgehogs and fox a fox knows many hedgehogs and fox a fox knows many things a little bit but a hedgehog knows things a little bit but a hedgehog knows things a little bit but a hedgehog knows one thing very very well. Um and in one thing very very well. Um and in one thing very very well. Um and in mathematics there's definitely both mathematics there's definitely both mathematics there's definitely both hedgehogs and foxes. Um and then there's hedgehogs and foxes. Um and then there's hedgehogs and foxes. Um and then there's people who are kind of uh who can play people who are kind of uh who can play people who are kind of uh who can play both roles. Um and I think like ideal both roles. Um and I think like ideal both roles. Um and I think like ideal collaboration between mathematicians collaboration between mathematicians collaboration between mathematicians involves a very you need some diversity involves a very you need some diversity involves a very you need some diversity like um a fox working with many like um a fox working with many like um a fox working with many hedgehogs or or vice versa. So yeah but hedgehogs or or vice versa. So yeah but hedgehogs or or vice versa. So yeah but I identify mostly as a fox certainly I I I identify mostly as a fox certainly I I I identify mostly as a fox certainly I I like uh arbitrage somehow you like like

like uh arbitrage somehow you like like like uh arbitrage somehow you like like um learning how one field works learning um learning how one field works learning um learning how one field works learning the tricks of that field and then going the tricks of that field and then going the tricks of that field and then going to another field which people don't to another field which people don't to another field which people don't think is related but I can I can adapt think is related but I can I can adapt think is related but I can I can adapt the tricks. So see the connections the tricks. So see the connections the tricks. So see the connections between the fields. Yeah. So there are between the fields. Yeah. So there are between the fields. Yeah. So there are other mathematicians who are far deeper other mathematicians who are far deeper other mathematicians who are far deeper than I am. Like who really they're than I am. Like who really they're than I am. Like who really they're really hedgehogs. They they know really hedgehogs. They they know really hedgehogs. They they know everything about one field and they're everything about one field and they're everything about one field and they're much faster and and and more effective much faster and and and more effective much faster and and and more effective in that field. But I can I can give them in that field. But I can I can give them in that field. But I can I can give them these extra tools. I mean you said that these extra tools. I mean you said that these extra tools. I mean you said that you can be both the hedgehog and and the you can be both the hedgehog and and the you can be both the hedgehog and and the fox depending on the context depending fox depending on the context depending fox depending on the context depending on the collaboration. So what can you if on the collaboration. So what can you if on the collaboration. So what can you if it's at all possible speak to the it's at all possible speak to the it's at all possible speak to the difference between those two ways of difference between those two ways of difference between those two ways of thinking about a problem? say you're thinking about a problem? say you're thinking about a problem? say you're encountering a new problem, you know, encountering a new problem, you know, encountering a new problem, you know, searching for the connections versus searching for the connections versus searching for the connections versus like very singular focus. I'm much more like very singular focus. I'm much more like very singular focus. I'm much more comfortable with with the uh the uh the comfortable with with the uh the uh the comfortable with with the uh the uh the fox paradigm. Yeah. So, um yeah, I I fox paradigm. Yeah. So, um yeah, I I fox paradigm. Yeah. So, um yeah, I I like looking for analogies, narratives. like looking for analogies, narratives. like looking for analogies, narratives. Um I I spend a lot of time if there's a Um I I spend a lot of time if there's a Um I I spend a lot of time if there's a result I see in one field and I like the result I see in one field and I like the result I see in one field and I like the result, it's a cool result, but I don't result, it's a cool result, but I don't result, it's a cool result, but I don't like the proof. like it uses types of like the proof. like it uses types of like the proof. like it uses types of mathematics that I'm not super familiar mathematics that I'm not super familiar mathematics that I'm not super familiar with. Um I often try to reprove it with. Um I often try to reprove it with. Um I often try to reprove it myself using the tools that I favor. Um myself using the tools that I favor. Um myself using the tools that I favor. Um often my proof is worse. Um but um by often my proof is worse. Um but um by often my proof is worse. Um but um by the exercise of doing so um I can say oh the exercise of doing so um I can say oh the exercise of doing so um I can say oh now I can see what the other proof was now I can see what the other proof was now I can see what the other proof was trying to do. Um and from that I can get trying to do. Um and from that I can get trying to do. Um and from that I can get some understanding of of the tools that some understanding of of the tools that some understanding of of the tools that are used in in that field. So it's very are used in in that field. So it's very are used in in that field. So it's very exploratory, very doing crazy things in exploratory, very doing crazy things in exploratory, very doing crazy things in crazy fields and like reinventing the

crazy fields and like reinventing the crazy fields and like reinventing the wheel a lot. Yeah. Whereas the hedgehog wheel a lot. Yeah. Whereas the hedgehog wheel a lot. Yeah. Whereas the hedgehog style is uh I think much more scholarly, style is uh I think much more scholarly, style is uh I think much more scholarly, you know, you you you're very knowledge you know, you you you're very knowledge you know, you you you're very knowledge based. You you you you stay up to speed based. You you you you stay up to speed based. You you you you stay up to speed on like all the developments in this on like all the developments in this on like all the developments in this field. You you know all the history. Um field. You you know all the history. Um field. You you know all the history. Um you have a very good understanding of of you have a very good understanding of of you have a very good understanding of of exactly the strengths and weaknesses of exactly the strengths and weaknesses of exactly the strengths and weaknesses of of each particular uh technique. Um of each particular uh technique. Um of each particular uh technique. Um yeah uh I think you you rely a lot more yeah uh I think you you rely a lot more yeah uh I think you you rely a lot more on sort of calculation than sort of on sort of calculation than sort of on sort of calculation than sort of trying to find narratives. Um so yeah I trying to find narratives. Um so yeah I trying to find narratives. Um so yeah I mean I can do that too but uh there are mean I can do that too but uh there are mean I can do that too but uh there are other people who are extremely good at other people who are extremely good at other people who are extremely good at that. Let's step back and uh that. Let's step back and uh that. Let's step back and uh uh maybe look at the the a bit of a uh maybe look at the the a bit of a uh maybe look at the the a bit of a romanticized version of mathematics. romanticized version of mathematics. romanticized version of mathematics. Mhm. So, uh I think you've said that Mhm. So, uh I think you've said that Mhm. So, uh I think you've said that early on in your life, uh math was more early on in your life, uh math was more early on in your life, uh math was more like a puzzle solving activity when you like a puzzle solving activity when you like a puzzle solving activity when you were uh young. When did you first were uh young. When did you first were uh young. When did you first encounter a problem or proof where you encounter a problem or proof where you encounter a problem or proof where you realize math can have a kind of elegance realize math can have a kind of elegance realize math can have a kind of elegance and beauty to it?

That's a good question. Um when I came That's a good question. Um when I came to graduate school uh in Princeton, um to graduate school uh in Princeton, um to graduate school uh in Princeton, um so John Conway was there at the time. He so John Conway was there at the time. He so John Conway was there at the time. He he passed away a few years ago. But uh I he passed away a few years ago. But uh I he passed away a few years ago. But uh I remember one of the very first research remember one of the very first research remember one of the very first research talks I I went to was a talk by Conway talks I I went to was a talk by Conway talks I I went to was a talk by Conway on what he called extreme proof. So on what he called extreme proof. So on what he called extreme proof. So Conway had just had this this amazing Conway had just had this this amazing Conway had just had this this amazing way of of thinking about all kinds of way of of thinking about all kinds of way of of thinking about all kinds of things in in a way that you would things in in a way that you would things in in a way that you would normally think of. So um he thought of normally think of. So um he thought of normally think of. So um he thought of proofs themselves as occupying some sort proofs themselves as occupying some sort proofs themselves as occupying some sort of space, you know. So, so um if you of space, you know. So, so um if you of space, you know. So, so um if you want to prove something, let's say that want to prove something, let's say that want to prove something, let's say that there's infinitely many primes, okay, there's infinitely many primes, okay, there's infinitely many primes, okay, you avoid different proofs, but you you avoid different proofs, but you you avoid different proofs, but you could you could rank them in different could you could rank them in different could you could rank them in different axes like some proofs are elegant, some axes like some proofs are elegant, some axes like some proofs are elegant, some are long, some proofs are are um are long, some proofs are are um are long, some proofs are are um elementary and so forth. Um and so elementary and so forth. Um and so elementary and so forth. Um and so there's this cloud. So the space of all there's this cloud. So the space of all there's this cloud. So the space of all proofs itself has some sort of shape. Um proofs itself has some sort of shape. Um proofs itself has some sort of shape. Um and so he was interested in in extreme and so he was interested in in extreme and so he was interested in in extreme points of this shape like out of all all points of this shape like out of all all points of this shape like out of all all these proofs what is one that is the these proofs what is one that is the these proofs what is one that is the shortest at the the extent of every shortest at the the extent of every shortest at the the extent of every everything else or or the most everything else or or the most everything else or or the most elementary or or whatever. Um and so he elementary or or whatever. Um and so he elementary or or whatever. Um and so he gave some examples of well-known gave some examples of well-known gave some examples of well-known theorems and then he would give what he theorems and then he would give what he theorems and then he would give what he thought was was the extreme proof um in thought was was the extreme proof um in thought was was the extreme proof um in these different aspects. Um and I I just these different aspects. Um and I I just these different aspects. Um and I I just found that really eye opening um that found that really eye opening um that found that really eye opening um that that um you know it's not just getting a that um you know it's not just getting a that um you know it's not just getting a proof for a result was interesting but proof for a result was interesting but proof for a result was interesting but but once you have that proof you know but once you have that proof you know but once you have that proof you know trying to to uh to optimize it in trying to to uh to optimize it in trying to to uh to optimize it in various ways. Um that that proof um uh various ways. Um that that proof um uh various ways. Um that that proof um uh proofing itself had some craftsmanship proofing itself had some craftsmanship proofing itself had some craftsmanship to it. Um it it certainly informed my to it. Um it it certainly informed my to it. Um it it certainly informed my writing style. Um but you know like when writing style. Um but you know like when writing style. Um but you know like when you do your your math assignments and as you do your your math assignments and as you do your your math assignments and as undergraduate your homework and so

undergraduate your homework and so undergraduate your homework and so forth, you you're sort of encouraged to forth, you you're sort of encouraged to forth, you you're sort of encouraged to just write down any proof that works, just write down any proof that works, just write down any proof that works, okay, and hand it in and get a get as okay, and hand it in and get a get as okay, and hand it in and get a get as long as it gets a tick mark, you you long as it gets a tick mark, you you long as it gets a tick mark, you you move on. Um but if you want your your move on. Um but if you want your your move on. Um but if you want your your results to actually be influential and results to actually be influential and results to actually be influential and be read by people, um it can't just be be read by people, um it can't just be be read by people, um it can't just be correct. It should also um be a pleasure correct. It should also um be a pleasure correct. It should also um be a pleasure to read, you know, um motivated um be to read, you know, um motivated um be to read, you know, um motivated um be adaptable to to generalize to other um adaptable to to generalize to other um adaptable to to generalize to other um things. Um it's the same in many other things. Um it's the same in many other things. Um it's the same in many other disciplines like like coding. It's a disciplines like like coding. It's a disciplines like like coding. It's a there's a lot of analogies between math there's a lot of analogies between math there's a lot of analogies between math and coding. I like analogies if you and coding. I like analogies if you and coding. I like analogies if you haven't noticed. Um but um you know like haven't noticed. Um but um you know like haven't noticed. Um but um you know like you can code something spaghetti code you can code something spaghetti code you can code something spaghetti code that works for a certain task and it's that works for a certain task and it's that works for a certain task and it's quick and dirty and it works. But uh quick and dirty and it works. But uh quick and dirty and it works. But uh there's lots of good principles for for there's lots of good principles for for there's lots of good principles for for um writing code well so that other um writing code well so that other um writing code well so that other people can use it build upon it and so people can use it build upon it and so people can use it build upon it and so on and has fewer bugs and whatever. Um on and has fewer bugs and whatever. Um on and has fewer bugs and whatever. Um and there's similar things with mathemat and there's similar things with mathemat and there's similar things with mathemat mathematics. So yeah the first of all mathematics. So yeah the first of all mathematics. So yeah the first of all there's so many beautiful things there there's so many beautiful things there there's so many beautiful things there and and is one of the great minds uh in and and is one of the great minds uh in and and is one of the great minds uh in mathematics ever and computer science.

mathematics ever and computer science. mathematics ever and computer science. Uh just even considering the space of Uh just even considering the space of Uh just even considering the space of proofs. Yeah. and saying, "Okay, what proofs. Yeah. and saying, "Okay, what proofs. Yeah. and saying, "Okay, what does this space look like and what are does this space look like and what are does this space look like and what are the extremes?" Uh, like you mentioned, the extremes?" Uh, like you mentioned, the extremes?" Uh, like you mentioned, coding as an analogy is interesting coding as an analogy is interesting coding as an analogy is interesting because there's also this activity because there's also this activity because there's also this activity called the code golf. Oh, yeah. Yeah. called the code golf. Oh, yeah. Yeah. called the code golf. Oh, yeah. Yeah. Yeah. Which I also find beautiful and Yeah. Which I also find beautiful and Yeah. Which I also find beautiful and fun where people use different fun where people use different fun where people use different programming languages to try to write programming languages to try to write programming languages to try to write the shortest possible program that the shortest possible program that the shortest possible program that accomplishes a particular tasks. Then I accomplishes a particular tasks. Then I accomplishes a particular tasks. Then I believe there's even competitions on believe there's even competitions on believe there's even competitions on this. Yeah. And uh it's also a nice way this. Yeah. And uh it's also a nice way this. Yeah. And uh it's also a nice way to stress test not just the to stress test not just the to stress test not just the sort of the programs or in this case the sort of the programs or in this case the sort of the programs or in this case the proofs but also the different languages proofs but also the different languages proofs but also the different languages maybe that's the different notation or maybe that's the different notation or maybe that's the different notation or whatever to use to to accomplish a whatever to use to to accomplish a whatever to use to to accomplish a different task. Yeah, you learn a lot. I different task. Yeah, you learn a lot. I different task. Yeah, you learn a lot. I mean it may seem like a frivolous mean it may seem like a frivolous mean it may seem like a frivolous exercise but it can generate all these exercise but it can generate all these exercise but it can generate all these insights which if you didn't have this insights which if you didn't have this insights which if you didn't have this artificial um objective to to to pursue artificial um objective to to to pursue artificial um objective to to to pursue you might not see. What to you is the you might not see. What to you is the you might not see. What to you is the most beautiful or elegant equation in most beautiful or elegant equation in most beautiful or elegant equation in mathematics? I mean one of the things mathematics? I mean one of the things mathematics? I mean one of the things that people often look to in in beauty that people often look to in in beauty that people often look to in in beauty is the simplicity. So if you look at E= is the simplicity. So if you look at E= is the simplicity. So if you look at E= MC² so when when a few concepts come MC² so when when a few concepts come MC² so when when a few concepts come together that's why the oiler identity together that's why the oiler identity together that's why the oiler identity is often considered uh the most is often considered uh the most is often considered uh the most beautiful equation in mathematics. Do beautiful equation in mathematics. Do beautiful equation in mathematics. Do you do you find beauty in that one and you do you find beauty in that one and you do you find beauty in that one and the oil identity? Yeah. Well, as I said, the oil identity? Yeah. Well, as I said, the oil identity? Yeah. Well, as I said, I mean, what I find most appealing is is I mean, what I find most appealing is is I mean, what I find most appealing is is connections between different things connections between different things connections between different things that um so the if ei= minus one um so

that um so the if ei= minus one um so that um so the if ei= minus one um so yeah people oh uses all the fundamental yeah people oh uses all the fundamental yeah people oh uses all the fundamental constants okay that that's I mean that's constants okay that that's I mean that's constants okay that that's I mean that's cute um but but to me so the exponential cute um but but to me so the exponential cute um but but to me so the exponential function was interested by oil to function was interested by oil to function was interested by oil to measure exponential growth you know so measure exponential growth you know so measure exponential growth you know so compound interest or decay anything compound interest or decay anything compound interest or decay anything which is continuously growing which is continuously growing which is continuously growing continuously decreasing growth and decay continuously decreasing growth and decay continuously decreasing growth and decay or dilation or contraction is modeled by or dilation or contraction is modeled by or dilation or contraction is modeled by the exponential function Um whereas pi the exponential function Um whereas pi the exponential function Um whereas pi uh comes around from circles and uh comes around from circles and uh comes around from circles and rotation right if you want to rotate a rotation right if you want to rotate a rotation right if you want to rotate a needle for example 180° you need to needle for example 180° you need to needle for example 180° you need to rotate by pi radians and i complex rotate by pi radians and i complex rotate by pi radians and i complex numbers represents the swing between numbers represents the swing between numbers represents the swing between imagine axis of a 90° rotation so a imagine axis of a 90° rotation so a imagine axis of a 90° rotation so a change in direction so the x function change in direction so the x function change in direction so the x function represents growth and decay in the represents growth and decay in the represents growth and decay in the direction where you really are um when direction where you really are um when direction where you really are um when you stick an i in the exponential it now you stick an i in the exponential it now you stick an i in the exponential it now it's it's instead of motion in the same it's it's instead of motion in the same it's it's instead of motion in the same direction as your current position it's direction as your current position it's direction as your current position it's the motion has right angles to the motion has right angles to the motion has right angles to composition. So rotation um and then so composition. So rotation um and then so composition. So rotation um and then so e e pi equ= minus 1 tells you that if e e pi equ= minus 1 tells you that if e e pi equ= minus 1 tells you that if you rotate for time pi you end up at the you rotate for time pi you end up at the you rotate for time pi you end up at the other direction. So it unifies geometry other direction. So it unifies geometry other direction. So it unifies geometry through dilation and exponential growth through dilation and exponential growth through dilation and exponential growth or dynamics through this act of of or dynamics through this act of of or dynamics through this act of of complexification rotation by by i. So it complexification rotation by by i. So it complexification rotation by by i. So it connects together all these tools connects together all these tools connects together all these tools mathematics. Yeah. Yeah. dynamic mathematics. Yeah. Yeah. dynamic mathematics. Yeah. Yeah. dynamic structure and complex and complex and um structure and complex and complex and um structure and complex and complex and um the complex numbers they all considered the complex numbers they all considered the complex numbers they all considered almost yeah they were all next door almost yeah they were all next door almost yeah they were all next door neighbors in mathematics because of this neighbors in mathematics because of this neighbors in mathematics because of this identity. Do do you think the thing you identity. Do do you think the thing you identity. Do do you think the thing you mentioned is cute the the the collision mentioned is cute the the the collision mentioned is cute the the the collision of notations from these disperate of notations from these disperate of notations from these disperate fields?

fields? fields? Um it's just a frivolous side effect or Um it's just a frivolous side effect or Um it's just a frivolous side effect or do you think there is legitimate like do you think there is legitimate like do you think there is legitimate like value in when the notation all the our value in when the notation all the our value in when the notation all the our old friends come together old friends come together old friends come together night? Well, it's it's it's confirmation night? Well, it's it's it's confirmation night? Well, it's it's it's confirmation that you have the right concepts. Um so that you have the right concepts. Um so that you have the right concepts. Um so when you first study anything um you you when you first study anything um you you when you first study anything um you you have to measure things and give them have to measure things and give them have to measure things and give them names. Um and initially sometimes your names. Um and initially sometimes your names. Um and initially sometimes your because your your model is again too far because your your model is again too far because your your model is again too far off from reality you give the wrong off from reality you give the wrong off from reality you give the wrong things the best names and you only find things the best names and you only find things the best names and you only find out later what's what's really important out later what's what's really important out later what's what's really important physicists can do this sometimes I mean physicists can do this sometimes I mean physicists can do this sometimes I mean but it turns out okay so actually with but it turns out okay so actually with but it turns out okay so actually with physics okay so E= MC² okay so one of physics okay so E= MC² okay so one of physics okay so E= MC² okay so one of the the big things was the E right so the the big things was the E right so the the big things was the E right so when when Aristotle first came up with when when Aristotle first came up with when when Aristotle first came up with his laws of of motion and then and then his laws of of motion and then and then his laws of of motion and then and then um Galileo or Newton and so forth you um Galileo or Newton and so forth you um Galileo or Newton and so forth you know they saw the things they could they know they saw the things they could they know they saw the things they could they could measure they could measure mass could measure they could measure mass could measure they could measure mass and acceleration and force and so forth and acceleration and force and so forth and acceleration and force and so forth and so Newtonian mechanics for example and so Newtonian mechanics for example and so Newtonian mechanics for example F= ma was the famous Newton second law F= ma was the famous Newton second law F= ma was the famous Newton second law of motion so those were the the primary of motion so those were the the primary of motion so those were the the primary objects so they gave them the central objects so they gave them the central objects so they gave them the central building in the theory it was only later building in the theory it was only later building in the theory it was only later after people started analyzing these after people started analyzing these after people started analyzing these equations that there always seemed to be equations that there always seemed to be equations that there always seemed to be these quantities that were conserved um these quantities that were conserved um these quantities that were conserved um so momentum and energy um uh and it's so momentum and energy um uh and it's so momentum and energy um uh and it's not obvious that things happen energy not obvious that things happen energy not obvious that things happen energy like it's not something you can directly like it's not something you can directly like it's not something you can directly measure the same way you can measure measure the same way you can measure measure the same way you can measure mass and and and velocity so forth but mass and and and velocity so forth but mass and and and velocity so forth but over time people realize is that this over time people realize is that this over time people realize is that this was actually a really fundamental was actually a really fundamental was actually a really fundamental concept. Hamilton eventually in 19th concept. Hamilton eventually in 19th concept. Hamilton eventually in 19th century reformulated Newton's laws of century reformulated Newton's laws of century reformulated Newton's laws of physics into what's called Hamiltonian physics into what's called Hamiltonian physics into what's called Hamiltonian mechanics where the energy which is now mechanics where the energy which is now mechanics where the energy which is now called the Hamiltonian was the dominant

called the Hamiltonian was the dominant called the Hamiltonian was the dominant object once you know how to measure the object once you know how to measure the object once you know how to measure the Hamiltonian of any system. You can Hamiltonian of any system. You can Hamiltonian of any system. You can describe completely the dynamics like describe completely the dynamics like describe completely the dynamics like what happens to to all the states like what happens to to all the states like what happens to to all the states like it's um it it really was a central actor it's um it it really was a central actor it's um it it really was a central actor which was not obvious initially. Um and which was not obvious initially. Um and which was not obvious initially. Um and this uh helped actually uh this change this uh helped actually uh this change this uh helped actually uh this change of perspective really helped when of perspective really helped when of perspective really helped when quantum mechanics came along. Uh because quantum mechanics came along. Uh because quantum mechanics came along. Uh because um the early physicists who studied um the early physicists who studied um the early physicists who studied quantum mechanics, they had a lot of quantum mechanics, they had a lot of quantum mechanics, they had a lot of trouble trying to adapt their Newtonian trouble trying to adapt their Newtonian trouble trying to adapt their Newtonian thinking because everything was a thinking because everything was a thinking because everything was a particle and so forth to to to quantum particle and so forth to to to quantum particle and so forth to to to quantum mechanics, you know, because I think mechanics, you know, because I think mechanics, you know, because I think because it was a wave. It just looked because it was a wave. It just looked because it was a wave. It just looked really really weird. Um like you ask really really weird. Um like you ask really really weird. Um like you ask what is the quantum version of F equals what is the quantum version of F equals what is the quantum version of F equals MA? And it's really really hard to to MA? And it's really really hard to to MA? And it's really really hard to to give an answer to that. Um but it turns give an answer to that. Um but it turns give an answer to that. Um but it turns out that the Hamiltonian which was so um out that the Hamiltonian which was so um out that the Hamiltonian which was so um secretly behind the scenes in classical secretly behind the scenes in classical secretly behind the scenes in classical mechanics also is the key uh object in mechanics also is the key uh object in mechanics also is the key uh object in um um in quantum mechanics that there's um um in quantum mechanics that there's um um in quantum mechanics that there's there's also an object called there's also an object called there's also an object called Hamiltonian. It's a different type of Hamiltonian. It's a different type of Hamiltonian. It's a different type of object. It's what's called an operator object. It's what's called an operator object. It's what's called an operator rather than than a function. But um and rather than than a function. But um and rather than than a function. But um and um but again once you specify it you um but again once you specify it you um but again once you specify it you specify the entire dynamics. So there's specify the entire dynamics. So there's specify the entire dynamics. So there's something called Shingers equation that something called Shingers equation that something called Shingers equation that tells you exactly how quantum systems tells you exactly how quantum systems tells you exactly how quantum systems evolve once you have a Hamiltonian. So evolve once you have a Hamiltonian. So evolve once you have a Hamiltonian. So side by side they look completely side by side they look completely side by side they look completely different objects you know like so one different objects you know like so one different objects you know like so one involves particles one involves waves involves particles one involves waves involves particles one involves waves and so forth but with this centrality and so forth but with this centrality and so forth but with this centrality you could start actually transferring a you could start actually transferring a you could start actually transferring a lot of intuition and facts from lot of intuition and facts from lot of intuition and facts from classical mechanics to quantum classical mechanics to quantum classical mechanics to quantum mechanics.

mechanics. mechanics. For example, in classical mechanics, For example, in classical mechanics, For example, in classical mechanics, there's this thing called ner's theorem. there's this thing called ner's theorem. there's this thing called ner's theorem. Every time there's a symmetry in a Every time there's a symmetry in a Every time there's a symmetry in a physical system, there is a conservation physical system, there is a conservation physical system, there is a conservation law. So the laws of physics are law. So the laws of physics are law. So the laws of physics are translation invariant. Like if I move 10 translation invariant. Like if I move 10 translation invariant. Like if I move 10 steps to the left, I experience the same steps to the left, I experience the same steps to the left, I experience the same laws of physics as if I was here. And laws of physics as if I was here. And laws of physics as if I was here. And that corresponds to conservation that corresponds to conservation that corresponds to conservation momentum. Um if I turn around by by some momentum. Um if I turn around by by some momentum. Um if I turn around by by some angle again, I experience the same laws angle again, I experience the same laws angle again, I experience the same laws of physics. This corresponds to of physics. This corresponds to of physics. This corresponds to conservation angular momentum. If I wait conservation angular momentum. If I wait conservation angular momentum. If I wait for 10 minutes, um I still have the same for 10 minutes, um I still have the same for 10 minutes, um I still have the same laws of physics. Um so this time laws of physics. Um so this time laws of physics. Um so this time translation variance. this corresponds translation variance. this corresponds translation variance. this corresponds to the low conservation of energy. Um, to the low conservation of energy. Um, to the low conservation of energy. Um, so there's this fundamental connection so there's this fundamental connection so there's this fundamental connection between symmetry and conservation. Um, between symmetry and conservation. Um, between symmetry and conservation. Um, and that's also true in quantum and that's also true in quantum and that's also true in quantum mechanics. Even though the equations are mechanics. Even though the equations are mechanics. Even though the equations are completely different, but because completely different, but because completely different, but because they're both coming from the they're both coming from the they're both coming from the Hamiltonian, the Hamiltonian controls Hamiltonian, the Hamiltonian controls Hamiltonian, the Hamiltonian controls everything. Um, every time the everything. Um, every time the everything. Um, every time the Hamiltonian has a symmetry, the Hamiltonian has a symmetry, the Hamiltonian has a symmetry, the equations will will have a conservation equations will will have a conservation equations will will have a conservation law. Um, so it's it's it's it's once you law. Um, so it's it's it's it's once you law. Um, so it's it's it's it's once you have the right language, it actually have the right language, it actually have the right language, it actually makes things um a lot a lot cleaner. One makes things um a lot a lot cleaner. One makes things um a lot a lot cleaner. One of the problems why we can't unify of the problems why we can't unify of the problems why we can't unify quantum mechanics and general relativity quantum mechanics and general relativity quantum mechanics and general relativity yet we haven't figured out what the yet we haven't figured out what the yet we haven't figured out what the fundamental objects are like for example fundamental objects are like for example fundamental objects are like for example we have to give up the notion of space we have to give up the notion of space we have to give up the notion of space and time being these almost uklidian and time being these almost uklidian and time being these almost uklidian type spaces and there has to be um you type spaces and there has to be um you type spaces and there has to be um you know and you know we kind of know that know and you know we kind of know that know and you know we kind of know that at very tiny scales um there's going to at very tiny scales um there's going to at very tiny scales um there's going to be quite fluctuations of space be quite fluctuations of space be quite fluctuations of space space-time foam um and trying to to use space-time foam um and trying to to use space-time foam um and trying to to use cartigian coord xyz is going to be it's cartigian coord xyz is going to be it's cartigian coord xyz is going to be it's it's just it's it's a non-starter but we it's just it's it's a non-starter but we it's just it's it's a non-starter but we don't know how to what to replace it don't know how to what to replace it don't know how to what to replace it with um We don't actually have the with um We don't actually have the with um We don't actually have the mathematical um um concepts the analog

mathematical um um concepts the analog mathematical um um concepts the analog Hamiltonian that sort of organized Hamiltonian that sort of organized Hamiltonian that sort of organized everything. Does your gut say that there everything. Does your gut say that there everything. Does your gut say that there is a theory of everything. So this is is a theory of everything. So this is is a theory of everything. So this is even possible to unify to find this even possible to unify to find this even possible to unify to find this language that unifies general relativity language that unifies general relativity language that unifies general relativity and quantum mechanics. I believe so. I and quantum mechanics. I believe so. I and quantum mechanics. I believe so. I mean the history of physics has been out mean the history of physics has been out mean the history of physics has been out of unification much like mathematics um of unification much like mathematics um of unification much like mathematics um over the years. You know electricity and over the years. You know electricity and over the years. You know electricity and magnetism were separate theories and magnetism were separate theories and magnetism were separate theories and then Maxwell unified them. you know, then Maxwell unified them. you know, then Maxwell unified them. you know, Newton unified the the motions of the Newton unified the the motions of the Newton unified the the motions of the heavens with the motions on of objects heavens with the motions on of objects heavens with the motions on of objects on the earth and so forth. So, it should on the earth and so forth. So, it should on the earth and so forth. So, it should happen. It's just that the um u again to happen. It's just that the um u again to happen. It's just that the um u again to go back to this model of the go back to this model of the go back to this model of the observations and and theory. Part of our observations and and theory. Part of our observations and and theory. Part of our problem is that physics is a victim's problem is that physics is a victim's problem is that physics is a victim's own success that our two big theories of own success that our two big theories of own success that our two big theories of of of physics general relativity and of of physics general relativity and of of physics general relativity and quantum mechanics are so are so good now quantum mechanics are so are so good now quantum mechanics are so are so good now that together they cover 99.9% of sort that together they cover 99.9% of sort that together they cover 99.9% of sort of all the observations we can make. Um, of all the observations we can make. Um, of all the observations we can make. Um, and you have to like either go to and you have to like either go to and you have to like either go to extremely insane particle accelerations extremely insane particle accelerations extremely insane particle accelerations or or the early universe or or or things or or the early universe or or or things or or the early universe or or or things that are really hard to measure um in that are really hard to measure um in that are really hard to measure um in order to get any deviation from either order to get any deviation from either order to get any deviation from either of these two theories to the point where of these two theories to the point where of these two theories to the point where you can actually figure out how to how you can actually figure out how to how you can actually figure out how to how to combine them together. Um, but I have to combine them together. Um, but I have to combine them together. Um, but I have faith that we, you know, we've we've faith that we, you know, we've we've faith that we, you know, we've we've been doing this for centuries and we've been doing this for centuries and we've been doing this for centuries and we've made progress before. There's no reason made progress before. There's no reason made progress before. There's no reason why we should stop. Do you think it will why we should stop. Do you think it will why we should stop. Do you think it will be a mathematician that develops uh be a mathematician that develops uh be a mathematician that develops uh theory of everything? What often happens theory of everything? What often happens theory of everything? What often happens is that when the physicists need uh um is that when the physicists need uh um is that when the physicists need uh um some of mathematics, there's often some some of mathematics, there's often some some of mathematics, there's often some precursor that the mathematicians um precursor that the mathematicians um precursor that the mathematicians um worked out earlier. So when Einstein worked out earlier. So when Einstein worked out earlier. So when Einstein started realizing that space was curved,

started realizing that space was curved, started realizing that space was curved, he went to some mathematician and asked he went to some mathematician and asked he went to some mathematician and asked is there is there some theory of curved is there is there some theory of curved is there is there some theory of curved space that the mathematicians already space that the mathematicians already space that the mathematicians already came up with that could be useful and he came up with that could be useful and he came up with that could be useful and he said oh yeah there's I think Reman came said oh yeah there's I think Reman came said oh yeah there's I think Reman came up with something um and so yeah Reman up with something um and so yeah Reman up with something um and so yeah Reman had developed remmaning geometry um had developed remmaning geometry um had developed remmaning geometry um which is precisely you know a theory of which is precisely you know a theory of which is precisely you know a theory of spaces that occurred in various general spaces that occurred in various general spaces that occurred in various general ways which turned out to be almost ways which turned out to be almost ways which turned out to be almost exactly what was needed um for exactly what was needed um for exactly what was needed um for Einstein's theory. This is going back to Einstein's theory. This is going back to Einstein's theory. This is going back to Dwick's unreasonable effectiveness of Dwick's unreasonable effectiveness of Dwick's unreasonable effectiveness of mathematics. I think the theories that mathematics. I think the theories that mathematics. I think the theories that work well to explain the universe tend work well to explain the universe tend work well to explain the universe tend to also involve the same mathematical to also involve the same mathematical to also involve the same mathematical objects that work well to solve objects that work well to solve objects that work well to solve mathematical problems. Ultimately, mathematical problems. Ultimately, mathematical problems. Ultimately, they're just sort of both ways of they're just sort of both ways of they're just sort of both ways of organizing data um in in in useful ways. organizing data um in in in useful ways. organizing data um in in in useful ways. It just feels like you might need to go It just feels like you might need to go It just feels like you might need to go some weird land that's very hard to to some weird land that's very hard to to some weird land that's very hard to to intuit it like you know you have like intuit it like you know you have like intuit it like you know you have like string theory. Yeah, that that's that string theory. Yeah, that that's that string theory. Yeah, that that's that was that was a leading candidate for was that was a leading candidate for was that was a leading candidate for many decades. It's I think it's slowly many decades. It's I think it's slowly many decades. It's I think it's slowly falling out of fashion because it's it's falling out of fashion because it's it's falling out of fashion because it's it's not matching experiment. So one of the not matching experiment. So one of the not matching experiment. So one of the big challenges of course like you said big challenges of course like you said big challenges of course like you said is experiment is very tough. Yes. is experiment is very tough. Yes. is experiment is very tough. Yes. Because of the how effective both Because of the how effective both Because of the how effective both theories are. But the other is like just theories are. But the other is like just theories are. But the other is like just you know you're talking about you're not you know you're talking about you're not you know you're talking about you're not just deviating from spaceime. You're just deviating from spaceime. You're just deviating from spaceime. You're going into like some crazy number of going into like some crazy number of going into like some crazy number of dimensions. You're doing all kinds of dimensions. You're doing all kinds of dimensions. You're doing all kinds of weird stuff that to us we've gone so far weird stuff that to us we've gone so far weird stuff that to us we've gone so far from this flat earth that we started at from this flat earth that we started at from this flat earth that we started at like now we're just it's it's very hard like now we're just it's it's very hard like now we're just it's it's very hard to use our limited ape descendants of uh to use our limited ape descendants of uh to use our limited ape descendants of uh uh cognition to intuitit what that

uh cognition to intuitit what that uh cognition to intuitit what that reality really is like. This is why reality really is like. This is why reality really is like. This is why analogies are so important, you know. I analogies are so important, you know. I analogies are so important, you know. I mean, so yeah, the round earth is not mean, so yeah, the round earth is not mean, so yeah, the round earth is not intuitive because we're stuck on it, but intuitive because we're stuck on it, but intuitive because we're stuck on it, but you know, but you know, but round you know, but you know, but round you know, but you know, but round objects in general, we have pretty good objects in general, we have pretty good objects in general, we have pretty good intuition over uh and we have intuition intuition over uh and we have intuition intuition over uh and we have intuition about light works and so forth. And like about light works and so forth. And like about light works and so forth. And like it's it's actually a good exercise to it's it's actually a good exercise to it's it's actually a good exercise to actually work out how eclipses and actually work out how eclipses and actually work out how eclipses and phases of of the sun and the moon and so phases of of the sun and the moon and so phases of of the sun and the moon and so forth can be really easily explained by forth can be really easily explained by forth can be really easily explained by by by by round earth and round moon, you by by by round earth and round moon, you by by by round earth and round moon, you know, um and models. Um and and you can know, um and models. Um and and you can know, um and models. Um and and you can just take you know a basketball and a just take you know a basketball and a just take you know a basketball and a golf ball and and and a light source and golf ball and and and a light source and golf ball and and and a light source and actually do these things yourself. Um so actually do these things yourself. Um so actually do these things yourself. Um so the intuition is there. Um but yeah you the intuition is there. Um but yeah you the intuition is there. Um but yeah you have to transfer it. That is a big leap have to transfer it. That is a big leap have to transfer it. That is a big leap intellectually for us to go from flat to intellectually for us to go from flat to intellectually for us to go from flat to round earth because you know our life is round earth because you know our life is round earth because you know our life is mostly lived in flat land. Yeah. To load mostly lived in flat land. Yeah. To load mostly lived in flat land. Yeah. To load that information and we all like take it that information and we all like take it that information and we all like take it for granted. We take so many things for for granted. We take so many things for for granted. We take so many things for granted because science has established granted because science has established granted because science has established a lot of evidence for this kind of a lot of evidence for this kind of a lot of evidence for this kind of thing. But you know, we're on a round thing. But you know, we're on a round thing. But you know, we're on a round rock. Yeah. Flying through space. Yeah.

rock. Yeah. Flying through space. Yeah. rock. Yeah. Flying through space. Yeah. Yeah. And it's a big leap and you have Yeah. And it's a big leap and you have Yeah. And it's a big leap and you have to take a chain of those leaps the more to take a chain of those leaps the more to take a chain of those leaps the more and more and more we progress. Right. and more and more we progress. Right. and more and more we progress. Right. Yeah. So modern science is maybe again a Yeah. So modern science is maybe again a Yeah. So modern science is maybe again a victim of its own success is that you victim of its own success is that you victim of its own success is that you know in order to be more accurate it has know in order to be more accurate it has know in order to be more accurate it has to to move further and further away from to to move further and further away from to to move further and further away from your initial intuition. And so um for your initial intuition. And so um for your initial intuition. And so um for someone who hasn't gone through the someone who hasn't gone through the someone who hasn't gone through the whole process of science education it whole process of science education it whole process of science education it looks more more suspicious because of looks more more suspicious because of looks more more suspicious because of that. So, you know, we we need we need that. So, you know, we we need we need that. So, you know, we we need we need more grounding. I mean, I I think um I more grounding. I mean, I I think um I more grounding. I mean, I I think um I mean, you know, there are there are mean, you know, there are there are mean, you know, there are there are scientists who do excellent outreach. Um scientists who do excellent outreach. Um scientists who do excellent outreach. Um but there's this there this there's but there's this there this there's but there's this there this there's there there's lots of science things there there's lots of science things there there's lots of science things that you can do at home. There's lots of that you can do at home. There's lots of that you can do at home. There's lots of YouTube videos. I did a YouTube video YouTube videos. I did a YouTube video YouTube videos. I did a YouTube video recent of Grant Sanderson. We talked recent of Grant Sanderson. We talked recent of Grant Sanderson. We talked about this earlier that uh you know how about this earlier that uh you know how about this earlier that uh you know how the ancient Greeks were able to measure the ancient Greeks were able to measure the ancient Greeks were able to measure things like the distance to the moon, things like the distance to the moon, things like the distance to the moon, distance to the earth, and you know, distance to the earth, and you know, distance to the earth, and you know, using techniques that you you could also using techniques that you you could also using techniques that you you could also replicate yourself. Um it doesn't all replicate yourself. Um it doesn't all replicate yourself. Um it doesn't all have to be like fancy space telescopes have to be like fancy space telescopes have to be like fancy space telescopes and and very intimidating mathematics. and and very intimidating mathematics. and and very intimidating mathematics. Yeah, that's uh I highly recommend that. Yeah, that's uh I highly recommend that. Yeah, that's uh I highly recommend that. I believe you give a lecture and you I believe you give a lecture and you I believe you give a lecture and you also did an incredible video with Grant. also did an incredible video with Grant. also did an incredible video with Grant. It's a beautiful experience to try to It's a beautiful experience to try to It's a beautiful experience to try to put yourself in the mind of a person put yourself in the mind of a person put yourself in the mind of a person from that time. Mhm. Shrouded in from that time. Mhm. Shrouded in from that time. Mhm. Shrouded in mystery, right? You know, you're like on mystery, right? You know, you're like on mystery, right? You know, you're like on this planet, you don't know the shape of this planet, you don't know the shape of this planet, you don't know the shape of it, the size of it. You see some stars, it, the size of it. You see some stars, it, the size of it. You see some stars, you see some you see some things and you you see some you see some things and you you see some you see some things and you try to like localize yourself in this try to like localize yourself in this try to like localize yourself in this world. Yeah. Yeah. And try to make some world. Yeah. Yeah. And try to make some world. Yeah. Yeah. And try to make some kind of general statements about kind of general statements about kind of general statements about distance to places. Change your distance to places. Change your distance to places. Change your perspective is really important. You say perspective is really important. You say perspective is really important. You say travel bordens the mind. This is travel bordens the mind. This is travel bordens the mind. This is intellectual travel. You know put intellectual travel. You know put intellectual travel. You know put yourself in the mind of the ancient yourself in the mind of the ancient yourself in the mind of the ancient Greeks or or some other person some Greeks or or some other person some Greeks or or some other person some other time period. Make hypothesis other time period. Make hypothesis other time period. Make hypothesis spherical cows whatever you know

spherical cows whatever you know spherical cows whatever you know speculate. Um and you know this is this speculate. Um and you know this is this speculate. Um and you know this is this is what mathematicians do and some what is what mathematicians do and some what is what mathematicians do and some what artists do actually. It's just artists do actually. It's just artists do actually. It's just incredible that given the extreme incredible that given the extreme incredible that given the extreme constraints, you could still say very constraints, you could still say very constraints, you could still say very powerful things. That's why it's powerful things. That's why it's powerful things. That's why it's inspiring looking back in history. How inspiring looking back in history. How inspiring looking back in history. How much can be figured out right when you much can be figured out right when you much can be figured out right when you don't have much to figure out stuff like don't have much to figure out stuff like don't have much to figure out stuff like if you propose axioms then the if you propose axioms then the if you propose axioms then the mathematics lets you follow those a to mathematics lets you follow those a to mathematics lets you follow those a to their conclusions and sometimes you can their conclusions and sometimes you can their conclusions and sometimes you can get quite a quite a long way from you get quite a quite a long way from you get quite a quite a long way from you know initial hypothesis. If we can stay know initial hypothesis. If we can stay know initial hypothesis. If we can stay in the land of the weird, you mentioned in the land of the weird, you mentioned in the land of the weird, you mentioned general relativity. You've uh you've general relativity. You've uh you've general relativity. You've uh you've contributed uh to the mathematical contributed uh to the mathematical contributed uh to the mathematical understanding of Einstein's field understanding of Einstein's field understanding of Einstein's field equations. Can you explain this work and equations. Can you explain this work and equations. Can you explain this work and from a sort of mathematical standpoint from a sort of mathematical standpoint from a sort of mathematical standpoint uh what aspects of general relativity uh what aspects of general relativity uh what aspects of general relativity are intriguing to you, challenging to are intriguing to you, challenging to are intriguing to you, challenging to you? I have worked on some equations. you? I have worked on some equations. you? I have worked on some equations. There's something called the the wave There's something called the the wave There's something called the the wave maps equation or the sigma field model maps equation or the sigma field model maps equation or the sigma field model which is not quite the equation of which is not quite the equation of which is not quite the equation of space-time gravity itself but of certain space-time gravity itself but of certain space-time gravity itself but of certain fields that might exist on top of fields that might exist on top of fields that might exist on top of spaceime. Um so Einstein's equations of spaceime. Um so Einstein's equations of spaceime. Um so Einstein's equations of relativity just describes space and time relativity just describes space and time relativity just describes space and time itself. Um but then there's other fields itself. Um but then there's other fields itself. Um but then there's other fields that live on top of that. There's the that live on top of that. There's the that live on top of that. There's the electromagnetic field. Um there's electromagnetic field. Um there's electromagnetic field. Um there's control fields and there's this whole control fields and there's this whole control fields and there's this whole hierarchy of different equations of hierarchy of different equations of hierarchy of different equations of which Einstein is considered one of the which Einstein is considered one of the which Einstein is considered one of the most nonlinear and difficult. But most nonlinear and difficult. But most nonlinear and difficult. But relatively low in the hierarchy was this relatively low in the hierarchy was this relatively low in the hierarchy was this thing called the wave maps equation. So thing called the wave maps equation. So thing called the wave maps equation. So it's a wave which at any given point uh it's a wave which at any given point uh it's a wave which at any given point uh is fixed to be like on a sphere. Um so

is fixed to be like on a sphere. Um so is fixed to be like on a sphere. Um so uh I can think of a bunch of arrows in uh I can think of a bunch of arrows in uh I can think of a bunch of arrows in space and time and and the arrows space and time and and the arrows space and time and and the arrows pointing in in different directions. Um pointing in in different directions. Um pointing in in different directions. Um but they propagate like waves. If you but they propagate like waves. If you but they propagate like waves. If you wiggle an arrow it was it will propagate wiggle an arrow it was it will propagate wiggle an arrow it was it will propagate and make all the arrows move kind of and make all the arrows move kind of and make all the arrows move kind of like sheets of wheat in the wheat field. like sheets of wheat in the wheat field. like sheets of wheat in the wheat field. And I was interested in the global And I was interested in the global And I was interested in the global regularity problem again for this regularity problem again for this regularity problem again for this question like is it possible for for all question like is it possible for for all question like is it possible for for all the energy here to collect at a point. the energy here to collect at a point. the energy here to collect at a point. So the equation I considered was So the equation I considered was So the equation I considered was actually what's called a critical actually what's called a critical actually what's called a critical equation where it's actually the equation where it's actually the equation where it's actually the behavior at all scales is roughly the behavior at all scales is roughly the behavior at all scales is roughly the same. Um and I was able barely to show same. Um and I was able barely to show same. Um and I was able barely to show that um that you couldn't actually force that um that you couldn't actually force that um that you couldn't actually force a scenario where all the energy a scenario where all the energy a scenario where all the energy concentrated at one point that the concentrated at one point that the concentrated at one point that the energy had to disperse a little bit and energy had to disperse a little bit and energy had to disperse a little bit and the moment it dis little bit it it would the moment it dis little bit it it would the moment it dis little bit it it would it would stay regular. Yeah. This was it would stay regular. Yeah. This was it would stay regular. Yeah. This was back in 2000. That was part of why I got back in 2000. That was part of why I got back in 2000. That was part of why I got interested in narrows afterwards interested in narrows afterwards interested in narrows afterwards actually. Yeah. So I developed some actually. Yeah. So I developed some actually. Yeah. So I developed some techniques to um solve that problem. So techniques to um solve that problem. So techniques to um solve that problem. So part of it is it was um this problem is part of it is it was um this problem is part of it is it was um this problem is really nonlinear uh because of the really nonlinear uh because of the really nonlinear uh because of the curvature of the sphere. Um this there curvature of the sphere. Um this there curvature of the sphere. Um this there was a certain nonlinear effect which was was a certain nonlinear effect which was was a certain nonlinear effect which was a non-perturbative effect. It was when a non-perturbative effect. It was when a non-perturbative effect. It was when you sort of looked at it normally it you sort of looked at it normally it you sort of looked at it normally it looked larger than the linear effects of looked larger than the linear effects of looked larger than the linear effects of the wave equation. Um and so it was hard the wave equation. Um and so it was hard the wave equation. Um and so it was hard to to keep things under control even to to keep things under control even to to keep things under control even when the energy was small. But I when the energy was small. But I when the energy was small. But I developed what's called a gauge developed what's called a gauge developed what's called a gauge transformation. So the equation is kind transformation. So the equation is kind transformation. So the equation is kind of like an evolution of of of heaves of of like an evolution of of of heaves of of like an evolution of of of heaves of wheat and and they're all bending back wheat and and they're all bending back wheat and and they're all bending back and forth and so there's a lot of and forth and so there's a lot of and forth and so there's a lot of motion. Um but like if you imagine like motion. Um but like if you imagine like motion. Um but like if you imagine like stabilizing the flow by attaching little stabilizing the flow by attaching little stabilizing the flow by attaching little cameras at different points in space cameras at different points in space cameras at different points in space which are trying to move in a way that which are trying to move in a way that which are trying to move in a way that captures most of the motion and under

captures most of the motion and under captures most of the motion and under this stabilized flow the flow becomes a this stabilized flow the flow becomes a this stabilized flow the flow becomes a lot more linear. I discovered a way to lot more linear. I discovered a way to lot more linear. I discovered a way to transform the the equation to reduce the transform the the equation to reduce the transform the the equation to reduce the amount of of nonlinear effects. Um and amount of of nonlinear effects. Um and amount of of nonlinear effects. Um and then I was able to to to to solve the then I was able to to to to solve the then I was able to to to to solve the equation. I found this transformation equation. I found this transformation equation. I found this transformation while visiting my aunt in Australia and while visiting my aunt in Australia and while visiting my aunt in Australia and I was trying to understand the dynamics I was trying to understand the dynamics I was trying to understand the dynamics of all these fields and I I couldn't do of all these fields and I I couldn't do of all these fields and I I couldn't do it with pen and paper. Um and I had not it with pen and paper. Um and I had not it with pen and paper. Um and I had not enough facility of computers to do any enough facility of computers to do any enough facility of computers to do any computer simulations. So I ended up computer simulations. So I ended up computer simulations. So I ended up closing my eyes being on on the floor closing my eyes being on on the floor closing my eyes being on on the floor and just imagining myself to actually be and just imagining myself to actually be and just imagining myself to actually be this vector field and rolling around to this vector field and rolling around to this vector field and rolling around to try to to see how to change coordinates try to to see how to change coordinates try to to see how to change coordinates in such a way that somehow things in all in such a way that somehow things in all in such a way that somehow things in all directions would behave in a reasonably directions would behave in a reasonably directions would behave in a reasonably linear fashion. And yeah, my aunt walked linear fashion. And yeah, my aunt walked linear fashion. And yeah, my aunt walked in on me while I was doing that and she in on me while I was doing that and she in on me while I was doing that and she was asking what do I what am I doing was asking what do I what am I doing was asking what do I what am I doing doing this? It's complicated is the doing this? It's complicated is the doing this? It's complicated is the answer. Yeah. Yeah. And you know, okay, answer. Yeah. Yeah. And you know, okay, answer. Yeah. Yeah. And you know, okay, fine. You know, you're a young man. I fine. You know, you're a young man. I fine. You know, you're a young man. I don't ask questions. I I I have to ask don't ask questions. I I I have to ask don't ask questions. I I I have to ask about the you know um how do you about the you know um how do you about the you know um how do you approach solving difficult problems? approach solving difficult problems? approach solving difficult problems? What if it's possible What if it's possible What if it's possible to go inside your mind when you're to go inside your mind when you're to go inside your mind when you're thinking? Are you visualizing thinking? Are you visualizing thinking? Are you visualizing in your mind the mathematical objects in your mind the mathematical objects in your mind the mathematical objects symbols maybe what are you visualizing symbols maybe what are you visualizing symbols maybe what are you visualizing in your mind usually when you're in your mind usually when you're in your mind usually when you're thinking um a lot of pen and paper one thinking um a lot of pen and paper one thinking um a lot of pen and paper one thing you pick up as a mathematician is thing you pick up as a mathematician is thing you pick up as a mathematician is sort of uh I call it cheating sort of uh I call it cheating sort of uh I call it cheating strategically um so u the the beauty of strategically um so u the the beauty of strategically um so u the the beauty of mathematics is that is that you get to mathematics is that is that you get to mathematics is that is that you get to change the rule change the problem change the rule change the problem change the rule change the problem change the rules as you wish this you change the rules as you wish this you change the rules as you wish this you don't get to do this for any other field

don't get to do this for any other field don't get to do this for any other field like you know if if you're an engineer like you know if if you're an engineer like you know if if you're an engineer and someone says build a bridge over and someone says build a bridge over and someone says build a bridge over this this You can't say I want to build this this You can't say I want to build this this You can't say I want to build this up bridge over here instead or I this up bridge over here instead or I this up bridge over here instead or I want to build out of paper in instead of want to build out of paper in instead of want to build out of paper in instead of steel. Um but a mathematician you can steel. Um but a mathematician you can steel. Um but a mathematician you can you can do whatever you want. Um you can do whatever you want. Um you can do whatever you want. Um it's it's like trying to solve a it's it's like trying to solve a it's it's like trying to solve a computer game where you can there's computer game where you can there's computer game where you can there's unlimited cheat codes available. Uh and unlimited cheat codes available. Uh and unlimited cheat codes available. Uh and so you know you you can you can set so you know you you can you can set so you know you you can you can set this. So there's a dimension that's too this. So there's a dimension that's too this. So there's a dimension that's too large. I'll set it to one. I'd solve the large. I'll set it to one. I'd solve the large. I'll set it to one. I'd solve the one dimension problem first. So there's one dimension problem first. So there's one dimension problem first. So there's a main term and an error term. I'm going a main term and an error term. I'm going a main term and an error term. I'm going to make a spherical car assumption. I'll to make a spherical car assumption. I'll to make a spherical car assumption. I'll assume the error term is zero. And so assume the error term is zero. And so assume the error term is zero. And so the way you should solve these problems the way you should solve these problems the way you should solve these problems is is not in sort of this iron man mode is is not in sort of this iron man mode is is not in sort of this iron man mode where you make things maximally where you make things maximally where you make things maximally difficult. Um but actually the way you difficult. Um but actually the way you difficult. Um but actually the way you should you should approach any should you should approach any should you should approach any reasonable math problem is that you if reasonable math problem is that you if reasonable math problem is that you if if there are 10 things that are making if there are 10 things that are making if there are 10 things that are making your life difficult. Find a version of your life difficult. Find a version of your life difficult. Find a version of the problem that turns off nine of the the problem that turns off nine of the the problem that turns off nine of the difficulties but only keeps one of them. difficulties but only keeps one of them. difficulties but only keeps one of them. Um and so that um and then that just so Um and so that um and then that just so Um and so that um and then that just so you you you install nine cheats. Okay. you you you install nine cheats. Okay. you you you install nine cheats. Okay. You install 10 cheats then then the game You install 10 cheats then then the game You install 10 cheats then then the game is trivial. You saw nine cheats, you is trivial. You saw nine cheats, you is trivial. You saw nine cheats, you solve one problem that that that teaches solve one problem that that that teaches solve one problem that that that teaches you how how to deal with that particular you how how to deal with that particular you how how to deal with that particular difficulty and then you turn that one difficulty and then you turn that one difficulty and then you turn that one off and you turn someone else something off and you turn someone else something off and you turn someone else something else else on and then you solve that one else else on and then you solve that one else else on and then you solve that one and after you you know how to solve the and after you you know how to solve the and after you you know how to solve the 10 problems 10 difficulties separately 10 problems 10 difficulties separately 10 problems 10 difficulties separately then you have to start merging them a then you have to start merging them a then you have to start merging them a few at a time. Um I I as a kid I watched few at a time. Um I I as a kid I watched few at a time. Um I I as a kid I watched a lot of these Hong Kong action movies.

a lot of these Hong Kong action movies. a lot of these Hong Kong action movies. Um it's from a culture. Um and uh one Um it's from a culture. Um and uh one Um it's from a culture. Um and uh one thing is that every time there was a thing is that every time there was a thing is that every time there was a fight scene, you know, so maybe the the fight scene, you know, so maybe the the fight scene, you know, so maybe the the hero will get swarmed by a hundred bad hero will get swarmed by a hundred bad hero will get swarmed by a hundred bad guy goons or whatever. But it would guy goons or whatever. But it would guy goons or whatever. But it would always be choreographed so that he'd always be choreographed so that he'd always be choreographed so that he'd always be only fighting one person at a always be only fighting one person at a always be only fighting one person at a time and then he would defeat that time and then he would defeat that time and then he would defeat that person and move on and and because of person and move on and and because of person and move on and and because of that he could he could defeat all of that he could he could defeat all of that he could he could defeat all of them, right? But whereas if they had them, right? But whereas if they had them, right? But whereas if they had fought a bit more intelligently and just fought a bit more intelligently and just fought a bit more intelligently and just swarmed the guy at once, uh it would swarmed the guy at once, uh it would swarmed the guy at once, uh it would make for much much worse um cinema, but make for much much worse um cinema, but make for much much worse um cinema, but uh but they would win. Are you usually uh but they would win. Are you usually uh but they would win. Are you usually uh pen and paper? Are you working uh uh pen and paper? Are you working uh uh pen and paper? Are you working uh with computer and latte? I'm mostly pen with computer and latte? I'm mostly pen with computer and latte? I'm mostly pen and paper actually. So in my office, I and paper actually. So in my office, I and paper actually. So in my office, I have four giant blackboards. Um and have four giant blackboards. Um and have four giant blackboards. Um and sometimes I just have to write sometimes I just have to write sometimes I just have to write everything I know about the problem on everything I know about the problem on everything I know about the problem on the four blackboards and then sit my the four blackboards and then sit my the four blackboards and then sit my couch and just sort of see the whole couch and just sort of see the whole couch and just sort of see the whole thing. Is it all symbols like notation thing. Is it all symbols like notation thing. Is it all symbols like notation or is there some drawings? Oh, there's a or is there some drawings? Oh, there's a or is there some drawings? Oh, there's a lot of drawing and a lot of bespoke lot of drawing and a lot of bespoke lot of drawing and a lot of bespoke doodles that that only make sense to me. doodles that that only make sense to me. doodles that that only make sense to me. Um I mean and and the beauty of Um I mean and and the beauty of Um I mean and and the beauty of blackboard is you erase and it's it's blackboard is you erase and it's it's blackboard is you erase and it's it's very organic thing. Um I'm beginning to very organic thing. Um I'm beginning to very organic thing. Um I'm beginning to use more and more computers. Um partly use more and more computers. Um partly use more and more computers. Um partly because AI makes it much easier to do because AI makes it much easier to do because AI makes it much easier to do simple coding things that you know if I simple coding things that you know if I simple coding things that you know if I wanted to plot a function before which wanted to plot a function before which wanted to plot a function before which is moderately complicated as some is moderately complicated as some is moderately complicated as some iteration or something you know I'd have iteration or something you know I'd have iteration or something you know I'd have to to remember how to set up a Python to to remember how to set up a Python to to remember how to set up a Python program and and and and and how does a program and and and and and how does a program and and and and and how does a for loop work and and and debug it and for loop work and and and debug it and for loop work and and and debug it and it would take two hours and so forth and it would take two hours and so forth and it would take two hours and so forth and and now I can do it in 10 15 minutes is and now I can do it in 10 15 minutes is and now I can do it in 10 15 minutes is much um yeah I'm using more and more uh much um yeah I'm using more and more uh much um yeah I'm using more and more uh computers to do simple explorations.

computers to do simple explorations. computers to do simple explorations. Let's talk about AI a little bit if we Let's talk about AI a little bit if we Let's talk about AI a little bit if we could. So um maybe a good entry point is could. So um maybe a good entry point is could. So um maybe a good entry point is just talking about computer assisted just talking about computer assisted just talking about computer assisted proofs in general. Can you describe the proofs in general. Can you describe the proofs in general. Can you describe the lean formal proof programming language lean formal proof programming language lean formal proof programming language and how it can help as a proof assistant and how it can help as a proof assistant and how it can help as a proof assistant and maybe how you started using it and and maybe how you started using it and and maybe how you started using it and how uh it has helped you. So um we is a how uh it has helped you. So um we is a how uh it has helped you. So um we is a computer language um much like sort of computer language um much like sort of computer language um much like sort of standard languages like Python and C and standard languages like Python and C and standard languages like Python and C and so forth except that in most languages so forth except that in most languages so forth except that in most languages the focus is on producing executable the focus is on producing executable the focus is on producing executable code. Lines of code do things you know code. Lines of code do things you know code. Lines of code do things you know they they flip bits or or they make a they they flip bits or or they make a they they flip bits or or they make a robot move or or they they deliver you robot move or or they they deliver you robot move or or they they deliver you text on the internet or something. Um so text on the internet or something. Um so text on the internet or something. Um so lean is a language that can also do lean is a language that can also do lean is a language that can also do that. Uh it can also be run as a that. Uh it can also be run as a that. Uh it can also be run as a standard traditional language but it can standard traditional language but it can standard traditional language but it can also produce certificates. So a software also produce certificates. So a software also produce certificates. So a software like like Python might do a computation like like Python might do a computation like like Python might do a computation and give you that the answer is seven. and give you that the answer is seven. and give you that the answer is seven. Okay, that does a sum of 3+ 4 is equal Okay, that does a sum of 3+ 4 is equal Okay, that does a sum of 3+ 4 is equal to 7 but uh lean can produce not just to 7 but uh lean can produce not just to 7 but uh lean can produce not just the answer but but a proof that how it the answer but but a proof that how it the answer but but a proof that how it got the the answer of seven as 3+ 4 and got the the answer of seven as 3+ 4 and got the the answer of seven as 3+ 4 and all the steps involved in in so it all the steps involved in in so it all the steps involved in in so it creates these more complicated objects creates these more complicated objects creates these more complicated objects not just statements but statements with not just statements but statements with not just statements but statements with proofs attached to them. um and um every proofs attached to them. um and um every proofs attached to them. um and um every line of code is just a way of p piecing line of code is just a way of p piecing line of code is just a way of p piecing together previous statements to to together previous statements to to together previous statements to to create new ones. So the idea is not new.

create new ones. So the idea is not new. create new ones. So the idea is not new. These things are are called proof These things are are called proof These things are are called proof assistants and so they provide languages assistants and so they provide languages assistants and so they provide languages for which you you can create quite for which you you can create quite for which you you can create quite complicated um intricate mathematical complicated um intricate mathematical complicated um intricate mathematical proofs and um they produce these proofs and um they produce these proofs and um they produce these certificates that that give a 100% um certificates that that give a 100% um certificates that that give a 100% um guarantee that your arguments are guarantee that your arguments are guarantee that your arguments are correct if you trust the compiler of but correct if you trust the compiler of but correct if you trust the compiler of but they made the compiler really small and they made the compiler really small and they made the compiler really small and you can there are several different you can there are several different you can there are several different compilers available for the same for um compilers available for the same for um compilers available for the same for um can you give people some intuition about can you give people some intuition about can you give people some intuition about the the difference between writing on the the difference between writing on the the difference between writing on pen and paper versus using lean pen and paper versus using lean pen and paper versus using lean programming language How hard is it to programming language How hard is it to programming language How hard is it to formalize formalize formalize statement? So lean a lot of statement? So lean a lot of statement? So lean a lot of mathematicians were involved in the mathematicians were involved in the mathematicians were involved in the design of lean. So it's it's designed so design of lean. So it's it's designed so design of lean. So it's it's designed so that individual lines of code resemble that individual lines of code resemble that individual lines of code resemble individual lines of mathematical individual lines of mathematical individual lines of mathematical argument like you might want to argument like you might want to argument like you might want to introduce a variable. You want want to introduce a variable. You want want to introduce a variable. You want want to prove a contradiction. You you um there prove a contradiction. You you um there prove a contradiction. You you um there are various standard things that you can are various standard things that you can are various standard things that you can do and and it's it's written so ideally do and and it's it's written so ideally do and and it's it's written so ideally it should like a one correspondence. In it should like a one correspondence. In it should like a one correspondence. In practice, it isn't because lean is like practice, it isn't because lean is like practice, it isn't because lean is like explaining a proof to an extremely explaining a proof to an extremely explaining a proof to an extremely pedantic colleague who will will point pedantic colleague who will will point pedantic colleague who will will point out okay did you really mean this like out okay did you really mean this like out okay did you really mean this like what what happens if this is zero? Okay.

what what happens if this is zero? Okay. what what happens if this is zero? Okay. Um did you how do you justify this? Um Um did you how do you justify this? Um Um did you how do you justify this? Um so lean has a lot of automation in it um so lean has a lot of automation in it um so lean has a lot of automation in it um to try to to uh to be less annoying. Um to try to to uh to be less annoying. Um to try to to uh to be less annoying. Um so for example um every mathematical so for example um every mathematical so for example um every mathematical object has to come with a type like if I object has to come with a type like if I object has to come with a type like if I if I talk about X is X a real number or if I talk about X is X a real number or if I talk about X is X a real number or um a natural number or or a function or um a natural number or or a function or um a natural number or or a function or something um if you write things something um if you write things something um if you write things informally um it's up in terms of informally um it's up in terms of informally um it's up in terms of context you say you know um clearly x is context you say you know um clearly x is context you say you know um clearly x is equal to let x be the sum of y and z and equal to let x be the sum of y and z and equal to let x be the sum of y and z and y and z were already real numbers so x y and z were already real numbers so x y and z were already real numbers so x should also be a real number um so lean should also be a real number um so lean should also be a real number um so lean can do a lot of that um but every so can do a lot of that um but every so can do a lot of that um but every so often it it says wait a minute can you often it it says wait a minute can you often it it says wait a minute can you tell me more about what this object is tell me more about what this object is tell me more about what this object is uh what type of object it is. You see, uh what type of object it is. You see, uh what type of object it is. You see, you have to think more um at a you have to think more um at a you have to think more um at a philosophical level. Well, not just sort philosophical level. Well, not just sort philosophical level. Well, not just sort of computations you're doing, but sort of computations you're doing, but sort of computations you're doing, but sort of what each object actually um is in of what each object actually um is in of what each object actually um is in some sense. Is he using something like some sense. Is he using something like some sense. Is he using something like LLMs to do uh the type inference or like LLMs to do uh the type inference or like LLMs to do uh the type inference or like you mention the real number? It's it's you mention the real number? It's it's you mention the real number? It's it's using much more traditional what's using much more traditional what's using much more traditional what's called good old fashioned AI. Yeah, you called good old fashioned AI. Yeah, you called good old fashioned AI. Yeah, you can represent all these things as trees can represent all these things as trees can represent all these things as trees and there's always algorithm to match and there's always algorithm to match and there's always algorithm to match one tree to another tree. So it's one tree to another tree. So it's one tree to another tree. So it's actually doable to figure out if actually doable to figure out if actually doable to figure out if something is a a real number or a something is a a real number or a something is a a real number or a natural number. Yeah. Every object sort natural number. Yeah. Every object sort natural number. Yeah. Every object sort of comes with a history of where it came of comes with a history of where it came of comes with a history of where it came from and you can you can kind of trace.

from and you can you can kind of trace. from and you can you can kind of trace. Oh, I see. Um yeah, so it's it's Oh, I see. Um yeah, so it's it's Oh, I see. Um yeah, so it's it's designed for reliability. So uh modern designed for reliability. So uh modern designed for reliability. So uh modern AIs are not used in it's a disjoint AIs are not used in it's a disjoint AIs are not used in it's a disjoint technology. People are beginning to use technology. People are beginning to use technology. People are beginning to use AIS on top of lean. So when a AIS on top of lean. So when a AIS on top of lean. So when a mathematician tries to program um a mathematician tries to program um a mathematician tries to program um a proof in lean um often there's a step proof in lean um often there's a step proof in lean um often there's a step okay now I want to use um the okay now I want to use um the okay now I want to use um the fundamental thing of calculus say okay fundamental thing of calculus say okay fundamental thing of calculus say okay to do the next step so the lean to do the next step so the lean to do the next step so the lean developers have built this this massive developers have built this this massive developers have built this this massive project called methal liib a collection project called methal liib a collection project called methal liib a collection of tens of thousands of useful facts of tens of thousands of useful facts of tens of thousands of useful facts about mathematical objects and somewhere about mathematical objects and somewhere about mathematical objects and somewhere in there is the fundamental theme of in there is the fundamental theme of in there is the fundamental theme of calculus but you need to find it so a calculus but you need to find it so a calculus but you need to find it so a lot the bottleneck now is actually lema lot the bottleneck now is actually lema lot the bottleneck now is actually lema search you know there's a tool that that search you know there's a tool that that search you know there's a tool that that you know is in there somewhere and you you know is in there somewhere and you you know is in there somewhere and you need to find it um and so you can there need to find it um and so you can there need to find it um and so you can there are various search engines specialized are various search engines specialized are various search engines specialized for math loop that you can do um but for math loop that you can do um but for math loop that you can do um but there's now these large language models there's now these large language models there's now these large language models that you can say um I need the that you can say um I need the that you can say um I need the fundamental calculus at this point and fundamental calculus at this point and fundamental calculus at this point and it say okay uh um uh for example um when it say okay uh um uh for example um when it say okay uh um uh for example um when I code I have GitHub copilot installed I code I have GitHub copilot installed I code I have GitHub copilot installed as a plugin to my IDE and it scans my as a plugin to my IDE and it scans my as a plugin to my IDE and it scans my text and it sees what I need says you text and it sees what I need says you text and it sees what I need says you know I might even type here okay now I know I might even type here okay now I know I might even type here okay now I need to use the final thing with need to use the final thing with need to use the final thing with calculus okay and then it might suggest calculus okay and then it might suggest calculus okay and then it might suggest okay try this and like maybe 25% of the okay try this and like maybe 25% of the okay try this and like maybe 25% of the time it works exactly and then another time it works exactly and then another time it works exactly and then another 10 15% of the time it doesn't quite work 10 15% of the time it doesn't quite work 10 15% of the time it doesn't quite work but it it's close enough that I can say but it it's close enough that I can say but it it's close enough that I can say oh if I just change it here and here it oh if I just change it here and here it oh if I just change it here and here it it will work and then like half the time it will work and then like half the time it will work and then like half the time it gives me complete rubbish um so but it gives me complete rubbish um so but it gives me complete rubbish um so but people are beginning to use AI a little people are beginning to use AI a little people are beginning to use AI a little bit on top um mostly on the level of bit on top um mostly on the level of bit on top um mostly on the level of basically fancy autocomplete um but uh basically fancy autocomplete um but uh basically fancy autocomplete um but uh you can type half of one line of a proof you can type half of one line of a proof you can type half of one line of a proof and it will find it will tell you yeah

and it will find it will tell you yeah and it will find it will tell you yeah but a fancy especially fancy with the but a fancy especially fancy with the but a fancy especially fancy with the sort of capital letter F is uh uh sort of capital letter F is uh uh sort of capital letter F is uh uh removes some of the friction removes some of the friction removes some of the friction mathematician might feel when they move mathematician might feel when they move mathematician might feel when they move from pen and paper to formalizing. Yes. from pen and paper to formalizing. Yes. from pen and paper to formalizing. Yes. Yeah. So, right now I estimate that the Yeah. So, right now I estimate that the Yeah. So, right now I estimate that the effort time and effort taken to effort time and effort taken to effort time and effort taken to formalize a proof is about 10 times the formalize a proof is about 10 times the formalize a proof is about 10 times the amount taken to to write it out. Yeah. amount taken to to write it out. Yeah. amount taken to to write it out. Yeah. So, it's doable, but uh you don't it's So, it's doable, but uh you don't it's So, it's doable, but uh you don't it's it's annoying. But doesn't it like kill it's annoying. But doesn't it like kill it's annoying. But doesn't it like kill the whole vibe of being a mathematician? the whole vibe of being a mathematician? the whole vibe of being a mathematician? Yeah. So, I mean having a pedantic Yeah. So, I mean having a pedantic Yeah. So, I mean having a pedantic coworker, right? Yeah. If if that was coworker, right? Yeah. If if that was coworker, right? Yeah. If if that was the only aspect of it. Okay. But um the only aspect of it. Okay. But um the only aspect of it. Okay. But um Okay. there there are some there's some Okay. there there are some there's some Okay. there there are some there's some case it was actually more pleasant to do case it was actually more pleasant to do case it was actually more pleasant to do things formally. So there was there was things formally. So there was there was things formally. So there was there was a theorem I formalized and there was a a theorem I formalized and there was a a theorem I formalized and there was a certain constant 12 um that that came certain constant 12 um that that came certain constant 12 um that that came out at um in the final statement and so out at um in the final statement and so out at um in the final statement and so this 12 had to be carried all through this 12 had to be carried all through this 12 had to be carried all through the proof um and like everything had to the proof um and like everything had to the proof um and like everything had to be checked that it goes all the all be checked that it goes all the all be checked that it goes all the all these other numbers had to be consistent these other numbers had to be consistent these other numbers had to be consistent with this final number 12 and so we with this final number 12 and so we with this final number 12 and so we wrote a paper through this theorem with wrote a paper through this theorem with wrote a paper through this theorem with this number 12 and then a few weeks this number 12 and then a few weeks this number 12 and then a few weeks later someone said oh we can actually later someone said oh we can actually later someone said oh we can actually improve this 12 to an 11 by reworking improve this 12 to an 11 by reworking improve this 12 to an 11 by reworking some of these steps and when this some of these steps and when this some of these steps and when this happens with pen and paper um like every happens with pen and paper um like every happens with pen and paper um like every time you change a parameter you have to time you change a parameter you have to time you change a parameter you have to check line by line that every single check line by line that every single check line by line that every single line of your proof still works and there line of your proof still works and there line of your proof still works and there can be subtle things that you didn't can be subtle things that you didn't can be subtle things that you didn't quite realize. Some properties on the quite realize. Some properties on the quite realize. Some properties on the number 12 that you didn't even realize number 12 that you didn't even realize number 12 that you didn't even realize that you were taking advantage of. So a that you were taking advantage of. So a that you were taking advantage of. So a proof can break down at a subtle place.

proof can break down at a subtle place. proof can break down at a subtle place. Um so we had formalized the proof with Um so we had formalized the proof with Um so we had formalized the proof with this constant 12 and then when this this this constant 12 and then when this this this constant 12 and then when this this new paper came out uh we said okay let's new paper came out uh we said okay let's new paper came out uh we said okay let's so that took like 3 weeks to formalize so that took like 3 weeks to formalize so that took like 3 weeks to formalize and and like 20 people to formalize this and and like 20 people to formalize this and and like 20 people to formalize this this this original proof. I said oh but this this original proof. I said oh but this this original proof. I said oh but now now let's let's um uh uh let's now now let's let's um uh uh let's now now let's let's um uh uh let's update the 12 to 11. And what you can do update the 12 to 11. And what you can do update the 12 to 11. And what you can do with lean is that you just in your with lean is that you just in your with lean is that you just in your headline theorem you you change a 12 to headline theorem you you change a 12 to headline theorem you you change a 12 to 11. You run the compiler and like of the 11. You run the compiler and like of the 11. You run the compiler and like of the thousands of lines of code you have 90% thousands of lines of code you have 90% thousands of lines of code you have 90% of them still work and there's a couple of them still work and there's a couple of them still work and there's a couple that are lined in red. Now I can't that are lined in red. Now I can't that are lined in red. Now I can't justify this these steps but it it justify this these steps but it it justify this these steps but it it immediately isolates which steps you immediately isolates which steps you immediately isolates which steps you need to change but you can skip over need to change but you can skip over need to change but you can skip over everything which which works just fine. everything which which works just fine. everything which which works just fine. Um, and if you program things correctly, Um, and if you program things correctly, Um, and if you program things correctly, um, with sort of good programming um, with sort of good programming um, with sort of good programming practices, most of your lines will not practices, most of your lines will not practices, most of your lines will not be read. Um, and there'll just be a few be read. Um, and there'll just be a few be read. Um, and there'll just be a few places where you, I mean, if if you places where you, I mean, if if you places where you, I mean, if if you don't hard code your constants, but you don't hard code your constants, but you don't hard code your constants, but you sort of, uh, um, um, you use smart sort of, uh, um, um, you use smart sort of, uh, um, um, you use smart tactics and so forth. Yeah, you can tactics and so forth. Yeah, you can tactics and so forth. Yeah, you can localize um, the things you need to localize um, the things you need to localize um, the things you need to change to to a very small um, period of change to to a very small um, period of change to to a very small um, period of time. So like within a day or two, we time. So like within a day or two, we time. So like within a day or two, we had updated our proof to this is very had updated our proof to this is very had updated our proof to this is very quick process. You um, you make a quick process. You um, you make a quick process. You um, you make a change, there are 10 things now that change, there are 10 things now that change, there are 10 things now that don't work. for each one you make a don't work. for each one you make a don't work. for each one you make a change and now there's five more things change and now there's five more things change and now there's five more things that don't work but but the process that don't work but but the process that don't work but but the process converges much more smoothly than with converges much more smoothly than with converges much more smoothly than with pen and paper. So that's for writing are pen and paper. So that's for writing are pen and paper. So that's for writing are you able to read it like if somebody you able to read it like if somebody you able to read it like if somebody else sends a proof are you able to like else sends a proof are you able to like else sends a proof are you able to like how what's what's the uh versus paper how what's what's the uh versus paper how what's what's the uh versus paper and yeah so the proofs are longer but and yeah so the proofs are longer but and yeah so the proofs are longer but each individual piece is easier to read.

each individual piece is easier to read. each individual piece is easier to read. So, um, if you take a math paper and you So, um, if you take a math paper and you So, um, if you take a math paper and you jump to page 27 and you look at jump to page 27 and you look at jump to page 27 and you look at paragraph 6 and you have a line of of of paragraph 6 and you have a line of of of paragraph 6 and you have a line of of of text of math, I often can't read it text of math, I often can't read it text of math, I often can't read it immediately because it assumes various immediately because it assumes various immediately because it assumes various definitions which I have to to go back definitions which I have to to go back definitions which I have to to go back and and maybe 10 pages earlier this was and and maybe 10 pages earlier this was and and maybe 10 pages earlier this was defined and this um the proof is defined and this um the proof is defined and this um the proof is scattered all over the place and you scattered all over the place and you scattered all over the place and you basically are forced to read fairly basically are forced to read fairly basically are forced to read fairly sequentially. Um, it's it's not like say sequentially. Um, it's it's not like say sequentially. Um, it's it's not like say a novel where like you know in theory a novel where like you know in theory a novel where like you know in theory you could you open up a novel halfway you could you open up a novel halfway you could you open up a novel halfway through and start reading. there's a lot through and start reading. there's a lot through and start reading. there's a lot of context. But when a proven lean, if of context. But when a proven lean, if of context. But when a proven lean, if you put your cursor on a line of code, you put your cursor on a line of code, you put your cursor on a line of code, every single object there, you can hover every single object there, you can hover every single object there, you can hover over it and it would it would say what over it and it would it would say what over it and it would it would say what it is, where it came from, where stuff it is, where it came from, where stuff it is, where it came from, where stuff is justified. You can trace things back is justified. You can trace things back is justified. You can trace things back much easier than sort of flipping much easier than sort of flipping much easier than sort of flipping through a math paper. So, one thing that through a math paper. So, one thing that through a math paper. So, one thing that lean really enables is actually lean really enables is actually lean really enables is actually collaborating on proofs at a really collaborating on proofs at a really collaborating on proofs at a really atomic scale that you really couldn't do atomic scale that you really couldn't do atomic scale that you really couldn't do in the past. So traditionally with pen in the past. So traditionally with pen in the past. So traditionally with pen and paper um when you want to and paper um when you want to and paper um when you want to collaborate with another mathematician collaborate with another mathematician collaborate with another mathematician um either you do it as a blackboard um either you do it as a blackboard um either you do it as a blackboard where you um you can really interact but where you um you can really interact but where you um you can really interact but if you're doing it sort of by email or if you're doing it sort of by email or if you're doing it sort of by email or something um basically yeah you have to something um basically yeah you have to something um basically yeah you have to segment it say I'm going to I'm going to segment it say I'm going to I'm going to segment it say I'm going to I'm going to finish section three you do section four finish section three you do section four finish section three you do section four but uh you can't really sort of work on but uh you can't really sort of work on but uh you can't really sort of work on the same thing collaboratively at the the same thing collaboratively at the the same thing collaboratively at the same time but with lean you can be same time but with lean you can be same time but with lean you can be trying to formalize some portion of the trying to formalize some portion of the trying to formalize some portion of the proof and say I got stuck at line 67 proof and say I got stuck at line 67 proof and say I got stuck at line 67 here I need to prove this thing but it here I need to prove this thing but it here I need to prove this thing but it it doesn't quite work here is like the it doesn't quite work here is like the it doesn't quite work here is like the three lines of code I'm having trouble three lines of code I'm having trouble three lines of code I'm having trouble with. Um, but because all the context is with. Um, but because all the context is with. Um, but because all the context is there, someone else can say, "Oh, okay.

there, someone else can say, "Oh, okay. there, someone else can say, "Oh, okay. I recognize what you need to do. You I recognize what you need to do. You I recognize what you need to do. You need to to apply this trick or this tool need to to apply this trick or this tool need to to apply this trick or this tool and you can do extremely atomic level and you can do extremely atomic level and you can do extremely atomic level conversations. So, because of lean, I conversations. So, because of lean, I conversations. So, because of lean, I can collaborate, you know, with dozens can collaborate, you know, with dozens can collaborate, you know, with dozens of people across the world, most of whom of people across the world, most of whom of people across the world, most of whom I don't have never met in person. Um, I don't have never met in person. Um, I don't have never met in person. Um, and I may not know actually even whether and I may not know actually even whether and I may not know actually even whether they're um how reliable they are in in they're um how reliable they are in in they're um how reliable they are in in in their um um in in the process, but in their um um in in the process, but in their um um in in the process, but lean gives me a certificate of of of lean gives me a certificate of of of lean gives me a certificate of of of trust. Um, so I can do I can do trust. Um, so I can do I can do trust. Um, so I can do I can do trustless mathematics. So there's so trustless mathematics. So there's so trustless mathematics. So there's so many interesting questions there's. So many interesting questions there's. So many interesting questions there's. So one, you're you're known for being a one, you're you're known for being a one, you're you're known for being a great collaborator. So what is the right great collaborator. So what is the right great collaborator. So what is the right way to approach way to approach way to approach solving a difficult problem in solving a difficult problem in solving a difficult problem in mathematics? When you're collaborating, mathematics? When you're collaborating, mathematics? When you're collaborating, are you doing a divide and conquer type are you doing a divide and conquer type are you doing a divide and conquer type of thing or are you brains are you of thing or are you brains are you of thing or are you brains are you focusing on a particular part and you're focusing on a particular part and you're focusing on a particular part and you're brainstorming? There's always a brainstorming? There's always a brainstorming? There's always a brainstorming process first. Yeah. So brainstorming process first. Yeah. So brainstorming process first. Yeah. So math research projects sort of by their math research projects sort of by their math research projects sort of by their nature when you start you don't really nature when you start you don't really nature when you start you don't really know how to do the problem. Um it's not know how to do the problem. Um it's not know how to do the problem. Um it's not like an engineering project where like an engineering project where like an engineering project where somehow the theory has been established somehow the theory has been established somehow the theory has been established for decades and it's it's implementation for decades and it's it's implementation for decades and it's it's implementation is the main difficulty. You have to is the main difficulty. You have to is the main difficulty. You have to figure out even what is the right path.

figure out even what is the right path. figure out even what is the right path. So so this is what I said about about So so this is what I said about about So so this is what I said about about cheating first you know um it's like um cheating first you know um it's like um cheating first you know um it's like um to go back to the bridge building to go back to the bridge building to go back to the bridge building analogy you know so first assume you analogy you know so first assume you analogy you know so first assume you have infinite budget and and like have infinite budget and and like have infinite budget and and like unlimited amounts of of of workforce and unlimited amounts of of of workforce and unlimited amounts of of of workforce and so forth. Now can you can you build this so forth. Now can you can you build this so forth. Now can you can you build this bridge? Okay. Okay. now have infinite bridge? Okay. Okay. now have infinite bridge? Okay. Okay. now have infinite budget but only finite workforce right budget but only finite workforce right budget but only finite workforce right now can you do that and so forth um so now can you do that and so forth um so now can you do that and so forth um so uh I mean of course you know no engineer uh I mean of course you know no engineer uh I mean of course you know no engineer can actually do this like I say they can actually do this like I say they can actually do this like I say they have fixed requirements yes there's this have fixed requirements yes there's this have fixed requirements yes there's this sort of jam sessions always at the sort of jam sessions always at the sort of jam sessions always at the beginning where you try all kinds of beginning where you try all kinds of beginning where you try all kinds of crazy things and you you make all these crazy things and you you make all these crazy things and you you make all these assumptions that are unrealistic but you assumptions that are unrealistic but you assumptions that are unrealistic but you plan to fix later um and you try to see plan to fix later um and you try to see plan to fix later um and you try to see if there's even some skeleton of an if there's even some skeleton of an if there's even some skeleton of an approach that might work um and then approach that might work um and then approach that might work um and then hopefully that breaks up the problem hopefully that breaks up the problem hopefully that breaks up the problem into smaller sub problems which you into smaller sub problems which you into smaller sub problems which you don't know how to do but then you uh you don't know how to do but then you uh you don't know how to do but then you uh you focus on on sub ones and sometimes focus on on sub ones and sometimes focus on on sub ones and sometimes different collaborators are better at at different collaborators are better at at different collaborators are better at at working on on certain things. Um so one working on on certain things. Um so one working on on certain things. Um so one of my themes I'm known for is a theorem of my themes I'm known for is a theorem of my themes I'm known for is a theorem of Ben Green which called the green of Ben Green which called the green of Ben Green which called the green tower theorem. Um it's a statement that tower theorem. Um it's a statement that tower theorem. Um it's a statement that the primes contain arithmetic the primes contain arithmetic the primes contain arithmetic progressions of any length. So it was a progressions of any length. So it was a progressions of any length. So it was a modification of this theoret modification of this theoret modification of this theoret and the way we collaborated was that Ben and the way we collaborated was that Ben and the way we collaborated was that Ben had already proven a similar result for had already proven a similar result for had already proven a similar result for progressions of length three. Um he progressions of length three. Um he progressions of length three. Um he showed that sets like the primes contain showed that sets like the primes contain showed that sets like the primes contain lots and lots of progressions of length lots and lots of progressions of length lots and lots of progressions of length three. Um even and even um subsets of three. Um even and even um subsets of three. Um even and even um subsets of the prime certain subsets do um but his the prime certain subsets do um but his the prime certain subsets do um but his techniques only worked for um for length techniques only worked for um for length techniques only worked for um for length three progressions. They didn't work for three progressions. They didn't work for three progressions. They didn't work for longer progressions. Um but I had these longer progressions. Um but I had these longer progressions. Um but I had these techniques coming from agotic theory techniques coming from agotic theory techniques coming from agotic theory which is something that I had been which is something that I had been which is something that I had been playing with and and uh I knew better playing with and and uh I knew better playing with and and uh I knew better than Ben at the time. Um and so um if I

than Ben at the time. Um and so um if I than Ben at the time. Um and so um if I could justify certain randomness could justify certain randomness could justify certain randomness properties of some set relating to properties of some set relating to properties of some set relating to primes like there there's a certain primes like there there's a certain primes like there there's a certain technical condition which if I could technical condition which if I could technical condition which if I could have it if if Ben could supply me this have it if if Ben could supply me this have it if if Ben could supply me this fact I could I could conclude the fact I could I could conclude the fact I could I could conclude the theorem but I what I asked was a really theorem but I what I asked was a really theorem but I what I asked was a really difficult question in number theory difficult question in number theory difficult question in number theory which um he said there's no way we can which um he said there's no way we can which um he said there's no way we can prove this can so he said can you prove prove this can so he said can you prove prove this can so he said can you prove your part of the theorem using a weaker your part of the theorem using a weaker your part of the theorem using a weaker hypothesis that I have a chance to prove hypothesis that I have a chance to prove hypothesis that I have a chance to prove it and he proposed something which he it and he proposed something which he it and he proposed something which he could prove but it was too weak for me I could prove but it was too weak for me I could prove but it was too weak for me I can't use this. Um, so there's this can't use this. Um, so there's this can't use this. Um, so there's this there was this conversation going back there was this conversation going back there was this conversation going back and forth. Um, so different cheats to and forth. Um, so different cheats to and forth. Um, so different cheats to Yeah. Yeah. I want to cheat more, he Yeah. Yeah. I want to cheat more, he Yeah. Yeah. I want to cheat more, he wants to cheat less. But eventually we wants to cheat less. But eventually we wants to cheat less. But eventually we found a a a a a a a a a a a a a a a a a found a a a a a a a a a a a a a a a a a found a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a property which a he a a a a a a a a a a property which a he a a a a a a a a a a property which a he could prove and b I could use um and could prove and b I could use um and could prove and b I could use um and then we we could prove our view and um then we we could prove our view and um then we we could prove our view and um yeah so there's there's a there all yeah so there's there's a there all yeah so there's there's a there all kinds of dynamics you know I mean it's kinds of dynamics you know I mean it's kinds of dynamics you know I mean it's every every um collaboration has a has a every every um collaboration has a has a every every um collaboration has a has a has some story no two are the same. And has some story no two are the same. And has some story no two are the same. And then on on the flip side of that like then on on the flip side of that like then on on the flip side of that like you mentioned with lean programming now you mentioned with lean programming now you mentioned with lean programming now that's almost like a different story that's almost like a different story that's almost like a different story because you can do you can create I because you can do you can create I because you can do you can create I think you've mentioned a kind of a think you've mentioned a kind of a think you've mentioned a kind of a blueprint blueprint blueprint right for a problem and then you can right for a problem and then you can right for a problem and then you can really do a divide and conquer with lean really do a divide and conquer with lean really do a divide and conquer with lean where you're working on separate parts where you're working on separate parts where you're working on separate parts right and they're using the computer right and they're using the computer right and they're using the computer system proof checker essentially to make system proof checker essentially to make system proof checker essentially to make sure that everything is correct along sure that everything is correct along sure that everything is correct along the way. Yeah. So it makes everything the way. Yeah. So it makes everything the way. Yeah. So it makes everything compatible and uh yeah and trustable. Um compatible and uh yeah and trustable. Um compatible and uh yeah and trustable. Um yeah so currently only a few yeah so currently only a few yeah so currently only a few mathematical projects can be cut up in mathematical projects can be cut up in mathematical projects can be cut up in this way at the current state of the art

this way at the current state of the art this way at the current state of the art most of the lean activity is on most of the lean activity is on most of the lean activity is on formalizing boos that have already been formalizing boos that have already been formalizing boos that have already been proven by humans a math paper basically proven by humans a math paper basically proven by humans a math paper basically is a boop a blueprint in a sense it is is a boop a blueprint in a sense it is is a boop a blueprint in a sense it is taking a a difficult statement like big taking a a difficult statement like big taking a a difficult statement like big theorem and breaking up into 100 little theorem and breaking up into 100 little theorem and breaking up into 100 little lemas um but often not all written with lemas um but often not all written with lemas um but often not all written with enough detail that each one can be sort enough detail that each one can be sort enough detail that each one can be sort of directly formalized. A blueprint is of directly formalized. A blueprint is of directly formalized. A blueprint is like a really pedantically written like a really pedantically written like a really pedantically written version of a paper where every step is version of a paper where every step is version of a paper where every step is explained as to as much detail as as as explained as to as much detail as as as explained as to as much detail as as as possible and trying to make each step possible and trying to make each step possible and trying to make each step kind of self-contained um and or kind of self-contained um and or kind of self-contained um and or depending on only a very specific number depending on only a very specific number depending on only a very specific number of of previous statements that been of of previous statements that been of of previous statements that been proven so that each node of this proven so that each node of this proven so that each node of this blueprint graph that gets generated can blueprint graph that gets generated can blueprint graph that gets generated can be tackled independently of of the be tackled independently of of the be tackled independently of of the others and you don't even need to know others and you don't even need to know others and you don't even need to know how the whole thing works. Um so it's how the whole thing works. Um so it's how the whole thing works. Um so it's like a modern supply chain you know like like a modern supply chain you know like like a modern supply chain you know like if you want to create an iPhone or or if you want to create an iPhone or or if you want to create an iPhone or or some other complicated object um no one some other complicated object um no one some other complicated object um no one person can can build up um a single person can can build up um a single person can can build up um a single object but you can a specialist who who object but you can a specialist who who object but you can a specialist who who just if they're given some widgets from just if they're given some widgets from just if they're given some widgets from some other company they can combine them some other company they can combine them some other company they can combine them together to form a slightly bigger together to form a slightly bigger together to form a slightly bigger widget. I think that's a really exciting widget. I think that's a really exciting widget. I think that's a really exciting possibility because you can have if you possibility because you can have if you possibility because you can have if you can find problems that could be can find problems that could be can find problems that could be broken down this way then you can have broken down this way then you can have broken down this way then you can have you know thousands of contributors right you know thousands of contributors right you know thousands of contributors right distributed. So I told you before about distributed. So I told you before about distributed. So I told you before about the split between theoretical and the split between theoretical and the split between theoretical and experimental mathematics and right now experimental mathematics and right now experimental mathematics and right now most mathematics is theoretical and when most mathematics is theoretical and when most mathematics is theoretical and when you type it it's experimental. I think you type it it's experimental. I think you type it it's experimental. I think the platform that lean and and other the platform that lean and and other the platform that lean and and other software tools so um GitHub and things software tools so um GitHub and things software tools so um GitHub and things like that um allow they will allow like that um allow they will allow like that um allow they will allow experimental mathematics to be to scale experimental mathematics to be to scale experimental mathematics to be to scale up um to a much greater degree than we

up um to a much greater degree than we up um to a much greater degree than we can do now. So right now if you want to can do now. So right now if you want to can do now. So right now if you want to um um do any mathematical exploration of um um do any mathematical exploration of um um do any mathematical exploration of some mathematical pattern or something some mathematical pattern or something some mathematical pattern or something you need some code to write out the you need some code to write out the you need some code to write out the pattern and I mean sometimes there are pattern and I mean sometimes there are pattern and I mean sometimes there are some computer algebra packages that help some computer algebra packages that help some computer algebra packages that help but often it's just one mathematician but often it's just one mathematician but often it's just one mathematician coding lots and lots of Python or coding lots and lots of Python or coding lots and lots of Python or whatever and because coding is such an whatever and because coding is such an whatever and because coding is such an errorprone activity it's not practical errorprone activity it's not practical errorprone activity it's not practical to allow other people to collaborate to allow other people to collaborate to allow other people to collaborate with you on writing modules for your with you on writing modules for your with you on writing modules for your code because if one of the modules has a code because if one of the modules has a code because if one of the modules has a bug in it the whole thing is unreliable. bug in it the whole thing is unreliable. bug in it the whole thing is unreliable. Um, so it's these are uh so you get Um, so it's these are uh so you get Um, so it's these are uh so you get these bespoke uh spaghetti code that these bespoke uh spaghetti code that these bespoke uh spaghetti code that written by not not professional written by not not professional written by not not professional programmers but by mathematicians you programmers but by mathematicians you programmers but by mathematicians you know and they're clunky and and and slow know and they're clunky and and and slow know and they're clunky and and and slow and um and so because of that it's it's and um and so because of that it's it's and um and so because of that it's it's hard to to really massproduce hard to to really massproduce hard to to really massproduce experimental results um but um yeah but experimental results um but um yeah but experimental results um but um yeah but I think with lean I mean so I'm already I think with lean I mean so I'm already I think with lean I mean so I'm already starting some projects where we are not starting some projects where we are not starting some projects where we are not just experimenting with data but just experimenting with data but just experimenting with data but experimenting with proofs. So I have experimenting with proofs. So I have experimenting with proofs. So I have this project called the equation this project called the equation this project called the equation theories project. Basically we generated theories project. Basically we generated theories project. Basically we generated about 22 million little problems in about 22 million little problems in about 22 million little problems in abstract algebra. Maybe should back up abstract algebra. Maybe should back up abstract algebra. Maybe should back up and tell you what what the project is.

and tell you what what the project is. and tell you what what the project is. Okay. So abstract algebra studies Okay. So abstract algebra studies Okay. So abstract algebra studies operations like multiplication and operations like multiplication and operations like multiplication and addition and the abstract properties. addition and the abstract properties. addition and the abstract properties. Okay. So multiplication for example is Okay. So multiplication for example is Okay. So multiplication for example is commutive. X * Y is always Y * X at commutive. X * Y is always Y * X at commutive. X * Y is always Y * X at least for numbers. Um and it's also least for numbers. Um and it's also least for numbers. Um and it's also associative. X * Y * Z is the same as X associative. X * Y * Z is the same as X associative. X * Y * Z is the same as X * Y * Z. Um so um these operations obey * Y * Z. Um so um these operations obey * Y * Z. Um so um these operations obey some laws that don't obey others. For some laws that don't obey others. For some laws that don't obey others. For example, x * x is not always equal to x. example, x * x is not always equal to x. example, x * x is not always equal to x. So that law is not always true. So given So that law is not always true. So given So that law is not always true. So given any any operation, it obeys some laws any any operation, it obeys some laws any any operation, it obeys some laws and not others. Um, and so we generated and not others. Um, and so we generated and not others. Um, and so we generated about 4,000 of these possible laws of about 4,000 of these possible laws of about 4,000 of these possible laws of algebra that certain operations can algebra that certain operations can algebra that certain operations can satisfy. And our question is which laws satisfy. And our question is which laws satisfy. And our question is which laws imply which other ones? Um, so for imply which other ones? Um, so for imply which other ones? Um, so for example, does commutivity imply example, does commutivity imply example, does commutivity imply associativity? And the answer is no associativity? And the answer is no associativity? And the answer is no because it turns out you can describe an because it turns out you can describe an because it turns out you can describe an operation which obeys the commitive law operation which obeys the commitive law operation which obeys the commitive law but doesn't obey the associative law. So but doesn't obey the associative law. So but doesn't obey the associative law. So by producing an example you can you can by producing an example you can you can by producing an example you can you can show that commitivity does not imply show that commitivity does not imply show that commitivity does not imply associativity but some other laws do associativity but some other laws do associativity but some other laws do imply other laws by substitution and so imply other laws by substitution and so imply other laws by substitution and so forth and you can write down some some forth and you can write down some some forth and you can write down some some algebraic proof. So we look at all the algebraic proof. So we look at all the algebraic proof. So we look at all the pairs between these 4,000 laws and this pairs between these 4,000 laws and this pairs between these 4,000 laws and this 22 million of these pairs and for each 22 million of these pairs and for each 22 million of these pairs and for each pair we ask does this law imply this um pair we ask does this law imply this um pair we ask does this law imply this um law? If so give a give u give a proof.

law? If so give a give u give a proof. law? If so give a give u give a proof. If not give a counter example. Mhm. Um If not give a counter example. Mhm. Um If not give a counter example. Mhm. Um so 22 million problems each one of which so 22 million problems each one of which so 22 million problems each one of which you could give to like an undergraduate you could give to like an undergraduate you could give to like an undergraduate algebra student and they had a decent algebra student and they had a decent algebra student and they had a decent chance of solving the problem. Although chance of solving the problem. Although chance of solving the problem. Although there are a few of these 22 million there are a few of these 22 million there are a few of these 22 million there like 100 or so that are really there like 100 or so that are really there like 100 or so that are really quite hard. Okay. But a lot are easy and quite hard. Okay. But a lot are easy and quite hard. Okay. But a lot are easy and the project was just to to work out to the project was just to to work out to the project was just to to work out to determine the entire graph like like determine the entire graph like like determine the entire graph like like which ones imply which other ones. which ones imply which other ones. which ones imply which other ones. That's an incredible project by the way. That's an incredible project by the way. That's an incredible project by the way. Such a good idea. Such a good test of Such a good idea. Such a good test of Such a good idea. Such a good test of the very thing we've been talking about the very thing we've been talking about the very thing we've been talking about at a scale that's remarkable. Yeah. So at a scale that's remarkable. Yeah. So at a scale that's remarkable. Yeah. So it would not have been feasible. Yeah, I it would not have been feasible. Yeah, I it would not have been feasible. Yeah, I mean the state-of-the-art in the mean the state-of-the-art in the mean the state-of-the-art in the literature was like, you know, 15 literature was like, you know, 15 literature was like, you know, 15 equations and sort of how they apply. equations and sort of how they apply. equations and sort of how they apply. That's sort of at the limit of what a That's sort of at the limit of what a That's sort of at the limit of what a human repentant paper can do. So, so you human repentant paper can do. So, so you human repentant paper can do. So, so you need to scale it up. So, you need to need to scale it up. So, you need to need to scale it up. So, you need to crowdsource, but you also need to trust crowdsource, but you also need to trust crowdsource, but you also need to trust all the um I mean no one person can all the um I mean no one person can all the um I mean no one person can check 22 million of these proofs. You check 22 million of these proofs. You check 22 million of these proofs. You needed to be computerized and so it only needed to be computerized and so it only needed to be computerized and so it only became possible with with lean. Um we became possible with with lean. Um we became possible with with lean. Um we were hoping to use a lot of AI as well. were hoping to use a lot of AI as well. were hoping to use a lot of AI as well. Um so the project is almost complete. Um Um so the project is almost complete. Um Um so the project is almost complete. Um so of these 22 million all but two had so of these 22 million all but two had so of these 22 million all but two had been settled. Um wow and uh well been settled. Um wow and uh well been settled. Um wow and uh well actually and of those two we have a pen actually and of those two we have a pen actually and of those two we have a pen and paper proof of the two uh and we and paper proof of the two uh and we and paper proof of the two uh and we we're formalizing it. In fact I was this we're formalizing it. In fact I was this we're formalizing it. In fact I was this morning I was working on finishing it.

morning I was working on finishing it. morning I was working on finishing it. Um so we're almost done on this um Um so we're almost done on this um Um so we're almost done on this um incredible is yeah fantastic. How many incredible is yeah fantastic. How many incredible is yeah fantastic. How many people were able to get about 50 um people were able to get about 50 um people were able to get about 50 um which in mathematics is is considered a which in mathematics is is considered a which in mathematics is is considered a huge number. It's a huge number. That's huge number. It's a huge number. That's huge number. It's a huge number. That's crazy. Yeah. So we kind of have a paper crazy. Yeah. So we kind of have a paper crazy. Yeah. So we kind of have a paper with 50 authors uh and a big appendex of with 50 authors uh and a big appendex of with 50 authors uh and a big appendex of who contribute to what. Here's an who contribute to what. Here's an who contribute to what. Here's an interesting question. Now to maybe speak interesting question. Now to maybe speak interesting question. Now to maybe speak even more generally about it. When you even more generally about it. When you even more generally about it. When you have this pool of people, have this pool of people, have this pool of people, is there a way to uh organize the is there a way to uh organize the is there a way to uh organize the contributions by level of expertise of contributions by level of expertise of contributions by level of expertise of the people of the contributors? Now the people of the contributors? Now the people of the contributors? Now okay, uh I'm asking you a lot of pthead okay, uh I'm asking you a lot of pthead okay, uh I'm asking you a lot of pthead questions here, but I I'm imagining a questions here, but I I'm imagining a questions here, but I I'm imagining a bunch of humans and maybe in the future bunch of humans and maybe in the future bunch of humans and maybe in the future some AIS. Can there be like an ELO some AIS. Can there be like an ELO some AIS. Can there be like an ELO rating type of situation where rating type of situation where rating type of situation where like a gamification of this? The beauty like a gamification of this? The beauty like a gamification of this? The beauty of of these lean projects is is that of of these lean projects is is that of of these lean projects is is that automatically you get all this data, you automatically you get all this data, you automatically you get all this data, you know, so like like everything has to be know, so like like everything has to be know, so like like everything has to be uploaded for this GitHub and GitHub uploaded for this GitHub and GitHub uploaded for this GitHub and GitHub tracks who contributed what. Um so you tracks who contributed what. Um so you tracks who contributed what. Um so you could generate statistics from at any at could generate statistics from at any at could generate statistics from at any at any later point in time. You can say oh any later point in time. You can say oh any later point in time. You can say oh this person contributed this many this this person contributed this many this this person contributed this many this many lines of code or whatever. I mean many lines of code or whatever. I mean many lines of code or whatever. I mean these are very crude metrics. Um I would these are very crude metrics. Um I would these are very crude metrics. Um I would I would definitely not want this to I would definitely not want this to I would definitely not want this to become like you know part of your tenure become like you know part of your tenure become like you know part of your tenure review or something. Uh um but um I mean review or something. Uh um but um I mean review or something. Uh um but um I mean I think already in in in enterprise I think already in in in enterprise I think already in in in enterprise computing right people do use some of computing right people do use some of computing right people do use some of these metrics as part of of the these metrics as part of of the these metrics as part of of the assessment of of performance of a of an assessment of of performance of a of an assessment of of performance of a of an employee. Um again this is a direction employee. Um again this is a direction employee. Um again this is a direction which is a bit scary for academics to go which is a bit scary for academics to go which is a bit scary for academics to go down. We we don't like metrics so much down. We we don't like metrics so much down. We we don't like metrics so much and yet academics use metrics they just

and yet academics use metrics they just and yet academics use metrics they just use old ones. Number of papers. Yeah. use old ones. Number of papers. Yeah. use old ones. Number of papers. Yeah. Yeah. It's true. It's true that Yeah. I Yeah. It's true. It's true that Yeah. I Yeah. It's true. It's true that Yeah. I mean um it feels like this is a metric mean um it feels like this is a metric mean um it feels like this is a metric while flawed is is going in the more in while flawed is is going in the more in while flawed is is going in the more in the right direction. Right. Yeah. It's the right direction. Right. Yeah. It's the right direction. Right. Yeah. It's an interesting at least it's a very an interesting at least it's a very an interesting at least it's a very interesting metric. Yeah. I think it's interesting metric. Yeah. I think it's interesting metric. Yeah. I think it's interesting to study. I mean I I think interesting to study. I mean I I think interesting to study. I mean I I think you can you can do studies of of whether you can you can do studies of of whether you can you can do studies of of whether these are better predictors. Um there's these are better predictors. Um there's these are better predictors. Um there's this problem called good heart's law. If this problem called good heart's law. If this problem called good heart's law. If a statistic is actually used to a statistic is actually used to a statistic is actually used to incentivize performance, it becomes incentivize performance, it becomes incentivize performance, it becomes gained. Um and then it is no longer a gained. Um and then it is no longer a gained. Um and then it is no longer a useful measure. Oh, humans always. Yeah. useful measure. Oh, humans always. Yeah. useful measure. Oh, humans always. Yeah. Yeah. I know. It's rational. So what Yeah. I know. It's rational. So what Yeah. I know. It's rational. So what we've done for this project is is we've done for this project is is we've done for this project is is self-report. So um there are actually self-report. So um there are actually self-report. So um there are actually standard categories um from the sciences standard categories um from the sciences standard categories um from the sciences of what types of contributions people of what types of contributions people of what types of contributions people give. So there's there's concept and give. So there's there's concept and give. So there's there's concept and validation and resources and and and and validation and resources and and and and validation and resources and and and and coding and so forth. Um, so we we we coding and so forth. Um, so we we we coding and so forth. Um, so we we we there's a standard list of troll or so there's a standard list of troll or so there's a standard list of troll or so categories. Um, and we just ask each categories. Um, and we just ask each categories. Um, and we just ask each contributor to there's a big matrix of contributor to there's a big matrix of contributor to there's a big matrix of all the of all the authors in all the all the of all the authors in all the all the of all the authors in all the categories just to tick the boxes where categories just to tick the boxes where categories just to tick the boxes where they think that they contributed. Um, they think that they contributed. Um, they think that they contributed. Um, and just give a rough idea you know like and just give a rough idea you know like and just give a rough idea you know like oh so you did some coding and and uh and oh so you did some coding and and uh and oh so you did some coding and and uh and you provided some compute but you didn't you provided some compute but you didn't you provided some compute but you didn't do any of the pen and paper verification do any of the pen and paper verification do any of the pen and paper verification or whatever. And I think that that works or whatever. And I think that that works or whatever. And I think that that works out traditionally mathematicians just out traditionally mathematicians just out traditionally mathematicians just order alphabetically by surname. So we order alphabetically by surname. So we order alphabetically by surname. So we don't have this tradition as in the don't have this tradition as in the don't have this tradition as in the sciences of you know lead author and sciences of you know lead author and sciences of you know lead author and second author and so forth like which second author and so forth like which second author and so forth like which we're proud of you know we make all the we're proud of you know we make all the we're proud of you know we make all the authors equal status but it doesn't authors equal status but it doesn't authors equal status but it doesn't quite scale to this size so a decade ago quite scale to this size so a decade ago quite scale to this size so a decade ago I was involved in these things called I was involved in these things called I was involved in these things called polymath projects it was the crowd

polymath projects it was the crowd polymath projects it was the crowd sourcing mathematics but without the sourcing mathematics but without the sourcing mathematics but without the lean component so it was limited by you lean component so it was limited by you lean component so it was limited by you needed a human moderator to actually needed a human moderator to actually needed a human moderator to actually check that all the contributions coming check that all the contributions coming check that all the contributions coming in were actually valid and and this was in were actually valid and and this was in were actually valid and and this was a huge bottleneck actually um but still a huge bottleneck actually um but still a huge bottleneck actually um but still we had projects that were you know 10 we had projects that were you know 10 we had projects that were you know 10 author author author or so. But we had decided at the time um or so. But we had decided at the time um or so. But we had decided at the time um not to try to decide who did what um but not to try to decide who did what um but not to try to decide who did what um but to have a single pseudonym. So we to have a single pseudonym. So we to have a single pseudonym. So we created this fictional character called created this fictional character called created this fictional character called DHJ Polymath in the spirit of Bwaki. DHJ Polymath in the spirit of Bwaki. DHJ Polymath in the spirit of Bwaki. Baki is is the pseudonym for a famous Baki is is the pseudonym for a famous Baki is is the pseudonym for a famous group of mathematicians in the 20th group of mathematicians in the 20th group of mathematicians in the 20th century. But um and so the paper was a century. But um and so the paper was a century. But um and so the paper was a authored under the pseudonym. So none of authored under the pseudonym. So none of authored under the pseudonym. So none of us got the author credit. Um this us got the author credit. Um this us got the author credit. Um this actually turned out to be not so great actually turned out to be not so great actually turned out to be not so great for a couple of reasons. So, so one is for a couple of reasons. So, so one is for a couple of reasons. So, so one is that if you actually wanted to be that if you actually wanted to be that if you actually wanted to be considered for tenure or whatever, you considered for tenure or whatever, you considered for tenure or whatever, you could not use this paper in your uh uh could not use this paper in your uh uh could not use this paper in your uh uh as your submitted as one of your as your submitted as one of your as your submitted as one of your publications because it wasn't you publications because it wasn't you publications because it wasn't you didn't have the formal author credit. Um didn't have the formal author credit. Um didn't have the formal author credit. Um um but the other thing that we've um but the other thing that we've um but the other thing that we've recognized much later is that when recognized much later is that when recognized much later is that when people referred to these projects, they people referred to these projects, they people referred to these projects, they naturally refer to the most famous naturally refer to the most famous naturally refer to the most famous person who was involved in the project.

person who was involved in the project. person who was involved in the project. Oh, so this was Tim Gow's P project. Oh, so this was Tim Gow's P project. Oh, so this was Tim Gow's P project. This was ter project and not mention the This was ter project and not mention the This was ter project and not mention the the other 19 or whatever people that the other 19 or whatever people that the other 19 or whatever people that were involved. Yeah. So we're trying were involved. Yeah. So we're trying were involved. Yeah. So we're trying something different this time around something different this time around something different this time around where we have everyone's an author. Um where we have everyone's an author. Um where we have everyone's an author. Um but we will have an an appendix with but we will have an an appendix with but we will have an an appendix with this matrix and we'll see how that this matrix and we'll see how that this matrix and we'll see how that works. I mean uh so both projects are works. I mean uh so both projects are works. I mean uh so both projects are incredible just the fact that you're incredible just the fact that you're incredible just the fact that you're involved in such huge collaborations. involved in such huge collaborations. involved in such huge collaborations. But I think I saw a talk from Kevin But I think I saw a talk from Kevin But I think I saw a talk from Kevin Buzzard about uh the lean programming Buzzard about uh the lean programming Buzzard about uh the lean programming language just a few years ago and he was language just a few years ago and he was language just a few years ago and he was saying that uh this might be the future saying that uh this might be the future saying that uh this might be the future of mathematics. And so it's also of mathematics. And so it's also of mathematics. And so it's also exciting that you're embracing uh one of exciting that you're embracing uh one of exciting that you're embracing uh one of the greatest mathematicians in in the the greatest mathematicians in in the the greatest mathematicians in in the world embracing this world embracing this world embracing this what seems like the paving of the future what seems like the paving of the future what seems like the paving of the future of mathematics. Um so I have to ask you of mathematics. Um so I have to ask you of mathematics. Um so I have to ask you here about here about here about the integration of AI into this whole the integration of AI into this whole the integration of AI into this whole process. So deep mind's alpha proof was process. So deep mind's alpha proof was process. So deep mind's alpha proof was trained using reinforcement learning on trained using reinforcement learning on trained using reinforcement learning on both failed and successful formal lean both failed and successful formal lean both failed and successful formal lean proofs of IMO problems. So this is sort proofs of IMO problems. So this is sort proofs of IMO problems. So this is sort of highlevel high school oh very high of highlevel high school oh very high of highlevel high school oh very high level yes very high level high school level yes very high level high school level yes very high level high school level mathematics problems. What do you level mathematics problems. What do you level mathematics problems. What do you think about the system and maybe what is think about the system and maybe what is think about the system and maybe what is the gap between this system that is able the gap between this system that is able the gap between this system that is able to prove the high school level problems to prove the high school level problems to prove the high school level problems uh versus gradual level uh problems.

uh versus gradual level uh problems. uh versus gradual level uh problems. Yeah, the difficulty increases Yeah, the difficulty increases Yeah, the difficulty increases exponentially with the the number of exponentially with the the number of exponentially with the the number of steps involved in the proof. It's a steps involved in the proof. It's a steps involved in the proof. It's a commentatorial explosion, right? So the commentatorial explosion, right? So the commentatorial explosion, right? So the thing with large language models is is thing with large language models is is thing with large language models is is that they make mistakes. And so if a that they make mistakes. And so if a that they make mistakes. And so if a proof has got 20 steps and your model proof has got 20 steps and your model proof has got 20 steps and your model has a 10% failure rate um at each step has a 10% failure rate um at each step has a 10% failure rate um at each step um of of going in the wrong direction um of of going in the wrong direction um of of going in the wrong direction like u it's just extremely unlikely to like u it's just extremely unlikely to like u it's just extremely unlikely to actually um reach the end. Actually uh actually um reach the end. Actually uh actually um reach the end. Actually uh just to take a small tangent here is how just to take a small tangent here is how just to take a small tangent here is how hard is the problem of mapping from hard is the problem of mapping from hard is the problem of mapping from natural language to the formal program? natural language to the formal program? natural language to the formal program? Oh yeah it's extremely hard actually. Um Oh yeah it's extremely hard actually. Um Oh yeah it's extremely hard actually. Um natural language you know it's very natural language you know it's very natural language you know it's very fault tolerant. Um like you can make a fault tolerant. Um like you can make a fault tolerant. Um like you can make a few minor grammatical errors and a few minor grammatical errors and a few minor grammatical errors and a speaker in the second language can get speaker in the second language can get speaker in the second language can get some idea of what you're saying. Um yeah some idea of what you're saying. Um yeah some idea of what you're saying. Um yeah but but formal language yeah you if you but but formal language yeah you if you but but formal language yeah you if you get one little thing wrong um like the get one little thing wrong um like the get one little thing wrong um like the whole thing is is is nonsense. um even whole thing is is is nonsense. um even whole thing is is is nonsense. um even formal to formal is is is very hard. formal to formal is is is very hard. formal to formal is is is very hard. There there are different incompatible There there are different incompatible There there are different incompatible um uh proofist languages. Uh there's um uh proofist languages. Uh there's um uh proofist languages. Uh there's lean but also coaul and Isabel and so lean but also coaul and Isabel and so lean but also coaul and Isabel and so forth and actually even converting from forth and actually even converting from forth and actually even converting from a formal language to formal language um a formal language to formal language um a formal language to formal language um is is an unsolved basically unsolved is is an unsolved basically unsolved is is an unsolved basically unsolved problem. That is fascinating. Okay. So problem. That is fascinating. Okay. So problem. That is fascinating. Okay. So uh but once you have an informal uh but once you have an informal uh but once you have an informal language language language they're using um their RL train model.

they're using um their RL train model. they're using um their RL train model. So some something akin to alpha zero So some something akin to alpha zero So some something akin to alpha zero that they used to go to then try to come that they used to go to then try to come that they used to go to then try to come up with poos they also have a model I up with poos they also have a model I up with poos they also have a model I believe it's a separate model for believe it's a separate model for believe it's a separate model for geometric problems so what impresses you geometric problems so what impresses you geometric problems so what impresses you about the system and um what do you about the system and um what do you about the system and um what do you think is the gap yeah we talked earlier think is the gap yeah we talked earlier think is the gap yeah we talked earlier about things that are amazing over time about things that are amazing over time about things that are amazing over time become kind of normalized um so yeah now become kind of normalized um so yeah now become kind of normalized um so yeah now somehow it's oh of course geometry is a somehow it's oh of course geometry is a somehow it's oh of course geometry is a silver problem right that's true that's silver problem right that's true that's silver problem right that's true that's true I mean it's still beautiful yeah true I mean it's still beautiful yeah true I mean it's still beautiful yeah these are great works it shows what's these are great works it shows what's these are great works it shows what's possible I mean um it's it um the possible I mean um it's it um the possible I mean um it's it um the approach doesn't scale currently is yeah approach doesn't scale currently is yeah approach doesn't scale currently is yeah 3 days of Google's survey server time to 3 days of Google's survey server time to 3 days of Google's survey server time to solve one high school math problem. This solve one high school math problem. This solve one high school math problem. This is not a scalable uh prospect. Um is not a scalable uh prospect. Um is not a scalable uh prospect. Um especially with the exponential increase especially with the exponential increase especially with the exponential increase in um as as the complexity um increases. in um as as the complexity um increases. in um as as the complexity um increases. We should mention that they got a silver We should mention that they got a silver We should mention that they got a silver medal performance the equivalent of I medal performance the equivalent of I medal performance the equivalent of I mean yeah equivalent of a silver so mean yeah equivalent of a silver so mean yeah equivalent of a silver so first of all they took way more time first of all they took way more time first of all they took way more time than was allotted um and they had this than was allotted um and they had this than was allotted um and they had this assistance where where the humans assistance where where the humans assistance where where the humans started helped by by formalizing um but started helped by by formalizing um but started helped by by formalizing um but uh also they they're giving us those uh also they they're giving us those uh also they they're giving us those full marks for the solution which I full marks for the solution which I full marks for the solution which I guess is formally verified. So I guess guess is formally verified. So I guess guess is formally verified. So I guess that that's that's fair. Um yeah um that that's that's fair. Um yeah um that that's that's fair. Um yeah um there there are efforts there was there there there are efforts there was there there there are efforts there was there will be a proposal at some point to will be a proposal at some point to will be a proposal at some point to actually have an an AI math olympiate actually have an an AI math olympiate actually have an an AI math olympiate where at the same time as the human where at the same time as the human where at the same time as the human contestants get the the actual Olympia contestants get the the actual Olympia contestants get the the actual Olympia um problems AIS will also be given the um problems AIS will also be given the um problems AIS will also be given the same problems with the same time period same problems with the same time period same problems with the same time period um and the outputs will have to be um and the outputs will have to be um and the outputs will have to be graded by the same judges um um and

graded by the same judges um um and graded by the same judges um um and which means that will have be written in which means that will have be written in which means that will have be written in natural language rather than formal natural language rather than formal natural language rather than formal language. Oh I hope that happens. I hope language. Oh I hope that happens. I hope language. Oh I hope that happens. I hope that this IMO it happens. I hope I hope that this IMO it happens. I hope I hope that this IMO it happens. I hope I hope next one it won't happen this IMO the next one it won't happen this IMO the next one it won't happen this IMO the performance is not good enough in in the performance is not good enough in in the performance is not good enough in in the time period and and uh um but there are time period and and uh um but there are time period and and uh um but there are smaller competitions um there are smaller competitions um there are smaller competitions um there are competitions where the the answer is a competitions where the the answer is a competitions where the the answer is a is a number rather than a long form is a number rather than a long form is a number rather than a long form proof um and that's that's um AI are proof um and that's that's um AI are proof um and that's that's um AI are actually a lot better at um problems actually a lot better at um problems actually a lot better at um problems where there's a specific numerical where there's a specific numerical where there's a specific numerical answer um because it's it's easy to to answer um because it's it's easy to to answer um because it's it's easy to to to uh to reinforce do reinforcement to uh to reinforce do reinforcement to uh to reinforce do reinforcement learning on it. Yeah, you got the right learning on it. Yeah, you got the right learning on it. Yeah, you got the right answer, you got the wrong answer. It's answer, you got the wrong answer. It's answer, you got the wrong answer. It's it's a very clear signal. But a long it's a very clear signal. But a long it's a very clear signal. But a long form proof either has to be formal and form proof either has to be formal and form proof either has to be formal and then the lean can give it a thumbs up, then the lean can give it a thumbs up, then the lean can give it a thumbs up, thumbs down, or it's informal. Um, but thumbs down, or it's informal. Um, but thumbs down, or it's informal. Um, but then you need a human to grade it to then you need a human to grade it to then you need a human to grade it to tell uh and if you're trying to do tell uh and if you're trying to do tell uh and if you're trying to do billions of of reinforcement learning um billions of of reinforcement learning um billions of of reinforcement learning um you know um um runs, you're not you you know um um runs, you're not you you know um um runs, you're not you can't hire enough humans to uh to grade can't hire enough humans to uh to grade can't hire enough humans to uh to grade those. um it's already hard enough for those. um it's already hard enough for those. um it's already hard enough for for the last language to do for the last language to do for the last language to do reinforcement learning on on just the reinforcement learning on on just the reinforcement learning on on just the regular text that that people get. But regular text that that people get. But regular text that that people get. But now if you actually hire people not just now if you actually hire people not just now if you actually hire people not just give thumbs up, thumbs down, but give thumbs up, thumbs down, but give thumbs up, thumbs down, but actually check the the output actually check the the output actually check the the output mathematically. Yeah, that's too mathematically. Yeah, that's too mathematically. Yeah, that's too expensive. So if we uh just explore this expensive. So if we uh just explore this expensive. So if we uh just explore this possible future, possible future, possible future, what what what is the thing that humans what what what is the thing that humans what what what is the thing that humans do that's most special in um in do that's most special in um in do that's most special in um in mathematics? So that you could see AI mathematics? So that you could see AI mathematics? So that you could see AI uh not cracking for a while. So uh not cracking for a while. So uh not cracking for a while. So inventing new theories. So coming up inventing new theories. So coming up inventing new theories. So coming up with new conjectures versus uh proving

with new conjectures versus uh proving with new conjectures versus uh proving the conjectures, the conjectures, the conjectures, right? Building new abstractions, new right? Building new abstractions, new right? Building new abstractions, new representations, maybe uh an AI turn representations, maybe uh an AI turn representations, maybe uh an AI turn style with seeing new connections style with seeing new connections style with seeing new connections between disparate fields. It's a good between disparate fields. It's a good between disparate fields. It's a good question. Um I think the nature of what question. Um I think the nature of what question. Um I think the nature of what mathematicians do over time has changed mathematicians do over time has changed mathematicians do over time has changed a lot. um you know um so a thousand a lot. um you know um so a thousand a lot. um you know um so a thousand years ago mathematicians had to compute years ago mathematicians had to compute years ago mathematicians had to compute the date of Easter uh and there was the date of Easter uh and there was the date of Easter uh and there was really complicated uh calculations you really complicated uh calculations you really complicated uh calculations you know but it's all automated been know but it's all automated been know but it's all automated been automated for centuries we don't need automated for centuries we don't need automated for centuries we don't need that anymore you know they used to that anymore you know they used to that anymore you know they used to navigate to do spherical navigation navigate to do spherical navigation navigate to do spherical navigation spherical trigonometry to navigate how spherical trigonometry to navigate how spherical trigonometry to navigate how to get from from um the old world to the to get from from um the old world to the to get from from um the old world to the new or very complicated calculations new or very complicated calculations new or very complicated calculations again we've been automated um you know again we've been automated um you know again we've been automated um you know even a lot of undergraduate mathematics even a lot of undergraduate mathematics even a lot of undergraduate mathematics even before AI um like wolf from alpha even before AI um like wolf from alpha even before AI um like wolf from alpha for example It's not a language model, for example It's not a language model, for example It's not a language model, but it can solve a lot of undergraduate but it can solve a lot of undergraduate but it can solve a lot of undergraduate level math tasks. So on the level math tasks. So on the level math tasks. So on the computational side, verifying routine computational side, verifying routine computational side, verifying routine things like having a a problem and um things like having a a problem and um things like having a a problem and um and say here's a problem in partial and say here's a problem in partial and say here's a problem in partial equations. Could you solve it using any equations. Could you solve it using any equations. Could you solve it using any of the 20 standard techniques? Um and of the 20 standard techniques? Um and of the 20 standard techniques? Um and they say yes, I've tried all 20 and here they say yes, I've tried all 20 and here they say yes, I've tried all 20 and here are the 100 different permutations and are the 100 different permutations and are the 100 different permutations and and here's my results. Um and that type and here's my results. Um and that type and here's my results. Um and that type of thing I think it will work very well.

of thing I think it will work very well. of thing I think it will work very well. um type of scaling to once you solve one um type of scaling to once you solve one um type of scaling to once you solve one problem to to make the AI attack 100 problem to to make the AI attack 100 problem to to make the AI attack 100 adjacent problems. Um the things that adjacent problems. Um the things that adjacent problems. Um the things that humans do still Yeah. So so where the AI humans do still Yeah. So so where the AI humans do still Yeah. So so where the AI really struggles right now um is knowing really struggles right now um is knowing really struggles right now um is knowing when it's made a wrong turn. Um that it when it's made a wrong turn. Um that it when it's made a wrong turn. Um that it can say, "Oh, I'm going to solve this can say, "Oh, I'm going to solve this can say, "Oh, I'm going to solve this problem. I'm going to split up this problem. I'm going to split up this problem. I'm going to split up this problem into um into these two cases. problem into um into these two cases. problem into um into these two cases. I'm going to try this technique." And um I'm going to try this technique." And um I'm going to try this technique." And um sometimes if you're lucky and it's a sometimes if you're lucky and it's a sometimes if you're lucky and it's a simple problem, it's the right technique simple problem, it's the right technique simple problem, it's the right technique and you solve the problem and sometimes and you solve the problem and sometimes and you solve the problem and sometimes it it will get it will have a problem it it it will get it will have a problem it it it will get it will have a problem it would propose an approach which is just would propose an approach which is just would propose an approach which is just complete nonsense. Um and but like it complete nonsense. Um and but like it complete nonsense. Um and but like it looks like a proof. Um so this is one looks like a proof. Um so this is one looks like a proof. Um so this is one annoying thing about LM generated annoying thing about LM generated annoying thing about LM generated mathematics. So um yeah we we we've had mathematics. So um yeah we we we've had mathematics. So um yeah we we we've had human generated mathematics as very low human generated mathematics as very low human generated mathematics as very low quality um uh like you know submissions quality um uh like you know submissions quality um uh like you know submissions people who don't have the formal people who don't have the formal people who don't have the formal training and so forth. But if a human training and so forth. But if a human training and so forth. But if a human proof is bad, you can tell it's bad proof is bad, you can tell it's bad proof is bad, you can tell it's bad pretty quickly. It makes really basic pretty quickly. It makes really basic pretty quickly. It makes really basic mistakes. But the AI generated proofs, mistakes. But the AI generated proofs, mistakes. But the AI generated proofs, they can look superficially flawless. Uh they can look superficially flawless. Uh they can look superficially flawless. Uh and that's partly because that's what and that's partly because that's what and that's partly because that's what the reinforcement learning has actually the reinforcement learning has actually the reinforcement learning has actually trained them to do, right? To to make trained them to do, right? To to make trained them to do, right? To to make things to to produce text that looks things to to produce text that looks things to to produce text that looks like um what is correct, which for many like um what is correct, which for many like um what is correct, which for many applications is good enough. Um uh so applications is good enough. Um uh so applications is good enough. Um uh so the errors often really subtle and then the errors often really subtle and then the errors often really subtle and then when you spot them, they're really when you spot them, they're really when you spot them, they're really stupid. Um like you know like no human stupid. Um like you know like no human stupid. Um like you know like no human would have actually made that mistake.

would have actually made that mistake. would have actually made that mistake. Yeah, it's actually really frustrating Yeah, it's actually really frustrating Yeah, it's actually really frustrating in the programming context because I I in the programming context because I I in the programming context because I I program a lot and yeah, when a human program a lot and yeah, when a human program a lot and yeah, when a human makes when lowquality code, there's makes when lowquality code, there's makes when lowquality code, there's something called code smell, right? You something called code smell, right? You something called code smell, right? You can you can tell you can tell can you can tell you can tell can you can tell you can tell immediately like, okay, there's signs. immediately like, okay, there's signs. immediately like, okay, there's signs. But with with a generate code of and But with with a generate code of and But with with a generate code of and then you're right eventually you find an then you're right eventually you find an then you're right eventually you find an obvious dumb thing that just looks like obvious dumb thing that just looks like obvious dumb thing that just looks like good code. Yeah. So, um it's very tricky good code. Yeah. So, um it's very tricky good code. Yeah. So, um it's very tricky to and frustrating for some reason to to and frustrating for some reason to to and frustrating for some reason to Yeah. to work. Yeah. So the sense of Yeah. to work. Yeah. So the sense of Yeah. to work. Yeah. So the sense of smell. Okay, there you go. This is this smell. Okay, there you go. This is this smell. Okay, there you go. This is this is one thing that humans have. Um and is one thing that humans have. Um and is one thing that humans have. Um and there's a metaphorical mathematical there's a metaphorical mathematical there's a metaphorical mathematical smell that uh this we it's not clear how smell that uh this we it's not clear how smell that uh this we it's not clear how to get the AI to duplicate that to get the AI to duplicate that to get the AI to duplicate that eventually. Um I mean so the way um eventually. Um I mean so the way um eventually. Um I mean so the way um Alpha Zero and so forth make progress on Alpha Zero and so forth make progress on Alpha Zero and so forth make progress on go and and chess and so forth is is in go and and chess and so forth is is in go and and chess and so forth is is in some sense they have developed a sense some sense they have developed a sense some sense they have developed a sense of smell for go and chess positions you of smell for go and chess positions you of smell for go and chess positions you know that that this position is good for know that that this position is good for know that that this position is good for white is good for black. um they can't white is good for black. um they can't white is good for black. um they can't initiate why. Um but just having that initiate why. Um but just having that initiate why. Um but just having that that sense of smell lets them that sense of smell lets them that sense of smell lets them strategize. So if AIs gain that ability strategize. So if AIs gain that ability strategize. So if AIs gain that ability to sort of a sense of viability of to sort of a sense of viability of to sort of a sense of viability of certain proof strategies say so so you certain proof strategies say so so you certain proof strategies say so so you can say I'm going to try to break up can say I'm going to try to break up can say I'm going to try to break up this problem into two small subtasks and this problem into two small subtasks and this problem into two small subtasks and they can say well this looks good two they can say well this looks good two they can say well this looks good two tasks look like they're simpler tasks tasks look like they're simpler tasks tasks look like they're simpler tasks than than your main task and they still than than your main task and they still than than your main task and they still got a good chance of being true. Um so got a good chance of being true. Um so got a good chance of being true. Um so this is good to try or no you've you this is good to try or no you've you this is good to try or no you've you made the problem worse because each of made the problem worse because each of made the problem worse because each of the two sub problems is actually harder the two sub problems is actually harder the two sub problems is actually harder than your original problem which is than your original problem which is than your original problem which is actually what normally happens if you

actually what normally happens if you actually what normally happens if you try a random uh thing to try normally try a random uh thing to try normally try a random uh thing to try normally actually it's very easy to transform a actually it's very easy to transform a actually it's very easy to transform a problem into even harder problem. Mhm. problem into even harder problem. Mhm. problem into even harder problem. Mhm. Very rarely do you problem transport a Very rarely do you problem transport a Very rarely do you problem transport a simpler problem. Um yeah so if they can simpler problem. Um yeah so if they can simpler problem. Um yeah so if they can pick up a sense of smell then they could pick up a sense of smell then they could pick up a sense of smell then they could maybe start competing with human level maybe start competing with human level maybe start competing with human level mathematicians. So, this is a hard mathematicians. So, this is a hard mathematicians. So, this is a hard question, but not competing, but question, but not competing, but question, but not competing, but collaborating. Yeah. If Okay, collaborating. Yeah. If Okay, collaborating. Yeah. If Okay, hypothetical. hypothetical. hypothetical. If I gave you an oracle If I gave you an oracle If I gave you an oracle that was able to do some aspect of what that was able to do some aspect of what that was able to do some aspect of what you do, and you could just collaborate you do, and you could just collaborate you do, and you could just collaborate with it. Yeah. Yeah. What would that with it. Yeah. Yeah. What would that with it. Yeah. Yeah. What would that oracle What would you like that oracle oracle What would you like that oracle oracle What would you like that oracle to be able to do? Would you like it to to be able to do? Would you like it to to be able to do? Would you like it to uh maybe be a verifier? Like check Mhm. uh maybe be a verifier? Like check Mhm. uh maybe be a verifier? Like check Mhm. Do the codes like you're Yes. uh Do the codes like you're Yes. uh Do the codes like you're Yes. uh professor to this is the correct this is professor to this is the correct this is professor to this is the correct this is a good this is a promising fruitful a good this is a promising fruitful a good this is a promising fruitful direction. Yeah. Yeah. Yeah. Or or would direction. Yeah. Yeah. Yeah. Or or would direction. Yeah. Yeah. Yeah. Or or would you like it to you like it to you like it to uh generate possible proofs and then you uh generate possible proofs and then you uh generate possible proofs and then you see which one is the right one? Um or see which one is the right one? Um or see which one is the right one? Um or would you like it to maybe generate would you like it to maybe generate would you like it to maybe generate different representation different different representation different different representation different totally different ways of seeing this totally different ways of seeing this totally different ways of seeing this problem? Yeah, I think all of the above.

problem? Yeah, I think all of the above. problem? Yeah, I think all of the above. Um a lot of it is we don't know how to Um a lot of it is we don't know how to Um a lot of it is we don't know how to use these tools because it's a paradigm use these tools because it's a paradigm use these tools because it's a paradigm that is not um yeah we have not had in that is not um yeah we have not had in that is not um yeah we have not had in the past systems that are competent the past systems that are competent the past systems that are competent enough to understand complex enough to understand complex enough to understand complex instructions. Mhm. Um that can work at instructions. Mhm. Um that can work at instructions. Mhm. Um that can work at massive scale but are also unreliable. massive scale but are also unreliable. massive scale but are also unreliable. Uh like it's it's an interesting uh bit Uh like it's it's an interesting uh bit Uh like it's it's an interesting uh bit unreliable in subtle ways while we while unreliable in subtle ways while we while unreliable in subtle ways while we while providing sufficiently good output. Um providing sufficiently good output. Um providing sufficiently good output. Um it's a interesting combination. um you it's a interesting combination. um you it's a interesting combination. um you know I mean you have you have like know I mean you have you have like know I mean you have you have like graduate students that you work with who graduate students that you work with who graduate students that you work with who kind of like this but not at scale um kind of like this but not at scale um kind of like this but not at scale um you know and and and we have previous you know and and and we have previous you know and and and we have previous software tools that um can work at scale software tools that um can work at scale software tools that um can work at scale but but very narrow um so we have to but but very narrow um so we have to but but very narrow um so we have to figure out how to how to use um I mean figure out how to how to use um I mean figure out how to how to use um I mean um so Tim C actually imagine he actually um so Tim C actually imagine he actually um so Tim C actually imagine he actually foresaw like in in 2000 he was foresaw like in in 2000 he was foresaw like in in 2000 he was envisioning what mathematics would look envisioning what mathematics would look envisioning what mathematics would look like in in actually two and a half like in in actually two and a half like in in actually two and a half decades decades decades and that's funny yeah He he wrote in his and that's funny yeah He he wrote in his and that's funny yeah He he wrote in his in in his article like a a a in in his article like a a a in in his article like a a a hypothetical conversation between a hypothetical conversation between a hypothetical conversation between a mathematical assistant of the future um mathematical assistant of the future um mathematical assistant of the future um and himself you know trying to solve a and himself you know trying to solve a and himself you know trying to solve a problem and they would have have a problem and they would have have a problem and they would have have a conversation that sometimes the human conversation that sometimes the human conversation that sometimes the human would would propose an idea and the AI would would propose an idea and the AI would would propose an idea and the AI would would evaluate it and sometimes would would evaluate it and sometimes would would evaluate it and sometimes the AI would propose an idea um and u the AI would propose an idea um and u the AI would propose an idea um and u and sometimes that computation was and sometimes that computation was and sometimes that computation was required and a would just go and say required and a would just go and say required and a would just go and say okay I've checked the 100 cases needed okay I've checked the 100 cases needed okay I've checked the 100 cases needed here or um the first you you said this here or um the first you you said this here or um the first you you said this is true for all n I've checked for n up is true for all n I've checked for n up is true for all n I've checked for n up to 100 um and it looks good so far or to 100 um and it looks good so far or to 100 um and it looks good so far or hang on there's a problem at n equals 46

hang on there's a problem at n equals 46 hang on there's a problem at n equals 46 you so just a free form conversation you so just a free form conversation you so just a free form conversation where you don't know in advance where where you don't know in advance where where you don't know in advance where things are going to go but just based on things are going to go but just based on things are going to go but just based on on I think ideas get proposed on both on I think ideas get proposed on both on I think ideas get proposed on both sides calculations get proposed on both sides calculations get proposed on both sides calculations get proposed on both sides I've had conversations with AI sides I've had conversations with AI sides I've had conversations with AI where I say okay let's we're going to where I say okay let's we're going to where I say okay let's we're going to collaborate to solve this math problem collaborate to solve this math problem collaborate to solve this math problem and it's a problem that I already know and it's a problem that I already know and it's a problem that I already know the solution to so I I try to prompt it the solution to so I I try to prompt it the solution to so I I try to prompt it okay so here's the problem I suggest okay so here's the problem I suggest okay so here's the problem I suggest using this tool and then you'll find using this tool and then you'll find using this tool and then you'll find this this lovely argument using a this this lovely argument using a this this lovely argument using a totally different tool which eventually totally different tool which eventually totally different tool which eventually goes you know, into the weeds and say, goes you know, into the weeds and say, goes you know, into the weeds and say, "No, no, no. If I using this, okay, and "No, no, no. If I using this, okay, and "No, no, no. If I using this, okay, and it might start using this and then it'll it might start using this and then it'll it might start using this and then it'll go back to the tool that I wanted to to go back to the tool that I wanted to to go back to the tool that I wanted to to before." Um, and like you have to keep before." Um, and like you have to keep before." Um, and like you have to keep railroading it um onto the path you railroading it um onto the path you railroading it um onto the path you want. And like I I could eventually want. And like I I could eventually want. And like I I could eventually force it to give the proof I wanted. Um, force it to give the proof I wanted. Um, force it to give the proof I wanted. Um, but it was like hurting cats um like and but it was like hurting cats um like and but it was like hurting cats um like and the amount of personal effort I had to the amount of personal effort I had to the amount of personal effort I had to take to not just sort of prompt it, but take to not just sort of prompt it, but take to not just sort of prompt it, but also check it output because it like a also check it output because it like a also check it output because it like a lot of what it looked like was going to lot of what it looked like was going to lot of what it looked like was going to work. I know there's a problem on online work. I know there's a problem on online work. I know there's a problem on online 17 and basically arguing with it. um 17 and basically arguing with it. um 17 and basically arguing with it. um like it was more exhausting than doing like it was more exhausting than doing like it was more exhausting than doing it unassisted. So like it but that's the it unassisted. So like it but that's the it unassisted. So like it but that's the current state of the art. I wonder if current state of the art. I wonder if current state of the art. I wonder if there's there's a phase shift that there's there's a phase shift that there's there's a phase shift that happens to where it's no longer feels happens to where it's no longer feels happens to where it's no longer feels like hurting cats and like hurting cats and like hurting cats and maybe it'll surprise us how quickly that maybe it'll surprise us how quickly that maybe it'll surprise us how quickly that comes. I I believe so. Um so in comes. I I believe so. Um so in comes. I I believe so. Um so in formalization I I mentioned before that formalization I I mentioned before that formalization I I mentioned before that it takes 10 times longer to formalize a it takes 10 times longer to formalize a it takes 10 times longer to formalize a proof than to write it by hand with proof than to write it by hand with proof than to write it by hand with these modern AI tools is and also just these modern AI tools is and also just these modern AI tools is and also just better tooling um the lean um um better tooling um the lean um um better tooling um the lean um um developers are doing a great job adding developers are doing a great job adding developers are doing a great job adding more and more features and making it more and more features and making it more and more features and making it user friendly. It's going up from 9 to 8

user friendly. It's going up from 9 to 8 user friendly. It's going up from 9 to 8 to 7. Okay, no big deal. But one day it to 7. Okay, no big deal. But one day it to 7. Okay, no big deal. But one day it will drop below one. Um and that's a will drop below one. Um and that's a will drop below one. Um and that's a phase shift because suddenly um it makes phase shift because suddenly um it makes phase shift because suddenly um it makes sense when you write a paper to to write sense when you write a paper to to write sense when you write a paper to to write it in lean first or through a it in lean first or through a it in lean first or through a conversation with AI who is generally um conversation with AI who is generally um conversation with AI who is generally um on the fly with you and it becomes on the fly with you and it becomes on the fly with you and it becomes natural for journals to accept you know natural for journals to accept you know natural for journals to accept you know maybe they'll offer expedite refereeing maybe they'll offer expedite refereeing maybe they'll offer expedite refereeing you know if if a paper has already been you know if if a paper has already been you know if if a paper has already been formalized in in lean um they'll just formalized in in lean um they'll just formalized in in lean um they'll just ask the referee to comment on on the ask the referee to comment on on the ask the referee to comment on on the significance of the results and how it significance of the results and how it significance of the results and how it connects to literature and not worry so connects to literature and not worry so connects to literature and not worry so much about the correctness. much about the correctness. much about the correctness. um because that's been certified. Um um because that's been certified. Um um because that's been certified. Um papers are getting longer and longer in papers are getting longer and longer in papers are getting longer and longer in mathematics and actually it's harder and mathematics and actually it's harder and mathematics and actually it's harder and harder to get good refereeing for um the harder to get good refereeing for um the harder to get good refereeing for um the really long ones unless they're really really long ones unless they're really really long ones unless they're really important. It is actually an issue which important. It is actually an issue which important. It is actually an issue which and the formalization is coming in at and the formalization is coming in at and the formalization is coming in at just the right time for this to be and just the right time for this to be and just the right time for this to be and the easier and easier to guess because the easier and easier to guess because the easier and easier to guess because of the tooling and all the other factors of the tooling and all the other factors of the tooling and all the other factors then you're going to see much more like then you're going to see much more like then you're going to see much more like math lib will grow potentially math lib will grow potentially math lib will grow potentially exponentially. It's a it's a it's a exponentially. It's a it's a it's a exponentially. It's a it's a it's a virtuous uh cycle. Okay. I mean one virtuous uh cycle. Okay. I mean one virtuous uh cycle. Okay. I mean one facet of this type that happened in the facet of this type that happened in the facet of this type that happened in the past was the adoption of latte. So so past was the adoption of latte. So so past was the adoption of latte. So so latte is this type seting language that latte is this type seting language that latte is this type seting language that all mians use now. So in the past people all mians use now. So in the past people all mians use now. So in the past people use all kinds of word processors and use all kinds of word processors and use all kinds of word processors and typewriters and whatever but at some typewriters and whatever but at some typewriters and whatever but at some point latte became easier to use than point latte became easier to use than point latte became easier to use than all other competitors and that people all other competitors and that people all other competitors and that people just switched you know within a few just switched you know within a few just switched you know within a few years like it was just a dramatic um pay years like it was just a dramatic um pay years like it was just a dramatic um pay shift. It's a wild out there question, shift. It's a wild out there question, shift. It's a wild out there question, but what but what but what what year how far away are we from

what year how far away are we from what year how far away are we from a uh AI system being a collaborator a uh AI system being a collaborator a uh AI system being a collaborator on a proof that wins the Fields medal. on a proof that wins the Fields medal. on a proof that wins the Fields medal. So that level. Okay. Um well, it depends So that level. Okay. Um well, it depends So that level. Okay. Um well, it depends on the level of collaboration. I mean, on the level of collaboration. I mean, on the level of collaboration. I mean, no, like it deserves to be to get the no, like it deserves to be to get the no, like it deserves to be to get the Fields Medal. like so half and half Fields Medal. like so half and half Fields Medal. like so half and half already like I I can imagine if it was a already like I I can imagine if it was a already like I I can imagine if it was a winning paper having some AI systems in winning paper having some AI systems in winning paper having some AI systems in writing it you know uh just you know writing it you know uh just you know writing it you know uh just you know like the order complete alone is already like the order complete alone is already like the order complete alone is already I I use it like it speeds up my my own I I use it like it speeds up my my own I I use it like it speeds up my my own writing um um like you know you you can writing um um like you know you you can writing um um like you know you you can have a theorem you have a proof and the have a theorem you have a proof and the have a theorem you have a proof and the proof has three cases and I I write down proof has three cases and I I write down proof has three cases and I I write down the proof of the first case and the the proof of the first case and the the proof of the first case and the autocomplete just suggests all right now autocomplete just suggests all right now autocomplete just suggests all right now now here's how the proof of second case now here's how the proof of second case now here's how the proof of second case could work and like it was exactly could work and like it was exactly could work and like it was exactly correct that was great saved me like 5 correct that was great saved me like 5 correct that was great saved me like 5 10 minutes of uh of typing but in that 10 minutes of uh of typing but in that 10 minutes of uh of typing but in that case The AI system doesn't get the case The AI system doesn't get the case The AI system doesn't get the Fields medal. No. Uh are we talking 20 years, 50 years, 100 are we talking 20 years, 50 years, 100 years? What do you think? Okay. So I I years? What do you think? Okay. So I I years? What do you think? Okay. So I I gave a prediction in print. So by 2026, gave a prediction in print. So by 2026, gave a prediction in print. So by 2026, which is now next year, um there will be which is now next year, um there will be which is now next year, um there will be math collaborations, you know, where the math collaborations, you know, where the math collaborations, you know, where the AI, so not Fields Medal winning, but but AI, so not Fields Medal winning, but but AI, so not Fields Medal winning, but but like actual research level math like like actual research level math like like actual research level math like published ideas that in part generated published ideas that in part generated published ideas that in part generated by AI. Um maybe not the ideas but at by AI. Um maybe not the ideas but at by AI. Um maybe not the ideas but at least uh some of the computations um um least uh some of the computations um um least uh some of the computations um um the verifications. Yeah. I mean has that the verifications. Yeah. I mean has that the verifications. Yeah. I mean has that already happened? Has that already already happened? Has that already already happened? Has that already happened? Yeah. There are there are happened? Yeah. There are there are happened? Yeah. There are there are problems that were solved uh by a problems that were solved uh by a problems that were solved uh by a complicated process conversing with AI complicated process conversing with AI complicated process conversing with AI to propose things and the human goes and to propose things and the human goes and to propose things and the human goes and tries it and the contract doesn't work

tries it and the contract doesn't work tries it and the contract doesn't work but it might propose a different idea. but it might propose a different idea. but it might propose a different idea. Um it it's it's hard to disentangle Um it it's it's hard to disentangle Um it it's it's hard to disentangle exactly. Um there are certainly math exactly. Um there are certainly math exactly. Um there are certainly math results which could only have been results which could only have been results which could only have been accomplished because there was a math accomplished because there was a math accomplished because there was a math method human mathematician and an AI method human mathematician and an AI method human mathematician and an AI involved. Um but it's hard to sort of involved. Um but it's hard to sort of involved. Um but it's hard to sort of disentangle credit. Um disentangle credit. Um disentangle credit. Um I mean these tools they they do not uh I mean these tools they they do not uh I mean these tools they they do not uh replicate all the skills needed to do replicate all the skills needed to do replicate all the skills needed to do mathematics but they can replicate sort mathematics but they can replicate sort mathematics but they can replicate sort of some non-trivial percentage of them of some non-trivial percentage of them of some non-trivial percentage of them you know 30 40%. they can fill in gaps. you know 30 40%. they can fill in gaps. you know 30 40%. they can fill in gaps. Um, you know, so, uh, coding is is is a Um, you know, so, uh, coding is is is a Um, you know, so, uh, coding is is is a is a good example, you know. So, I I um is a good example, you know. So, I I um is a good example, you know. So, I I um um it's annoying for me to code in um it's annoying for me to code in um it's annoying for me to code in Python. I'm not I'm not a native um I'm Python. I'm not I'm not a native um I'm Python. I'm not I'm not a native um I'm not a professional um programmer. Um, not a professional um programmer. Um, not a professional um programmer. Um, but um the with AI that the the friction but um the with AI that the the friction but um the with AI that the the friction cost of of doing it is is is much cost of of doing it is is is much cost of of doing it is is is much reduced. Uh so it it fills in that gap reduced. Uh so it it fills in that gap reduced. Uh so it it fills in that gap for me. Um for me. Um for me. Um AI is getting quite good at literature AI is getting quite good at literature AI is getting quite good at literature review. Um I mean there's still a review. Um I mean there's still a review. Um I mean there's still a problem with um hallucinating you know problem with um hallucinating you know problem with um hallucinating you know the references that don't exist. Um but the references that don't exist. Um but the references that don't exist. Um but this I think is a civil war problem if this I think is a civil war problem if this I think is a civil war problem if you train in the right way and so forth you train in the right way and so forth you train in the right way and so forth you can you can and um and verify um you you can you can and um and verify um you you can you can and um and verify um you know using the internet um you know um know using the internet um you know um know using the internet um you know um you should in a few years get to the you should in a few years get to the you should in a few years get to the point where you you have a a lema that point where you you have a a lema that point where you you have a a lema that you need and uh we say has anyone proven you need and uh we say has anyone proven you need and uh we say has anyone proven this lema before and it will do this lema before and it will do this lema before and it will do basically a fancy web search AI basically a fancy web search AI basically a fancy web search AI assistant and say yeah yeah there are assistant and say yeah yeah there are assistant and say yeah yeah there are these six papers where something similar these six papers where something similar these six papers where something similar has happened and I mean it you can ask

has happened and I mean it you can ask has happened and I mean it you can ask it right now and it'll give you six it right now and it'll give you six it right now and it'll give you six papers of which maybe one is is papers of which maybe one is is papers of which maybe one is is legitimate and relevant. One exists but legitimate and relevant. One exists but legitimate and relevant. One exists but is not relevant and four are is not relevant and four are is not relevant and four are hallucinated. Um it has a non-zero hallucinated. Um it has a non-zero hallucinated. Um it has a non-zero success rate right now, but uh it's success rate right now, but uh it's success rate right now, but uh it's there's so much garbage. Uh so much the there's so much garbage. Uh so much the there's so much garbage. Uh so much the signal to noise ratio is so poor that signal to noise ratio is so poor that signal to noise ratio is so poor that it's it's um it's most helpful when you it's it's um it's most helpful when you it's it's um it's most helpful when you already somewhat know the literature. Um already somewhat know the literature. Um already somewhat know the literature. Um and you just need to be prompted to be and you just need to be prompted to be and you just need to be prompted to be reminded of a paper that was already reminded of a paper that was already reminded of a paper that was already subconsciously in your memory versus subconsciously in your memory versus subconsciously in your memory versus helping you discover new you were not helping you discover new you were not helping you discover new you were not even aware of but is the correct even aware of but is the correct even aware of but is the correct citation. Yeah, that's yeah, that it can citation. Yeah, that's yeah, that it can citation. Yeah, that's yeah, that it can sometimes do. But but when it does, it's sometimes do. But but when it does, it's sometimes do. But but when it does, it's it's buried in in a list of options for it's buried in in a list of options for it's buried in in a list of options for which the other that are bad. Yeah. I which the other that are bad. Yeah. I which the other that are bad. Yeah. I mean, being able to automatically mean, being able to automatically mean, being able to automatically generate a related work section that is generate a related work section that is generate a related work section that is correct. Yeah. That's actually a correct. Yeah. That's actually a correct. Yeah. That's actually a beautiful thing that might be another beautiful thing that might be another beautiful thing that might be another phase shift because it assigns credit phase shift because it assigns credit phase shift because it assigns credit correctly. Yeah. It does. It breaks you correctly. Yeah. It does. It breaks you correctly. Yeah. It does. It breaks you out of the silos of Yeah. Yeah. Yeah. out of the silos of Yeah. Yeah. Yeah. out of the silos of Yeah. Yeah. Yeah. thought, you know. Yeah. No, there's a thought, you know. Yeah. No, there's a thought, you know. Yeah. No, there's a big hump to overcome right now. I mean, big hump to overcome right now. I mean, big hump to overcome right now. I mean, it's it's like self-driving cars, you it's it's like self-driving cars, you it's it's like self-driving cars, you know. the the safety margin has to be know. the the safety margin has to be know. the the safety margin has to be really high for it to be um uh to be really high for it to be um uh to be really high for it to be um uh to be feasible. So yeah, so there's a last feasible. So yeah, so there's a last feasible. So yeah, so there's a last mile problem um with a lot of AI mile problem um with a lot of AI mile problem um with a lot of AI applications um that uh you know they applications um that uh you know they applications um that uh you know they can develop tools that work 20% 80% of can develop tools that work 20% 80% of can develop tools that work 20% 80% of the time but it's still not good enough the time but it's still not good enough the time but it's still not good enough um and in fact even worse than good some um and in fact even worse than good some um and in fact even worse than good some ways. I mean another way of asking the ways. I mean another way of asking the ways. I mean another way of asking the Fields metal question is what year do Fields metal question is what year do Fields metal question is what year do you think you'll wake up and be like you think you'll wake up and be like you think you'll wake up and be like real surprised? you read the headline, real surprised? you read the headline, real surprised? you read the headline, the news of something happened that AI

the news of something happened that AI the news of something happened that AI did like you know real breakthrough did like you know real breakthrough did like you know real breakthrough something it doesn't you know like feels something it doesn't you know like feels something it doesn't you know like feels metal even hypothesis it could be like metal even hypothesis it could be like metal even hypothesis it could be like really just really just really just this alpha zero moment with go that kind this alpha zero moment with go that kind this alpha zero moment with go that kind of thing right um yeah this this decade of thing right um yeah this this decade of thing right um yeah this this decade I can I can see it like making a I can I can see it like making a I can I can see it like making a conjecture conjecture conjecture between two unrelated two two things between two unrelated two two things between two unrelated two two things that people thought was unrelated oh that people thought was unrelated oh that people thought was unrelated oh interesting generating a conjecture interesting generating a conjecture interesting generating a conjecture that's a beautiful conjecture Yeah. And that's a beautiful conjecture Yeah. And that's a beautiful conjecture Yeah. And and actually has a real chance of being and actually has a real chance of being and actually has a real chance of being correct and and and meaningful and um correct and and and meaningful and um correct and and and meaningful and um because that's actually kind of doable I because that's actually kind of doable I because that's actually kind of doable I suppose but the word of the data is suppose but the word of the data is suppose but the word of the data is Yeah. No, that would be truly amazing. Yeah. No, that would be truly amazing. Yeah. No, that would be truly amazing. Um the current models struggle a lot. I Um the current models struggle a lot. I Um the current models struggle a lot. I mean so um a version of this is um I mean so um a version of this is um I mean so um a version of this is um I mean the physicists have a dream of mean the physicists have a dream of mean the physicists have a dream of getting the AI to discover new new laws getting the AI to discover new new laws getting the AI to discover new new laws of physics. Um you know the the dream is of physics. Um you know the the dream is of physics. Um you know the the dream is you just feed it all this data. Okay. you just feed it all this data. Okay. you just feed it all this data. Okay. and and this is here's a new patent that and and this is here's a new patent that and and this is here's a new patent that we didn't see before but it actually we didn't see before but it actually we didn't see before but it actually even struggle the current state of the even struggle the current state of the even struggle the current state of the art even struggles to discover old laws art even struggles to discover old laws art even struggles to discover old laws of physics um from the data uh or if it of physics um from the data uh or if it of physics um from the data uh or if it does there's a big concern contamination does there's a big concern contamination does there's a big concern contamination that that it did it only because like that that it did it only because like that that it did it only because like somewhere in this training data it some somewhere in this training data it some somewhere in this training data it some new um you know boils law or whatever new um you know boils law or whatever new um you know boils law or whatever ball you're trying to to to reconstruct ball you're trying to to to reconstruct ball you're trying to to to reconstruct um part of it is that we don't have the um part of it is that we don't have the um part of it is that we don't have the right type of training data for this um right type of training data for this um right type of training data for this um yeah so for laws of physics like we we yeah so for laws of physics like we we yeah so for laws of physics like we we don't have like a million different don't have like a million different don't have like a million different universes with a million infant laws of universes with a million infant laws of universes with a million infant laws of nature. Um nature. Um nature. Um and um like a lot of what we're missing and um like a lot of what we're missing and um like a lot of what we're missing in math is actually the negative space

in math is actually the negative space in math is actually the negative space of so we have published things of things of so we have published things of things of so we have published things of things that people have been able to prove um that people have been able to prove um that people have been able to prove um and conjectures that ended up being and conjectures that ended up being and conjectures that ended up being verified um or maybe counter examples verified um or maybe counter examples verified um or maybe counter examples produced but um we don't have data on on produced but um we don't have data on on produced but um we don't have data on on things that were proposed and they're things that were proposed and they're things that were proposed and they're kind of a good thing to try but then kind of a good thing to try but then kind of a good thing to try but then people quickly realized that it was the people quickly realized that it was the people quickly realized that it was the wrong conjecture and then they they said wrong conjecture and then they they said wrong conjecture and then they they said oh but we we should actually change um oh but we we should actually change um oh but we we should actually change um our claim to modify it in this way to our claim to modify it in this way to our claim to modify it in this way to actually make it more plausible. Um actually make it more plausible. Um actually make it more plausible. Um there's this there's a trial and error there's this there's a trial and error there's this there's a trial and error process which is a real integral part of process which is a real integral part of process which is a real integral part of human mathematical discovery which we human mathematical discovery which we human mathematical discovery which we don't record cuz it's embarrassing. Uh don't record cuz it's embarrassing. Uh don't record cuz it's embarrassing. Uh we make mistakes and and we only like to we make mistakes and and we only like to we make mistakes and and we only like to publish our wins. Um and uh the AI has publish our wins. Um and uh the AI has publish our wins. Um and uh the AI has no access to this data to train on. Um I no access to this data to train on. Um I no access to this data to train on. Um I sometimes joke that basically AI has to sometimes joke that basically AI has to sometimes joke that basically AI has to go through um grad school and actually go through um grad school and actually go through um grad school and actually you know go to grad courses, do the you know go to grad courses, do the you know go to grad courses, do the assignments, go to office hours, make assignments, go to office hours, make assignments, go to office hours, make mistakes, um get advice on how to mistakes, um get advice on how to mistakes, um get advice on how to correct the mistakes and learn from correct the mistakes and learn from correct the mistakes and learn from that. Let me uh ask you if I may about that. Let me uh ask you if I may about that. Let me uh ask you if I may about uh Gregori Pearlman. Mhm. You mentioned uh Gregori Pearlman. Mhm. You mentioned uh Gregori Pearlman. Mhm. You mentioned that you try to be careful in your work that you try to be careful in your work that you try to be careful in your work and not let a problem completely consume and not let a problem completely consume and not let a problem completely consume you. just you really fall in love with you. just you really fall in love with you. just you really fall in love with the problem and really cannot rest until the problem and really cannot rest until the problem and really cannot rest until you solve it. But you also hasted to add you solve it. But you also hasted to add you solve it. But you also hasted to add that sometimes this approach actually that sometimes this approach actually that sometimes this approach actually can be very successful. An example you can be very successful. An example you can be very successful. An example you gave is Gregoria Pearlman who proved the gave is Gregoria Pearlman who proved the gave is Gregoria Pearlman who proved the point conjecture and did so by working point conjecture and did so by working point conjecture and did so by working alone for 7 years with basically little alone for 7 years with basically little alone for 7 years with basically little contact with the outside world. Can you

contact with the outside world. Can you contact with the outside world. Can you explain this one millennial prize explain this one millennial prize explain this one millennial prize problem that's been solved point problem that's been solved point problem that's been solved point conjecture and maybe speak to the conjecture and maybe speak to the conjecture and maybe speak to the journey that Gagora Pearlman's been on. journey that Gagora Pearlman's been on. journey that Gagora Pearlman's been on. All right. So it's it's a question about All right. So it's it's a question about All right. So it's it's a question about curb spaces. Earth is a good example. So curb spaces. Earth is a good example. So curb spaces. Earth is a good example. So you can think of a 2D surface in being you can think of a 2D surface in being you can think of a 2D surface in being round could maybe be a Taurus with a round could maybe be a Taurus with a round could maybe be a Taurus with a hole in it or it can have many holes and hole in it or it can have many holes and hole in it or it can have many holes and there there are many different there there are many different there there are many different topologies up priori that that a surface topologies up priori that that a surface topologies up priori that that a surface could have. um even if you assume that could have. um even if you assume that could have. um even if you assume that it's it's bounded and and uh and smooth it's it's bounded and and uh and smooth it's it's bounded and and uh and smooth and so forth. So we have figured out how and so forth. So we have figured out how and so forth. So we have figured out how to classify surfaces as a first to classify surfaces as a first to classify surfaces as a first approximation everything is determined approximation everything is determined approximation everything is determined by something called the genus how many by something called the genus how many by something called the genus how many holes it has. So a sphere has genus 0 a holes it has. So a sphere has genus 0 a holes it has. So a sphere has genus 0 a donut has genus one and so forth and one donut has genus one and so forth and one donut has genus one and so forth and one way you can tell these surfaces apart way you can tell these surfaces apart way you can tell these surfaces apart probably the sphere has which is called probably the sphere has which is called probably the sphere has which is called simply connected if you take any closed simply connected if you take any closed simply connected if you take any closed loop on the sphere like a big closed loop on the sphere like a big closed loop on the sphere like a big closed little rope you can contract it to a little rope you can contract it to a little rope you can contract it to a point and while staying on the surface point and while staying on the surface point and while staying on the surface and the sphere has this property but a and the sphere has this property but a and the sphere has this property but a taurus doesn't if on a taurus and you taurus doesn't if on a taurus and you taurus doesn't if on a taurus and you take a rope that goes around say the the take a rope that goes around say the the take a rope that goes around say the the outer diameter taurus there's no way it outer diameter taurus there's no way it outer diameter taurus there's no way it can't get through the hole there's no can't get through the hole there's no can't get through the hole there's no way to to contract it to a point so it way to to contract it to a point so it way to to contract it to a point so it turns out that the this the sphere is turns out that the this the sphere is turns out that the this the sphere is the only surface with this property of the only surface with this property of the only surface with this property of contractability up to like continuous contractability up to like continuous contractability up to like continuous deformationations of the sphere. So um deformationations of the sphere. So um deformationations of the sphere. So um things that I want to call topologically things that I want to call topologically things that I want to call topologically um equivalent of the sphere. So point um equivalent of the sphere. So point um equivalent of the sphere. So point asked the same question in higher asked the same question in higher asked the same question in higher dimensions. Um so this it becomes hard dimensions. Um so this it becomes hard dimensions. Um so this it becomes hard to visualize because um surface you can to visualize because um surface you can to visualize because um surface you can think of as embedded in three dimensions think of as embedded in three dimensions think of as embedded in three dimensions but a curved free space we don't have but a curved free space we don't have but a curved free space we don't have good intuition of 4D space to to to live good intuition of 4D space to to to live good intuition of 4D space to to to live and and there are also 3D spaces that

and and there are also 3D spaces that and and there are also 3D spaces that can't even fit into four dimensions. you can't even fit into four dimensions. you can't even fit into four dimensions. you need five or six or or higher. But need five or six or or higher. But need five or six or or higher. But anyway, uh mathematically you can still anyway, uh mathematically you can still anyway, uh mathematically you can still pose this question that if you have a pose this question that if you have a pose this question that if you have a bounded threedimensional space now which bounded threedimensional space now which bounded threedimensional space now which is also has this simply connected is also has this simply connected is also has this simply connected property that every loop can be property that every loop can be property that every loop can be contracted. Can you turn it into a contracted. Can you turn it into a contracted. Can you turn it into a threedimensional version of a sphere? threedimensional version of a sphere? threedimensional version of a sphere? And so this is the point conjecture. And so this is the point conjecture. And so this is the point conjecture. Weirdly in higher dimensions four and Weirdly in higher dimensions four and Weirdly in higher dimensions four and five it was actually easier. So uh it five it was actually easier. So uh it five it was actually easier. So uh it was solved first in higher dimensions. was solved first in higher dimensions. was solved first in higher dimensions. There's somehow more room to do the There's somehow more room to do the There's somehow more room to do the deformation. It's easier to to to move deformation. It's easier to to to move deformation. It's easier to to to move things around to a sphere. But three was things around to a sphere. But three was things around to a sphere. But three was really hard. So people tried many really hard. So people tried many really hard. So people tried many approaches. There sort of commentary approaches. There sort of commentary approaches. There sort of commentary approaches where you chop up the the approaches where you chop up the the approaches where you chop up the the surface into little triangles or or surface into little triangles or or surface into little triangles or or tetrahedra and you you just try to argue tetrahedra and you you just try to argue tetrahedra and you you just try to argue based on how the faces interact each based on how the faces interact each based on how the faces interact each other. Um there were um algebraic other. Um there were um algebraic other. Um there were um algebraic approaches. There's there's various approaches. There's there's various approaches. There's there's various algebraic objects like things called the algebraic objects like things called the algebraic objects like things called the fundamental group that you can attach to fundamental group that you can attach to fundamental group that you can attach to these homology and coology and and and these homology and coology and and and these homology and coology and and and all these very fancy tools. Um they also all these very fancy tools. Um they also all these very fancy tools. Um they also didn't quite work. Um but Richard didn't quite work. Um but Richard didn't quite work. Um but Richard Hamilton's proposed a um partial Hamilton's proposed a um partial Hamilton's proposed a um partial differential equations approach. So you differential equations approach. So you differential equations approach. So you take um you take so the problem is that take um you take so the problem is that take um you take so the problem is that you so you have this object which is so you so you have this object which is so you so you have this object which is so secretly is a sphere but it's given to secretly is a sphere but it's given to secretly is a sphere but it's given to you in a in a really um in in a weird you in a in a really um in in a weird you in a in a really um in in a weird way. So like like think of a ball that's way. So like like think of a ball that's way. So like like think of a ball that's been kind of crumpled up and twisted and been kind of crumpled up and twisted and been kind of crumpled up and twisted and it's not obvious that it's a ball. Um it's not obvious that it's a ball. Um it's not obvious that it's a ball. Um but um like if you if you have some sort but um like if you if you have some sort but um like if you if you have some sort of surface which is which is a deformed of surface which is which is a deformed of surface which is which is a deformed sphere, you could um u you could for sphere, you could um u you could for sphere, you could um u you could for example think of it as a surface of a example think of it as a surface of a example think of it as a surface of a balloon. You could try to inflate it.

balloon. You could try to inflate it. balloon. You could try to inflate it. You you blow it up. Um and naturally as You you blow it up. Um and naturally as You you blow it up. Um and naturally as you fill it with air um the the wrinkles you fill it with air um the the wrinkles you fill it with air um the the wrinkles will sort of smooth out and it will turn will sort of smooth out and it will turn will sort of smooth out and it will turn into um um a nice round sphere. Um into um um a nice round sphere. Um into um um a nice round sphere. Um unless of course it was a Taurus or unless of course it was a Taurus or unless of course it was a Taurus or something in which case it would get something in which case it would get something in which case it would get stuck at some point like if you instead stuck at some point like if you instead stuck at some point like if you instead of Taurus it would there'll be a point of Taurus it would there'll be a point of Taurus it would there'll be a point in the middle when the inner ring in the middle when the inner ring in the middle when the inner ring shrinks to zero you get you get a shrinks to zero you get you get a shrinks to zero you get you get a singularity and you can't blow up any singularity and you can't blow up any singularity and you can't blow up any further. You can't flow any further. So further. You can't flow any further. So further. You can't flow any further. So he created this flow which is called he created this flow which is called he created this flow which is called Richie flow which is a way of taking an Richie flow which is a way of taking an Richie flow which is a way of taking an arbitrary surface or or space and arbitrary surface or or space and arbitrary surface or or space and smoothing it out to make it rounder and smoothing it out to make it rounder and smoothing it out to make it rounder and rounder to make it look like a sphere. rounder to make it look like a sphere. rounder to make it look like a sphere. And he wanted to show that that either And he wanted to show that that either And he wanted to show that that either uh this process would give you a sphere uh this process would give you a sphere uh this process would give you a sphere or it would create a singularity. Um or it would create a singularity. Um or it would create a singularity. Um actually very much like how PDS either actually very much like how PDS either actually very much like how PDS either they have global regularity or finite they have global regularity or finite they have global regularity or finite blow like basically it's almost exactly blow like basically it's almost exactly blow like basically it's almost exactly the same thing. It's all connected. Um the same thing. It's all connected. Um the same thing. It's all connected. Um and so and and he showed that for two and so and and he showed that for two and so and and he showed that for two dimensions two dimensional services dimensions two dimensional services dimensions two dimensional services surfaces um if you started simply surfaces um if you started simply surfaces um if you started simply connected no singularities ever formed connected no singularities ever formed connected no singularities ever formed um you never ran into trouble and you um you never ran into trouble and you um you never ran into trouble and you could flow and it would give you a could flow and it would give you a could flow and it would give you a sphere and it so he he got a new proof sphere and it so he he got a new proof sphere and it so he he got a new proof of the two dimensional result but by the of the two dimensional result but by the of the two dimensional result but by the way that's a beautiful explanation of way that's a beautiful explanation of way that's a beautiful explanation of reach flow and its application in this reach flow and its application in this reach flow and its application in this context how difficult is the mathematics context how difficult is the mathematics context how difficult is the mathematics here like for the 2D case is it yeah here like for the 2D case is it yeah here like for the 2D case is it yeah these are quite sophisticated equations these are quite sophisticated equations these are quite sophisticated equations on par with the Einstein equations on par with the Einstein equations on par with the Einstein equations slightly simpler but um Um yeah but but slightly simpler but um Um yeah but but slightly simpler but um Um yeah but but they were considered hard nonlinear they were considered hard nonlinear they were considered hard nonlinear equations to solve um and there's lots equations to solve um and there's lots equations to solve um and there's lots of special tricks in 2D that that that of special tricks in 2D that that that of special tricks in 2D that that that helped but in 3D the problem was that uh helped but in 3D the problem was that uh helped but in 3D the problem was that uh this equation was actually super this equation was actually super this equation was actually super critical the same problems as Nabia critical the same problems as Nabia critical the same problems as Nabia Stokes as you blow up um maybe the

Stokes as you blow up um maybe the Stokes as you blow up um maybe the curvature could get constraint in finer curvature could get constraint in finer curvature could get constraint in finer smaller smaller regions and it um it smaller smaller regions and it um it smaller smaller regions and it um it looked more and more nonlinear and looked more and more nonlinear and looked more and more nonlinear and things just look worse and worse and things just look worse and worse and things just look worse and worse and there could be all kinds of there could be all kinds of there could be all kinds of singularities that showed up. um some singularities that showed up. um some singularities that showed up. um some singularities um like if there's these singularities um like if there's these singularities um like if there's these things called neck pinches where where things called neck pinches where where things called neck pinches where where the uh the surface sort of creates the uh the surface sort of creates the uh the surface sort of creates behaves like like a like a a barbell and behaves like like a like a a barbell and behaves like like a like a a barbell and it it pinches at a point. Some some it it pinches at a point. Some some it it pinches at a point. Some some singularities are simple enough that you singularities are simple enough that you singularities are simple enough that you can sort of see what to do next. You can sort of see what to do next. You can sort of see what to do next. You just make a snip and then you can turn just make a snip and then you can turn just make a snip and then you can turn one surface into two and evol them one surface into two and evol them one surface into two and evol them separately. But there was there was a separately. But there was there was a separately. But there was there was a the prospect that there's some really the prospect that there's some really the prospect that there's some really nasty like knotted singularities showed nasty like knotted singularities showed nasty like knotted singularities showed up that you you couldn't see how to um up that you you couldn't see how to um up that you you couldn't see how to um resolve in any way that you couldn't do resolve in any way that you couldn't do resolve in any way that you couldn't do any surgery to. Um so you need to any surgery to. Um so you need to any surgery to. Um so you need to classify all the singularities like what classify all the singularities like what classify all the singularities like what are all the possible ways that things are all the possible ways that things are all the possible ways that things can go wrong. Um so what Pearlman did can go wrong. Um so what Pearlman did can go wrong. Um so what Pearlman did was first of all he he made the problem was first of all he he made the problem was first of all he he made the problem he turned the problem a super critical he turned the problem a super critical he turned the problem a super critical problem to a critical problem. Um I said problem to a critical problem. Um I said problem to a critical problem. Um I said before about how um the invention of the before about how um the invention of the before about how um the invention of the of of energy the Hamiltonian like really of of energy the Hamiltonian like really of of energy the Hamiltonian like really clarified um Newtonian mechanics. Um uh clarified um Newtonian mechanics. Um uh clarified um Newtonian mechanics. Um uh so he introduced something which is now so he introduced something which is now so he introduced something which is now called permanence reduced volume and called permanence reduced volume and called permanence reduced volume and permanence entropy. He introduced new permanence entropy. He introduced new permanence entropy. He introduced new quantities kind of like energy that look quantities kind of like energy that look quantities kind of like energy that look the same at every single scale and the same at every single scale and the same at every single scale and turned the problem into a critical one turned the problem into a critical one turned the problem into a critical one where the nonlinearities actually where the nonlinearities actually where the nonlinearities actually suddenly looked a lot less scary than suddenly looked a lot less scary than suddenly looked a lot less scary than they did before. Um and then he had to they did before. Um and then he had to they did before. Um and then he had to solve he still had to analyze the solve he still had to analyze the solve he still had to analyze the singularities of this critical problem.

singularities of this critical problem. singularities of this critical problem. uh and that itself was a problem similar uh and that itself was a problem similar uh and that itself was a problem similar to this wake up thing I worked on to this wake up thing I worked on to this wake up thing I worked on actually um so on the on the level of actually um so on the on the level of actually um so on the on the level of difficulty of that. So he managed to difficulty of that. So he managed to difficulty of that. So he managed to classify all the singularities of this classify all the singularities of this classify all the singularities of this problem and show how to apply surgery to problem and show how to apply surgery to problem and show how to apply surgery to each of these and through that was able each of these and through that was able each of these and through that was able to to resolve the point Cray conjecture. to to resolve the point Cray conjecture. to to resolve the point Cray conjecture. um quite like a lot of really ambitious um quite like a lot of really ambitious um quite like a lot of really ambitious steps um and like like nothing that a steps um and like like nothing that a steps um and like like nothing that a large language model today for example large language model today for example large language model today for example could I mean um at best uh I could could I mean um at best uh I could could I mean um at best uh I could imagine model proposing this idea as one imagine model proposing this idea as one imagine model proposing this idea as one of hundreds of different things to try of hundreds of different things to try of hundreds of different things to try um but the other 99 would be complete um but the other 99 would be complete um but the other 99 would be complete dead ends but you'd only find out after dead ends but you'd only find out after dead ends but you'd only find out after months of work he must have had some months of work he must have had some months of work he must have had some sense that this was the right track to sense that this was the right track to sense that this was the right track to pursue because you know I it takes years pursue because you know I it takes years pursue because you know I it takes years to get them from A to B so you've done to get them from A to B so you've done to get them from A to B so you've done like you said Actually you see even like you said Actually you see even like you said Actually you see even strictly mathematically but more broadly strictly mathematically but more broadly strictly mathematically but more broadly in terms of the process he's done in terms of the process he's done in terms of the process he's done similarly difficult similarly difficult similarly difficult things what what can you infer from the things what what can you infer from the things what what can you infer from the process he was going through because he process he was going through because he process he was going through because he was doing it alone what are some low was doing it alone what are some low was doing it alone what are some low points in a process like that when you points in a process like that when you points in a process like that when you start to like you've mentioned hardship start to like you've mentioned hardship start to like you've mentioned hardship like uh AI doesn't know when it's like uh AI doesn't know when it's like uh AI doesn't know when it's failing what happens to you you're failing what happens to you you're failing what happens to you you're sitting in your office when you realize sitting in your office when you realize sitting in your office when you realize the thing you did for the last few days the thing you did for the last few days the thing you did for the last few days maybe weeks weeks. Yeah. Is a failure.

maybe weeks weeks. Yeah. Is a failure. maybe weeks weeks. Yeah. Is a failure. Well, for me, I switch to different Well, for me, I switch to different Well, for me, I switch to different problem. Uh so, uh as said, I'm I'm a problem. Uh so, uh as said, I'm I'm a problem. Uh so, uh as said, I'm I'm a fox. I'm not a hedgehog. But you fox. I'm not a hedgehog. But you fox. I'm not a hedgehog. But you legitimately that is a break that you legitimately that is a break that you legitimately that is a break that you can take is is to step away and look at can take is is to step away and look at can take is is to step away and look at a different problem. Yeah, you can a different problem. Yeah, you can a different problem. Yeah, you can modify the problem too. Um I mean um modify the problem too. Um I mean um modify the problem too. Um I mean um yeah, you can ask some cheat if if yeah, you can ask some cheat if if yeah, you can ask some cheat if if there's a specific thing that's blocking there's a specific thing that's blocking there's a specific thing that's blocking you that this um some bad case keeps you that this um some bad case keeps you that this um some bad case keeps showing up that that that for which your showing up that that that for which your showing up that that that for which your tool doesn't work, you can just assume tool doesn't work, you can just assume tool doesn't work, you can just assume by fiat this this bad case doesn't by fiat this this bad case doesn't by fiat this this bad case doesn't occur. So you you do some magical occur. So you you do some magical occur. So you you do some magical thinking um for the but but but thinking um for the but but but thinking um for the but but but strategically okay for the point to see strategically okay for the point to see strategically okay for the point to see if the rest of the argument goes through if the rest of the argument goes through if the rest of the argument goes through um if there's multiple problems uh with um if there's multiple problems uh with um if there's multiple problems uh with with with your approach then maybe you with with your approach then maybe you with with your approach then maybe you just give up okay but if this is the just give up okay but if this is the just give up okay but if this is the only problem that you know but only problem that you know but only problem that you know but everything else checks out then it's everything else checks out then it's everything else checks out then it's still worth fighting um so yeah you have still worth fighting um so yeah you have still worth fighting um so yeah you have to do some some sort of forward to do some some sort of forward to do some some sort of forward reconnaissance sometimes to uh you know reconnaissance sometimes to uh you know reconnaissance sometimes to uh you know and that is sometimes productive to and that is sometimes productive to and that is sometimes productive to assume like okay we'll figure it out oh assume like okay we'll figure it out oh assume like okay we'll figure it out oh yeah yeah eventually um Sometimes yeah yeah eventually um Sometimes yeah yeah eventually um Sometimes actually it's even productive to make actually it's even productive to make actually it's even productive to make mistakes. So um one of the I mean um mistakes. So um one of the I mean um mistakes. So um one of the I mean um there was a project which actually u we there was a project which actually u we there was a project which actually u we won some prizes for actually won some prizes for actually won some prizes for actually four other people. Um we worked on this four other people. Um we worked on this four other people. Um we worked on this PD problem again actually this blow of PD problem again actually this blow of PD problem again actually this blow of regularity type problem. Um and it was regularity type problem. Um and it was regularity type problem. Um and it was considered very hard. Um Sean Bain who considered very hard. Um Sean Bain who considered very hard. Um Sean Bain who was another field methodist who worked was another field methodist who worked was another field methodist who worked on a special case of this but he could on a special case of this but he could on a special case of this but he could not solve the general case. Um and we not solve the general case. Um and we not solve the general case. Um and we worked on this problem for two months worked on this problem for two months worked on this problem for two months and we found we thought we solved it. We and we found we thought we solved it. We and we found we thought we solved it. We we had this this cute argument that if we had this this cute argument that if we had this this cute argument that if everything fit and we were excited uh we everything fit and we were excited uh we everything fit and we were excited uh we were planning celebrationally um to all

were planning celebrationally um to all were planning celebrationally um to all get together and have champagne or get together and have champagne or get together and have champagne or something. Um and we started writing it something. Um and we started writing it something. Um and we started writing it up. Um and one of one of us, not me up. Um and one of one of us, not me up. Um and one of one of us, not me actually, but another co-author said, actually, but another co-author said, actually, but another co-author said, "Oh, um in this in this lema here, we um "Oh, um in this in this lema here, we um "Oh, um in this in this lema here, we um we have to estimate these 13 terms that we have to estimate these 13 terms that we have to estimate these 13 terms that that show up in this expansion." And we that show up in this expansion." And we that show up in this expansion." And we estimate 12 of them, but in our notes, I estimate 12 of them, but in our notes, I estimate 12 of them, but in our notes, I can't find the estimation of the 13th. can't find the estimation of the 13th. can't find the estimation of the 13th. Can you can someone supply that? And I Can you can someone supply that? And I Can you can someone supply that? And I said, "Sure, I'll look at this." and said, "Sure, I'll look at this." and said, "Sure, I'll look at this." and actually yeah we didn't cover we actually yeah we didn't cover we actually yeah we didn't cover we completely omitted this term and this completely omitted this term and this completely omitted this term and this term turned out to be worse than the term turned out to be worse than the term turned out to be worse than the other 12 terms put together um in fact other 12 terms put together um in fact other 12 terms put together um in fact we could not estimate this term um and we could not estimate this term um and we could not estimate this term um and we tried for a few more months and all we tried for a few more months and all we tried for a few more months and all different permutations and there was different permutations and there was different permutations and there was always this one thing one term that we always this one thing one term that we always this one thing one term that we could not control um and so like um this could not control um and so like um this could not control um and so like um this was very frustrating um but because we was very frustrating um but because we was very frustrating um but because we had already invested months and months had already invested months and months had already invested months and months of effort into this already um we stuck of effort into this already um we stuck of effort into this already um we stuck at this we we tried increasingly at this we we tried increasingly at this we we tried increasingly desperate things and and crazy things um desperate things and and crazy things um desperate things and and crazy things um and after two is we found an approach and after two is we found an approach and after two is we found an approach which was actually somewhat different by which was actually somewhat different by which was actually somewhat different by quite a bit from our initial um strategy quite a bit from our initial um strategy quite a bit from our initial um strategy which did actually didn't generate these which did actually didn't generate these which did actually didn't generate these problematic terms and and and actually problematic terms and and and actually problematic terms and and and actually solve the problem. So we we solve a solve the problem. So we we solve a solve the problem. So we we solve a problem after 2 years but if we hadn't problem after 2 years but if we hadn't problem after 2 years but if we hadn't had that initial false dawn of nearly had that initial false dawn of nearly had that initial false dawn of nearly solving a problem we would have given up solving a problem we would have given up solving a problem we would have given up by month two or something and and worked by month two or something and and worked by month two or something and and worked on an easier problem. Um yeah if we had on an easier problem. Um yeah if we had on an easier problem. Um yeah if we had known it would take two years not sure known it would take two years not sure known it would take two years not sure we would have started the project. Yeah we would have started the project. Yeah we would have started the project. Yeah sometimes actually having the incorrect sometimes actually having the incorrect sometimes actually having the incorrect you know it's like Columbus New you know it's like Columbus New you know it's like Columbus New incorrect version of measurement of the incorrect version of measurement of the incorrect version of measurement of the size of the earth. He thought he was size of the earth. He thought he was size of the earth. He thought he was going to find a new trade route to India going to find a new trade route to India going to find a new trade route to India or at least that was how he sold it in

or at least that was how he sold it in or at least that was how he sold it in his perspectus. I mean it could be that his perspectus. I mean it could be that his perspectus. I mean it could be that he actually secretly knew but just on he actually secretly knew but just on he actually secretly knew but just on the psychological element. the psychological element. the psychological element. Do you have like emotional or Do you have like emotional or Do you have like emotional or like self-doubt that just overwhelms you like self-doubt that just overwhelms you like self-doubt that just overwhelms you moments like that? You know, because moments like that? You know, because moments like that? You know, because this stuff it feels like math is is so this stuff it feels like math is is so this stuff it feels like math is is so engrossing engrossing engrossing that like it can break you when you like that like it can break you when you like that like it can break you when you like invest so much yourself in the problem invest so much yourself in the problem invest so much yourself in the problem and then it turns out wrong. You could and then it turns out wrong. You could and then it turns out wrong. You could start to start to start to similar way chess has broken some similar way chess has broken some similar way chess has broken some people. Yeah. Um I I think different people. Yeah. Um I I think different people. Yeah. Um I I think different mathematicians have different levels of mathematicians have different levels of mathematicians have different levels of emotional investment in what they do. I emotional investment in what they do. I emotional investment in what they do. I mean I think for some people it's just a mean I think for some people it's just a mean I think for some people it's just a job. you know you you have a problem and job. you know you you have a problem and job. you know you you have a problem and if it doesn't work out you you you go on if it doesn't work out you you you go on if it doesn't work out you you you go on the next one. Um yeah so the fact that the next one. Um yeah so the fact that the next one. Um yeah so the fact that you can always move on to another you can always move on to another you can always move on to another problem um it reduces the emotional problem um it reduces the emotional problem um it reduces the emotional connection. I mean connection. I mean connection. I mean there are cases you know so there are there are cases you know so there are there are cases you know so there are certain problems that are what I call certain problems that are what I call certain problems that are what I call back diseases where where where just back diseases where where where just back diseases where where where just latch on to that one problem and they latch on to that one problem and they latch on to that one problem and they spend years and years thinking about spend years and years thinking about spend years and years thinking about nothing but that one problem and um you nothing but that one problem and um you nothing but that one problem and um you know maybe their career suffers and so know maybe their career suffers and so know maybe their career suffers and so forth but okay this big win this will forth but okay this big win this will forth but okay this big win this will you know once I once I finish this you know once I once I finish this you know once I once I finish this problem I will make up for all the years problem I will make up for all the years problem I will make up for all the years of of of lost opportunity but that's of of of lost opportunity but that's of of of lost opportunity but that's that's I mean occasionally occasionally that's I mean occasionally occasionally that's I mean occasionally occasionally it works But I I um I really don't it works But I I um I really don't it works But I I um I really don't recommend it for people without the the recommend it for people without the the recommend it for people without the the right fortitude. Yeah. So I I've never right fortitude. Yeah. So I I've never right fortitude. Yeah. So I I've never been super invested in any one problem.

been super invested in any one problem. been super invested in any one problem. Um one thing that helps is that we don't Um one thing that helps is that we don't Um one thing that helps is that we don't need to call our problems in advance. Uh need to call our problems in advance. Uh need to call our problems in advance. Uh um well uh when we do grant proposals we um well uh when we do grant proposals we um well uh when we do grant proposals we s say we we will study this set of s say we we will study this set of s say we we will study this set of problems. But even then we don't promise problems. But even then we don't promise problems. But even then we don't promise definitely by 5 years I will supply a definitely by 5 years I will supply a definitely by 5 years I will supply a proof of all these things. you know, or proof of all these things. you know, or proof of all these things. you know, or um you promise to make some progress or um you promise to make some progress or um you promise to make some progress or discover some interesting phenomena. Uh discover some interesting phenomena. Uh discover some interesting phenomena. Uh and maybe you don't solve the problem, and maybe you don't solve the problem, and maybe you don't solve the problem, but you find some related problem that but you find some related problem that but you find some related problem that you you can say something new about and you you can say something new about and you you can say something new about and that's that's a much more feasible task. that's that's a much more feasible task. that's that's a much more feasible task. But I'm sure for you there's problems But I'm sure for you there's problems But I'm sure for you there's problems like this. You have you have like this. You have you have like this. You have you have um made so much progress towards the um made so much progress towards the um made so much progress towards the hardest problems in the history of hardest problems in the history of hardest problems in the history of mathematics. So is there is there a mathematics. So is there is there a mathematics. So is there is there a problem that just haunts you? It sits problem that just haunts you? It sits problem that just haunts you? It sits there in the dark corners, you know, there in the dark corners, you know, there in the dark corners, you know, twin prime conjecture, reman hypothesis, twin prime conjecture, reman hypothesis, twin prime conjecture, reman hypothesis, global conjecture. Twin prime that global conjecture. Twin prime that global conjecture. Twin prime that sounds again. So, I mean, the problem is sounds again. So, I mean, the problem is sounds again. So, I mean, the problem is like a reman hypothesis, those are so like a reman hypothesis, those are so like a reman hypothesis, those are so far out of reach. Why do you think so? far out of reach. Why do you think so? far out of reach. Why do you think so? Yeah. there's no even viable strate like Yeah. there's no even viable strate like Yeah. there's no even viable strate like even if I activate all my all the cheats even if I activate all my all the cheats even if I activate all my all the cheats that I know of in this problem like it that I know of in this problem like it that I know of in this problem like it there's just still no way to get me to there's just still no way to get me to there's just still no way to get me to be um like it's um I think it needs a be um like it's um I think it needs a be um like it's um I think it needs a breakthrough in another area of breakthrough in another area of breakthrough in another area of mathematics to happen first and for mathematics to happen first and for mathematics to happen first and for someone to recognize that it that would someone to recognize that it that would someone to recognize that it that would be a useful thing to transport into this be a useful thing to transport into this be a useful thing to transport into this problem. So we we should maybe step back problem. So we we should maybe step back problem. So we we should maybe step back for a little bit and just talk about for a little bit and just talk about for a little bit and just talk about prime numbers. Okay. So they're often prime numbers. Okay. So they're often prime numbers. Okay. So they're often referred to as the atoms of mathematics.

referred to as the atoms of mathematics. referred to as the atoms of mathematics. Can you just speak to the structure that Can you just speak to the structure that Can you just speak to the structure that these uh atoms the natural numbers have these uh atoms the natural numbers have these uh atoms the natural numbers have two basic operations attached to them? two basic operations attached to them? two basic operations attached to them? Addition and multiplication. Um so if Addition and multiplication. Um so if Addition and multiplication. Um so if you want to generate the natural you want to generate the natural you want to generate the natural numbers, you can do one of two things. numbers, you can do one of two things. numbers, you can do one of two things. You can just start with one and add one You can just start with one and add one You can just start with one and add one to itself over and over again and that to itself over and over again and that to itself over and over again and that generates you the natural numbers. So generates you the natural numbers. So generates you the natural numbers. So additively they're very easy to generate additively they're very easy to generate additively they're very easy to generate 1 2 3 4 5. Or you can take the prime if 1 2 3 4 5. Or you can take the prime if 1 2 3 4 5. Or you can take the prime if you want to generate multiplicatively you want to generate multiplicatively you want to generate multiplicatively you can take all the prime numbers 2 3 you can take all the prime numbers 2 3 you can take all the prime numbers 2 3 57 and multiply them all together. um 57 and multiply them all together. um 57 and multiply them all together. um and together that gives you all the the and together that gives you all the the and together that gives you all the the natural numbers except maybe for one. So natural numbers except maybe for one. So natural numbers except maybe for one. So there these two separate ways of there these two separate ways of there these two separate ways of thinking about the natural numbers from thinking about the natural numbers from thinking about the natural numbers from an additive point of view and point of an additive point of view and point of an additive point of view and point of view. Um and separately they're not so view. Um and separately they're not so view. Um and separately they're not so bad. Um so like any question about that bad. Um so like any question about that bad. Um so like any question about that only was addition is relatively easy to only was addition is relatively easy to only was addition is relatively easy to solve and any question that only was solve and any question that only was solve and any question that only was multiplication is easy to solve. Um but multiplication is easy to solve. Um but multiplication is easy to solve. Um but what has been frustrating is that you what has been frustrating is that you what has been frustrating is that you combine the two together. Um and combine the two together. Um and combine the two together. Um and suddenly you get this extremely rich I suddenly you get this extremely rich I suddenly you get this extremely rich I mean we know that there are statements mean we know that there are statements mean we know that there are statements in number theory that are actually as in number theory that are actually as in number theory that are actually as undecidable. There are certain polomials undecidable. There are certain polomials undecidable. There are certain polomials in some number of variables. You know is in some number of variables. You know is in some number of variables. You know is there a solution in the natural numbers there a solution in the natural numbers there a solution in the natural numbers and the answer depends on on an and the answer depends on on an and the answer depends on on an undecidable statement um like like undecidable statement um like like undecidable statement um like like whether um the aims of of mathematics whether um the aims of of mathematics whether um the aims of of mathematics are consistent or not. Um are consistent or not. Um are consistent or not. Um but um yeah but even this the simplest but um yeah but even this the simplest but um yeah but even this the simplest problems that combine something problems that combine something problems that combine something multiplicative such as the primes with multiplicative such as the primes with multiplicative such as the primes with something additive such as shifting by something additive such as shifting by something additive such as shifting by two uh separately we understand both of two uh separately we understand both of two uh separately we understand both of them well but if you ask when you shift them well but if you ask when you shift them well but if you ask when you shift the prime by two do you can you get a the prime by two do you can you get a the prime by two do you can you get a how often can you get another prime we how often can you get another prime we how often can you get another prime we it's been amazingly hard to relate the

it's been amazingly hard to relate the it's been amazingly hard to relate the two and we should say that the twin two and we should say that the twin two and we should say that the twin prime conjecture is just that it posits prime conjecture is just that it posits prime conjecture is just that it posits that there are infinitely many pairs of that there are infinitely many pairs of that there are infinitely many pairs of prime numbers that differ by do. Yes. prime numbers that differ by do. Yes. prime numbers that differ by do. Yes. Now the interesting thing is that you Now the interesting thing is that you Now the interesting thing is that you have been very successful at pushing have been very successful at pushing have been very successful at pushing forward the field in answering these forward the field in answering these forward the field in answering these complicated questions uh of this variety complicated questions uh of this variety complicated questions uh of this variety like you mentioned the green tile like you mentioned the green tile like you mentioned the green tile theorem. It proves that prime numbers theorem. It proves that prime numbers theorem. It proves that prime numbers contain arithmetic progressions of any contain arithmetic progressions of any contain arithmetic progressions of any length, right? Which is mind-blowing length, right? Which is mind-blowing length, right? Which is mind-blowing that you can prove something like that, that you can prove something like that, that you can prove something like that, right? Yeah. So, what we've realized right? Yeah. So, what we've realized right? Yeah. So, what we've realized because of this this this type of of because of this this this type of of because of this this this type of of research is that there's different research is that there's different research is that there's different patterns have different levels of uh patterns have different levels of uh patterns have different levels of uh indestructibility. Um so, so what makes indestructibility. Um so, so what makes indestructibility. Um so, so what makes the twin prime problem hard is that if the twin prime problem hard is that if the twin prime problem hard is that if you take all the primes in the world, you take all the primes in the world, you take all the primes in the world, you know, 3, 5, 7, 11, so forth, there you know, 3, 5, 7, 11, so forth, there you know, 3, 5, 7, 11, so forth, there are some twins in there. 11 and 13 is a are some twins in there. 11 and 13 is a are some twins in there. 11 and 13 is a twin prime pair of twin primes and so twin prime pair of twin primes and so twin prime pair of twin primes and so forth. But you could easily if you forth. But you could easily if you forth. But you could easily if you wanted to um redact the primes to get wanted to um redact the primes to get wanted to um redact the primes to get rid of to get rid of the um these twins rid of to get rid of the um these twins rid of to get rid of the um these twins like the twins they show up and they're like the twins they show up and they're like the twins they show up and they're infinitely many of them but they're infinitely many of them but they're infinitely many of them but they're actually reasonably sparse. Um not there actually reasonably sparse. Um not there actually reasonably sparse. Um not there there's not I mean initially there's there's not I mean initially there's there's not I mean initially there's quite a few but once you got to the quite a few but once you got to the quite a few but once you got to the millions the trillions they become rarer millions the trillions they become rarer millions the trillions they become rarer and rarer and you could actually just and rarer and you could actually just and rarer and you could actually just you know if if someone was given access you know if if someone was given access you know if if someone was given access to the database of primes you just edit to the database of primes you just edit to the database of primes you just edit out a a few primes here and there they out a a few primes here and there they out a a few primes here and there they could make the trim pan conjure false by could make the trim pan conjure false by could make the trim pan conjure false by just removing like 01% of the primes. or just removing like 01% of the primes. or just removing like 01% of the primes. or something um just well well chosen to to something um just well well chosen to to something um just well well chosen to to um to do this. And so you could present um to do this. And so you could present um to do this. And so you could present a censored database of the primes which

a censored database of the primes which a censored database of the primes which passes all of the statistical tests of passes all of the statistical tests of passes all of the statistical tests of the primes. You know that it it obeys the primes. You know that it it obeys the primes. You know that it it obeys things like the paralle theorem and and things like the paralle theorem and and things like the paralle theorem and and other texts about the primes but doesn't other texts about the primes but doesn't other texts about the primes but doesn't contain any true primes anymore. Um and contain any true primes anymore. Um and contain any true primes anymore. Um and this is a real obstacle for the twin this is a real obstacle for the twin this is a real obstacle for the twin prime conjecture. It means that any prime conjecture. It means that any prime conjecture. It means that any proof strategy to actually find twin proof strategy to actually find twin proof strategy to actually find twin primes in the ecto primes must fail when primes in the ecto primes must fail when primes in the ecto primes must fail when applied to these slightly edited primes. applied to these slightly edited primes. applied to these slightly edited primes. And so it must be some very um subtle And so it must be some very um subtle And so it must be some very um subtle delicate feature of the primes that you delicate feature of the primes that you delicate feature of the primes that you can't just get from like like aggregate can't just get from like like aggregate can't just get from like like aggregate statistical analysis. Okay. So that's statistical analysis. Okay. So that's statistical analysis. Okay. So that's all yeah on the other hand progressions all yeah on the other hand progressions all yeah on the other hand progressions has turned out to be much more robust. has turned out to be much more robust. has turned out to be much more robust. um like you can take the primes and you um like you can take the primes and you um like you can take the primes and you can eliminate 99% of the primes actually can eliminate 99% of the primes actually can eliminate 99% of the primes actually you know and you can take take any 99% you know and you can take take any 99% you know and you can take take any 99% you want and uh it turns out and another you want and uh it turns out and another you want and uh it turns out and another thing we prove is that you still get thing we prove is that you still get thing we prove is that you still get arithmetic progressions um arithmetic arithmetic progressions um arithmetic arithmetic progressions um arithmetic progressions are much you know they're progressions are much you know they're progressions are much you know they're like cockroaches of arbitrary length yes like cockroaches of arbitrary length yes like cockroaches of arbitrary length yes that's crazy I mean so so this for for that's crazy I mean so so this for for that's crazy I mean so so this for for people who don't know arithmetic people who don't know arithmetic people who don't know arithmetic progressions is a sequence of numbers progressions is a sequence of numbers progressions is a sequence of numbers that differ by some fixed amount yeah that differ by some fixed amount yeah that differ by some fixed amount yeah but it's again like it's infinite monkey but it's again like it's infinite monkey but it's again like it's infinite monkey type phenomenon for any fixed length of type phenomenon for any fixed length of type phenomenon for any fixed length of your set. You don't get arbitrary as your set. You don't get arbitrary as your set. You don't get arbitrary as progressions. You only get quite short progressions. You only get quite short progressions. You only get quite short progressions. But you're saying twin progressions. But you're saying twin progressions. But you're saying twin prime is not an infinite monkey prime is not an infinite monkey prime is not an infinite monkey phenomena. I mean, it's a very subtle phenomena. I mean, it's a very subtle phenomena. I mean, it's a very subtle monkey. It's still an infinite monkey monkey. It's still an infinite monkey monkey. It's still an infinite monkey phenomen. Yeah. If the primes were phenomen. Yeah. If the primes were phenomen. Yeah. If the primes were really genuinely random, if the primes really genuinely random, if the primes really genuinely random, if the primes were generated by monkeys, um then yes, were generated by monkeys, um then yes, were generated by monkeys, um then yes, in fact, the infinite monkey theorem in fact, the infinite monkey theorem in fact, the infinite monkey theorem would Oh, but you're saying that twin would Oh, but you're saying that twin would Oh, but you're saying that twin prime is it doesn't you can't use the

prime is it doesn't you can't use the prime is it doesn't you can't use the same tools like the it doesn't appear same tools like the it doesn't appear same tools like the it doesn't appear random almost. Well, we don't know. random almost. Well, we don't know. random almost. Well, we don't know. Yeah, we we we we believe the prior Yeah, we we we we believe the prior Yeah, we we we we believe the prior behave like a random set. And so the behave like a random set. And so the behave like a random set. And so the reason why we care about the trim reason why we care about the trim reason why we care about the trim conjecture is is a test case for whether conjecture is is a test case for whether conjecture is is a test case for whether we can genuinely confidently say with we can genuinely confidently say with we can genuinely confidently say with with 0% chance of error that the primes with 0% chance of error that the primes with 0% chance of error that the primes behave like a random set. Okay. Random. behave like a random set. Okay. Random. behave like a random set. Okay. Random. Yeah. Random versions of the primes we Yeah. Random versions of the primes we Yeah. Random versions of the primes we know contain twins. Um at least with know contain twins. Um at least with know contain twins. Um at least with with 100% probability or probably with 100% probability or probably with 100% probability or probably tending to 100% as you go out further tending to 100% as you go out further tending to 100% as you go out further and further. Um yeah. So the primes we and further. Um yeah. So the primes we and further. Um yeah. So the primes we believe that they're random. Um the believe that they're random. Um the believe that they're random. Um the reason why arithmetic progressions are reason why arithmetic progressions are reason why arithmetic progressions are indestructible is that regardless of indestructible is that regardless of indestructible is that regardless of whether you looks random or looks um whether you looks random or looks um whether you looks random or looks um structured like periodic in both cases structured like periodic in both cases structured like periodic in both cases um arithmetic regressions appear but for um arithmetic regressions appear but for um arithmetic regressions appear but for different reasons. Um and this is different reasons. Um and this is different reasons. Um and this is basically all the ways in which the basically all the ways in which the basically all the ways in which the there are many proofs of of these sort there are many proofs of of these sort there are many proofs of of these sort of arithmetic region epithems and of arithmetic region epithems and of arithmetic region epithems and they're all proven by some sort of they're all proven by some sort of they're all proven by some sort of dichotomy where your set is either dichotomy where your set is either dichotomy where your set is either structured or random and in both cases structured or random and in both cases structured or random and in both cases you can say something and then you put you can say something and then you put you can say something and then you put the two together. Um but in twin primes the two together. Um but in twin primes the two together. Um but in twin primes if if the primes are random then you're if if the primes are random then you're if if the primes are random then you're happy you win. But if your primes are happy you win. But if your primes are happy you win. But if your primes are structured they could be structured in structured they could be structured in structured they could be structured in in a specific way that eliminates the in a specific way that eliminates the in a specific way that eliminates the twin the twins. Uh and we can't rule out twin the twins. Uh and we can't rule out twin the twins. Uh and we can't rule out that one conspiracy and yet you were that one conspiracy and yet you were that one conspiracy and yet you were able to make a as I understand progress able to make a as I understand progress able to make a as I understand progress on the Kupal version. Right. Yeah. So um on the Kupal version. Right. Yeah. So um on the Kupal version. Right. Yeah. So um the one funny thing about conspiracies the one funny thing about conspiracies the one funny thing about conspiracies is that any one conspiracy theory is is that any one conspiracy theory is is that any one conspiracy theory is really hard to disprove that you know if really hard to disprove that you know if really hard to disprove that you know if if you believe the water is won by if you believe the water is won by if you believe the water is won by lizards you say here's some evidence lizards you say here's some evidence lizards you say here's some evidence that that it it's not run by lizards

that that it it's not run by lizards that that it it's not run by lizards well that that evidence was planted by well that that evidence was planted by well that that evidence was planted by the lizards. Yeah. Right. You may have the lizards. Yeah. Right. You may have the lizards. Yeah. Right. You may have encountered this kind of phenomen. Yeah. encountered this kind of phenomen. Yeah. encountered this kind of phenomen. Yeah. So like like um a pure like there's So like like um a pure like there's So like like um a pure like there's there's almost no way to um definitively there's almost no way to um definitively there's almost no way to um definitively rule out a con and the same is true in rule out a con and the same is true in rule out a con and the same is true in mathematics that a con is to solely mathematics that a con is to solely mathematics that a con is to solely devote devoted to learning twin primes devote devoted to learning twin primes devote devoted to learning twin primes you know like it would you have to also you know like it would you have to also you know like it would you have to also infiltrate other areas of mathematics to infiltrate other areas of mathematics to infiltrate other areas of mathematics to sort of but but like it could be made sort of but but like it could be made sort of but but like it could be made consistent at least as far as we know consistent at least as far as we know consistent at least as far as we know but there's a weird phenomenon that you but there's a weird phenomenon that you but there's a weird phenomenon that you can make one um one conspiracy rule out can make one um one conspiracy rule out can make one um one conspiracy rule out other conspiracies so you know if the if other conspiracies so you know if the if other conspiracies so you know if the if the world is is run by lizardist it the world is is run by lizardist it the world is is run by lizardist it can't also be run by Right. can't also be run by Right. can't also be run by Right. Right. So one unreasonable thing is is Right. So one unreasonable thing is is Right. So one unreasonable thing is is is is hard to dispute but but more than is is hard to dispute but but more than is is hard to dispute but but more than one there are there are tools. Um so one there are there are tools. Um so one there are there are tools. Um so yeah so for example we we know there's yeah so for example we we know there's yeah so for example we we know there's infinitely many primes that are um no infinitely many primes that are um no infinitely many primes that are um no two which are um so there infinite pair two which are um so there infinite pair two which are um so there infinite pair of primes which differ by at most um 246 of primes which differ by at most um 246 of primes which differ by at most um 246 actually is is a is the current. So actually is is a is the current. So actually is is a is the current. So there's like a bound yes on the right. there's like a bound yes on the right. there's like a bound yes on the right. So like there's twin primes this thing So like there's twin primes this thing So like there's twin primes this thing called cousin primes that differ by by called cousin primes that differ by by called cousin primes that differ by by four. Um there's called sexy primes that four. Um there's called sexy primes that four. Um there's called sexy primes that differ by six. Uh, what are sexy primes?

differ by six. Uh, what are sexy primes? differ by six. Uh, what are sexy primes? Primes that differ by six. The name is Primes that differ by six. The name is Primes that differ by six. The name is much less the concept is much less much less the concept is much less much less the concept is much less exciting than the name suggests. Got it. exciting than the name suggests. Got it. exciting than the name suggests. Got it. Um, so you can make a conspiracy rule Um, so you can make a conspiracy rule Um, so you can make a conspiracy rule out one of these, but like once you have out one of these, but like once you have out one of these, but like once you have like 50 of them, it turns out that you like 50 of them, it turns out that you like 50 of them, it turns out that you can't rule out all of them at once. It can't rule out all of them at once. It can't rule out all of them at once. It just it requires too much energy somehow just it requires too much energy somehow just it requires too much energy somehow in this conspiracy space. How do you do in this conspiracy space. How do you do in this conspiracy space. How do you do the bound part? How do you how do you the bound part? How do you how do you the bound part? How do you how do you develop a bound for the difference develop a bound for the difference develop a bound for the difference between the prize that okay so um that between the prize that okay so um that between the prize that okay so um that there's an infinite number of so it's there's an infinite number of so it's there's an infinite number of so it's ultimately based on what's called the ultimately based on what's called the ultimately based on what's called the pigeon hole principle um so the pigeon pigeon hole principle um so the pigeon pigeon hole principle um so the pigeon hole principle uh it's a statement that hole principle uh it's a statement that hole principle uh it's a statement that if you have a number of pigeons and they if you have a number of pigeons and they if you have a number of pigeons and they all have to go into into pigeon holes all have to go into into pigeon holes all have to go into into pigeon holes and you have more pigeons than pigeon and you have more pigeons than pigeon and you have more pigeons than pigeon holes then one of the pigeon holes has holes then one of the pigeon holes has holes then one of the pigeon holes has to have at least two pigeons in so there to have at least two pigeons in so there to have at least two pigeons in so there has to be two pigeons that that are has to be two pigeons that that are has to be two pigeons that that are close together. So for instance if you close together. So for instance if you close together. So for instance if you have 100 numbers and they all range from have 100 numbers and they all range from have 100 numbers and they all range from one to a thousand um two of them have to one to a thousand um two of them have to one to a thousand um two of them have to be at most 10 apart. Mhm. because you be at most 10 apart. Mhm. because you be at most 10 apart. Mhm. because you can divide up the numbers one to 100 can divide up the numbers one to 100 can divide up the numbers one to 100 into 100 pigeon holes. Let's let's say into 100 pigeon holes. Let's let's say into 100 pigeon holes. Let's let's say you have if you have 101 numbers 100 one you have if you have 101 numbers 100 one you have if you have 101 numbers 100 one numbers then two of them have to be numbers then two of them have to be numbers then two of them have to be distance less than 10 apart because two distance less than 10 apart because two distance less than 10 apart because two of them have to belong to the same of them have to belong to the same of them have to belong to the same pigeon hole. So it's a basic um basic pigeon hole. So it's a basic um basic pigeon hole. So it's a basic um basic feature of uh a basic principle in feature of uh a basic principle in feature of uh a basic principle in mathematics. Um so it doesn't quite work mathematics. Um so it doesn't quite work mathematics. Um so it doesn't quite work with the primes directly because the with the primes directly because the with the primes directly because the primes get sparer and sparser as you go primes get sparer and sparser as you go primes get sparer and sparser as you go out that fewer and fewer numbers are out that fewer and fewer numbers are out that fewer and fewer numbers are prime. But it turns out that there's a prime. But it turns out that there's a prime. But it turns out that there's a way to assign weights to the to to way to assign weights to the to to way to assign weights to the to to numbers like um so there are numbers numbers like um so there are numbers numbers like um so there are numbers that are kind of almost prime but that are kind of almost prime but that are kind of almost prime but they're not they they don't have no they're not they they don't have no they're not they they don't have no factors at all other than themselves in factors at all other than themselves in factors at all other than themselves in one but they have very few factors. Um one but they have very few factors. Um one but they have very few factors. Um and it turns out that we understand

and it turns out that we understand and it turns out that we understand almost primes a lot better than primes. almost primes a lot better than primes. almost primes a lot better than primes. Um and so for example it was known for a Um and so for example it was known for a Um and so for example it was known for a long time that there were twin almost long time that there were twin almost long time that there were twin almost primes. This has been worked out. So primes. This has been worked out. So primes. This has been worked out. So almost primes are something we can't almost primes are something we can't almost primes are something we can't understand. So you can actually restrict understand. So you can actually restrict understand. So you can actually restrict attention to a a suitable set of almost attention to a a suitable set of almost attention to a a suitable set of almost primes and uh whereas the primes are primes and uh whereas the primes are primes and uh whereas the primes are very sparse overall relative to the very sparse overall relative to the very sparse overall relative to the almost primes actually are much less almost primes actually are much less almost primes actually are much less sparse. They make um you can set up a sparse. They make um you can set up a sparse. They make um you can set up a set of almost primes where the primes set of almost primes where the primes set of almost primes where the primes have density like say 1%. Um and that have density like say 1%. Um and that have density like say 1%. Um and that gives you a shot at proving by applying gives you a shot at proving by applying gives you a shot at proving by applying some sort of original principle that some sort of original principle that some sort of original principle that that those pairs of primes are just only that those pairs of primes are just only that those pairs of primes are just only 100 100 apart. But in order to with the 100 100 apart. But in order to with the 100 100 apart. But in order to with the twin pan conjecture you need to get the twin pan conjecture you need to get the twin pan conjecture you need to get the density of primes inside the also size density of primes inside the also size density of primes inside the also size up to up to a first of 50%. Um once you up to up to a first of 50%. Um once you up to up to a first of 50%. Um once you get up to 50% you will get twin primes. get up to 50% you will get twin primes. get up to 50% you will get twin primes. But uh unfortunately there are barriers. But uh unfortunately there are barriers. But uh unfortunately there are barriers. Um we know that that no matter what kind Um we know that that no matter what kind Um we know that that no matter what kind of good set of almost primes you pick of good set of almost primes you pick of good set of almost primes you pick the density primes can never get above the density primes can never get above the density primes can never get above 50%. It's called the parody barrier. Um 50%. It's called the parody barrier. Um 50%. It's called the parody barrier. Um and I would love to find yes. So one of and I would love to find yes. So one of and I would love to find yes. So one of my long-term dreams is to find a way to my long-term dreams is to find a way to my long-term dreams is to find a way to breach that barrier because it would breach that barrier because it would breach that barrier because it would open up not only the trip conjecture the open up not only the trip conjecture the open up not only the trip conjecture the go back conjecture and many other go back conjecture and many other go back conjecture and many other problems in number theory are currently problems in number theory are currently problems in number theory are currently blocked because our current techniques blocked because our current techniques blocked because our current techniques would require improve going beyond this would require improve going beyond this would require improve going beyond this theoretical um parody barriers. It's theoretical um parody barriers. It's theoretical um parody barriers. It's like it's like pulling past the speed of like it's like pulling past the speed of like it's like pulling past the speed of light. Yeah. So we just say a twin prime light. Yeah. So we just say a twin prime light. Yeah. So we just say a twin prime conjecture one of the biggest problems conjecture one of the biggest problems conjecture one of the biggest problems in the history of mathematics go by in the history of mathematics go by in the history of mathematics go by conjecture also um they feel like conjecture also um they feel like conjecture also um they feel like nextdoor neighbors. Uh has there been nextdoor neighbors. Uh has there been nextdoor neighbors. Uh has there been days when you felt you saw the path? Oh days when you felt you saw the path? Oh days when you felt you saw the path? Oh yeah. Um um yeah uh sometimes you try

yeah. Um um yeah uh sometimes you try yeah. Um um yeah uh sometimes you try something and it it works super well. Um something and it it works super well. Um something and it it works super well. Um you you again again the sense of methac you you again again the sense of methac you you again again the sense of methac smell uh we talked about earlier uh you smell uh we talked about earlier uh you smell uh we talked about earlier uh you learn from experience when things are learn from experience when things are learn from experience when things are going too well going too well going too well because there are certain difficulties because there are certain difficulties because there are certain difficulties that you sort of have to encounter. Um that you sort of have to encounter. Um that you sort of have to encounter. Um um I think the way a colleague might put um I think the way a colleague might put um I think the way a colleague might put it is that um you know like if if you it is that um you know like if if you it is that um you know like if if you are on the streets in New York and you are on the streets in New York and you are on the streets in New York and you put in a blindfold and you put in a car put in a blindfold and you put in a car put in a blindfold and you put in a car and and um after some hours um you the and and um after some hours um you the and and um after some hours um you the blindfold's off and you're in Beijing. blindfold's off and you're in Beijing. blindfold's off and you're in Beijing. Um you know I mean that was too easy Um you know I mean that was too easy Um you know I mean that was too easy somehow like like there was no ocean somehow like like there was no ocean somehow like like there was no ocean being crossed. Even if you don't know being crossed. Even if you don't know being crossed. Even if you don't know exactly what how what what was done exactly what how what what was done exactly what how what what was done you're suspecting that that something you're suspecting that that something you're suspecting that that something wasn't right. But is that still in the wasn't right. But is that still in the wasn't right. But is that still in the back of your head to do you return to back of your head to do you return to back of your head to do you return to these to the prime do you return to the these to the prime do you return to the these to the prime do you return to the prime numbers every once in a while to prime numbers every once in a while to prime numbers every once in a while to see yeah when I have nothing better to see yeah when I have nothing better to see yeah when I have nothing better to do which is less and less tired which is do which is less and less tired which is do which is less and less tired which is I get busy with so many things these I get busy with so many things these I get busy with so many things these days but yeah when I have free time and days but yeah when I have free time and days but yeah when I have free time and I'm not and I'm too frustrated to to I'm not and I'm too frustrated to to I'm not and I'm too frustrated to to work on my sort of real research work on my sort of real research work on my sort of real research projects and I also don't want to do my projects and I also don't want to do my projects and I also don't want to do my administrative stuff I don't want to do administrative stuff I don't want to do administrative stuff I don't want to do some errands for my family um I can play some errands for my family um I can play some errands for my family um I can play with these these things um for fun uh with these these things um for fun uh with these these things um for fun uh and usually you get nowhere Yeah, you and usually you get nowhere Yeah, you and usually you get nowhere Yeah, you have you have to learn to just say okay have you have to learn to just say okay have you have to learn to just say okay fine once again nothing happened I I fine once again nothing happened I I fine once again nothing happened I I will move on. Um yeah very occasionally will move on. Um yeah very occasionally will move on. Um yeah very occasionally one of these problems I actually solved one of these problems I actually solved one of these problems I actually solved or sometimes as you say you think you or sometimes as you say you think you or sometimes as you say you think you solved it and then you're euphoric for solved it and then you're euphoric for solved it and then you're euphoric for maybe 15 minutes and then you think I maybe 15 minutes and then you think I maybe 15 minutes and then you think I should check this because this is too should check this because this is too should check this because this is too easy too good to be true and it usually easy too good to be true and it usually easy too good to be true and it usually is. What's your gut say about when these

is. What's your gut say about when these is. What's your gut say about when these problems would be uh solved when prime problems would be uh solved when prime problems would be uh solved when prime and go back? Prime I think we'll keep and go back? Prime I think we'll keep and go back? Prime I think we'll keep getting keep getting more partial getting keep getting more partial getting keep getting more partial results. Um results. Um results. Um it does need at least one this parody it does need at least one this parody it does need at least one this parody barrier is is the biggest remaining barrier is is the biggest remaining barrier is is the biggest remaining obstacle. Um there are simpler versions obstacle. Um there are simpler versions obstacle. Um there are simpler versions of the conjecture where we are getting of the conjecture where we are getting of the conjecture where we are getting really close. Um so I think we will in really close. Um so I think we will in really close. Um so I think we will in 10 years we will have many more much 10 years we will have many more much 10 years we will have many more much closer results. May not have the whole closer results. May not have the whole closer results. May not have the whole thing. Um yeah so trens is somewhat thing. Um yeah so trens is somewhat thing. Um yeah so trens is somewhat close reman hypothesis I have no I mean close reman hypothesis I have no I mean close reman hypothesis I have no I mean it has to happen by accident I think so it has to happen by accident I think so it has to happen by accident I think so the reman hypothesis is a kind of more the reman hypothesis is a kind of more the reman hypothesis is a kind of more general conjecture about the general conjecture about the general conjecture about the distribution of prime numbers right yeah distribution of prime numbers right yeah distribution of prime numbers right yeah it's it's states are sort of viewed it's it's states are sort of viewed it's it's states are sort of viewed multiplicatively like for questions only multiplicatively like for questions only multiplicatively like for questions only involving multiplication no addition the involving multiplication no addition the involving multiplication no addition the primes really do behave as randomly as primes really do behave as randomly as primes really do behave as randomly as as you could hope so there's a as you could hope so there's a as you could hope so there's a phenomenon in probability called square phenomenon in probability called square phenomenon in probability called square root cancellation that um you know like root cancellation that um you know like root cancellation that um you know like if you want to poll say America upon on if you want to poll say America upon on if you want to poll say America upon on on some issue. Um, and you you ask one on some issue. Um, and you you ask one on some issue. Um, and you you ask one or two voters and you may have sampled a or two voters and you may have sampled a or two voters and you may have sampled a bad sample and then you get you get a bad sample and then you get you get a bad sample and then you get you get a really imprecise um measurement of of really imprecise um measurement of of really imprecise um measurement of of the full average, but if you sample more the full average, but if you sample more the full average, but if you sample more and more people, the accuracy gets and more people, the accuracy gets and more people, the accuracy gets better and better and it actually better and better and it actually better and better and it actually improves like the square root of the improves like the square root of the improves like the square root of the number of people you you sample. So number of people you you sample. So number of people you you sample. So yeah, if you sample a thousand people, yeah, if you sample a thousand people, yeah, if you sample a thousand people, you can get like a 2 3% margin of error.

you can get like a 2 3% margin of error. you can get like a 2 3% margin of error. So in the same sense if you measure the So in the same sense if you measure the So in the same sense if you measure the primes in a certain multiplicative sense primes in a certain multiplicative sense primes in a certain multiplicative sense there's a certain type of statistic you there's a certain type of statistic you there's a certain type of statistic you can measure and it's called the reman's can measure and it's called the reman's can measure and it's called the reman's data function and it fluctuates up and data function and it fluctuates up and data function and it fluctuates up and down but in some sense um as you keep down but in some sense um as you keep down but in some sense um as you keep averaging more and more if you sample averaging more and more if you sample averaging more and more if you sample and more and more the fluctuation should and more and more the fluctuation should and more and more the fluctuation should go down as if they were random and go down as if they were random and go down as if they were random and there's a very precise way to quantify there's a very precise way to quantify there's a very precise way to quantify that and the reman hypothesis is a very that and the reman hypothesis is a very that and the reman hypothesis is a very elegant way that captures this but um as elegant way that captures this but um as elegant way that captures this but um as with many others in mathematics we have with many others in mathematics we have with many others in mathematics we have very few tools to show that something very few tools to show that something very few tools to show that something really genuinely behaves like really really genuinely behaves like really really genuinely behaves like really random And this is actually not just a random And this is actually not just a random And this is actually not just a little bit random but it's it's asking little bit random but it's it's asking little bit random but it's it's asking that it behaves as random as it actually that it behaves as random as it actually that it behaves as random as it actually random set this this square root random set this this square root random set this this square root cancellation and we know actually cancellation and we know actually cancellation and we know actually because of things related to the parity because of things related to the parity because of things related to the parity problem actually that most of us usual problem actually that most of us usual problem actually that most of us usual techniques cannot hope to settle this techniques cannot hope to settle this techniques cannot hope to settle this question. Um the proof has to come out question. Um the proof has to come out question. Um the proof has to come out of left field. Um of left field. Um of left field. Um yeah but uh what that is yeah no one has yeah but uh what that is yeah no one has yeah but uh what that is yeah no one has any serious proposal. Um yeah and and any serious proposal. Um yeah and and any serious proposal. Um yeah and and there's there's various ways to sort of there's there's various ways to sort of there's there's various ways to sort of as I said you can modify the primes a as I said you can modify the primes a as I said you can modify the primes a little bit and you can destroy the human little bit and you can destroy the human little bit and you can destroy the human hypothesis. Um so like it has to be very hypothesis. Um so like it has to be very hypothesis. Um so like it has to be very delicate. You can't apply something that delicate. You can't apply something that delicate. You can't apply something that has huge margins of error. It has to has huge margins of error. It has to has huge margins of error. It has to just barely work. Um and like um there's just barely work. Um and like um there's just barely work. Um and like um there's like all these pits pitfalls that you like all these pits pitfalls that you like all these pits pitfalls that you have like dodge very adeptly. The prime have like dodge very adeptly. The prime have like dodge very adeptly. The prime numbers are just fascinating. Yeah.

numbers are just fascinating. Yeah. numbers are just fascinating. Yeah. Yeah. What what to you is um most Yeah. What what to you is um most Yeah. What what to you is um most mysterious about the prime numbers. mysterious about the prime numbers. mysterious about the prime numbers. So that's a good question. So like So that's a good question. So like So that's a good question. So like conjecturally we have a good model of conjecturally we have a good model of conjecturally we have a good model of them. I mean like as I said I mean they them. I mean like as I said I mean they them. I mean like as I said I mean they have certain patterns like the primes have certain patterns like the primes have certain patterns like the primes are usually odd for instance but apart are usually odd for instance but apart are usually odd for instance but apart from this of obvious patterns they from this of obvious patterns they from this of obvious patterns they behave very randomly and just assuming behave very randomly and just assuming behave very randomly and just assuming that they behave so there's something that they behave so there's something that they behave so there's something called the crema random model of the called the crema random model of the called the crema random model of the primes that that after a certain point primes that that after a certain point primes that that after a certain point primes just behave like a random set. Um primes just behave like a random set. Um primes just behave like a random set. Um and there's various slight modifications and there's various slight modifications and there's various slight modifications this model but this has been a very good this model but this has been a very good this model but this has been a very good model. It matches the numeric. It tells model. It matches the numeric. It tells model. It matches the numeric. It tells us what to predict. Like I can tell you us what to predict. Like I can tell you us what to predict. Like I can tell you with complete certainty the truth is with complete certainty the truth is with complete certainty the truth is true. Uh the random model gives true. Uh the random model gives true. Uh the random model gives overwhelming odds it is true. I just overwhelming odds it is true. I just overwhelming odds it is true. I just can't prove it. Most of our mathematics can't prove it. Most of our mathematics can't prove it. Most of our mathematics is optimized for solving things with is optimized for solving things with is optimized for solving things with patterns in them. Um and the primes have patterns in them. Um and the primes have patterns in them. Um and the primes have this anti-attern um as do almost this anti-attern um as do almost this anti-attern um as do almost everything really. But we can't prove everything really. But we can't prove everything really. But we can't prove that. Yeah. I guess it's not mysterious that. Yeah. I guess it's not mysterious that. Yeah. I guess it's not mysterious that the prize be kind of random because that the prize be kind of random because that the prize be kind of random because there no reason for them to be um uh to there no reason for them to be um uh to there no reason for them to be um uh to have any kind of secret pattern but what have any kind of secret pattern but what have any kind of secret pattern but what is mysterious is what is the mechanism is mysterious is what is the mechanism is mysterious is what is the mechanism that really forces the randomness to that really forces the randomness to that really forces the randomness to happen. Uh and this is just absent.

happen. Uh and this is just absent. happen. Uh and this is just absent. Another incredibly surprisingly Another incredibly surprisingly Another incredibly surprisingly difficult problem is the colots's difficult problem is the colots's difficult problem is the colots's conjecture. Oh yes. simple to state, conjecture. Oh yes. simple to state, conjecture. Oh yes. simple to state, beautiful to visualize in its simplicity beautiful to visualize in its simplicity beautiful to visualize in its simplicity and yet extremely and yet extremely and yet extremely uh difficult to solve and yet you have uh difficult to solve and yet you have uh difficult to solve and yet you have been able to make progress. Uh Paular been able to make progress. Uh Paular been able to make progress. Uh Paular said about the coloss conjecture that said about the coloss conjecture that said about the coloss conjecture that mathematics may not be ready for such mathematics may not be ready for such mathematics may not be ready for such problems. Others have stated that it is problems. Others have stated that it is problems. Others have stated that it is an extraordinarily difficult problem an extraordinarily difficult problem an extraordinarily difficult problem completely out of reach this is in 2010 completely out of reach this is in 2010 completely out of reach this is in 2010 out of reach of present- day mathematics out of reach of present- day mathematics out of reach of present- day mathematics and yet you have made some progress. Why and yet you have made some progress. Why and yet you have made some progress. Why is it so difficult to make? Can you is it so difficult to make? Can you is it so difficult to make? Can you actually even explain what it is? Oh, actually even explain what it is? Oh, actually even explain what it is? Oh, yeah. So, it's it's it's a problem that yeah. So, it's it's it's a problem that yeah. So, it's it's it's a problem that you can explain. Um yeah, it um it helps you can explain. Um yeah, it um it helps you can explain. Um yeah, it um it helps with some um visual aids, but yeah, so with some um visual aids, but yeah, so with some um visual aids, but yeah, so you take any natural number like say 13. you take any natural number like say 13. you take any natural number like say 13. And you apply the the following And you apply the the following And you apply the the following procedure to it. So, if it's even, you procedure to it. So, if it's even, you procedure to it. So, if it's even, you divide it by two and if it's odd, you divide it by two and if it's odd, you divide it by two and if it's odd, you multiply by three and add one. So, even multiply by three and add one. So, even multiply by three and add one. So, even numbers get smaller, odd numbers get numbers get smaller, odd numbers get numbers get smaller, odd numbers get bigger. So, 13 will become 40 because 13 bigger. So, 13 will become 40 because 13 bigger. So, 13 will become 40 because 13 * 3 is 39. Add one, you get 40. So, it's * 3 is 39. Add one, you get 40. So, it's * 3 is 39. Add one, you get 40. So, it's a simple process for odd numbers and a simple process for odd numbers and a simple process for odd numbers and even numbers. They're both very easy even numbers. They're both very easy even numbers. They're both very easy operations. And then you put it operations. And then you put it operations. And then you put it together. It's still reasonably simple.

together. It's still reasonably simple. together. It's still reasonably simple. Um, but then you ask what happens when Um, but then you ask what happens when Um, but then you ask what happens when you iterate it. You take the output that you iterate it. You take the output that you iterate it. You take the output that you just got and feed it back in. So, 13 you just got and feed it back in. So, 13 you just got and feed it back in. So, 13 becomes 40. 40 is now even divide by 2 becomes 40. 40 is now even divide by 2 becomes 40. 40 is now even divide by 2 is 20. 20 is still even divide by 10 2 is 20. 20 is still even divide by 10 2 is 20. 20 is still even divide by 10 2 10 5 and then 5 * 3 + 1 is 16. And then 10 5 and then 5 * 3 + 1 is 16. And then 10 5 and then 5 * 3 + 1 is 16. And then 8 4 2 1. So, uh, and then from 1 it goes 8 4 2 1. So, uh, and then from 1 it goes 8 4 2 1. So, uh, and then from 1 it goes 1 4 2 1 421. It cycles forever. So this 1 4 2 1 421. It cycles forever. So this 1 4 2 1 421. It cycles forever. So this sequence I just described um yeah 13 40 sequence I just described um yeah 13 40 sequence I just described um yeah 13 40 20 10 so these are also called hailstone 20 10 so these are also called hailstone 20 10 so these are also called hailstone sequences because there's an sequences because there's an sequences because there's an oversimplified model of of hailstone oversimplified model of of hailstone oversimplified model of of hailstone formation yeah which is not actually formation yeah which is not actually formation yeah which is not actually quite correct but it's so somehow taught quite correct but it's so somehow taught quite correct but it's so somehow taught to high school students as a first to high school students as a first to high school students as a first approximation is that um like a a little approximation is that um like a a little approximation is that um like a a little nugget of ice gets gets an ice crystal nugget of ice gets gets an ice crystal nugget of ice gets gets an ice crystal forms in a cloud and it it goes up and forms in a cloud and it it goes up and forms in a cloud and it it goes up and down because of the wind and sometimes down because of the wind and sometimes down because of the wind and sometimes when it's cold it get acquires a bit when it's cold it get acquires a bit when it's cold it get acquires a bit more mass and maybe it melts a little more mass and maybe it melts a little more mass and maybe it melts a little bit and this process of going up down bit and this process of going up down bit and this process of going up down creates this s of partially melted ice creates this s of partially melted ice creates this s of partially melted ice which event hell stone and eventually it which event hell stone and eventually it which event hell stone and eventually it falls out the earth. So the conjecture falls out the earth. So the conjecture falls out the earth. So the conjecture is that no matter how high you start up is that no matter how high you start up is that no matter how high you start up like you take a number which is in the like you take a number which is in the like you take a number which is in the millions or billions you go this process millions or billions you go this process millions or billions you go this process that that goes up if you're odd and down that that goes up if you're odd and down that that goes up if you're odd and down if you're even eventually um goes down if you're even eventually um goes down if you're even eventually um goes down to to earth all the time no matter where to to earth all the time no matter where to to earth all the time no matter where you start with this very simple you start with this very simple you start with this very simple algorithm you end up at one and you algorithm you end up at one and you algorithm you end up at one and you might climb for a while right yeah so might climb for a while right yeah so might climb for a while right yeah so yeah if you plot it um these sequences yeah if you plot it um these sequences yeah if you plot it um these sequences they look like brownie in motion um they they look like brownie in motion um they they look like brownie in motion um they look like the stock market you know they look like the stock market you know they look like the stock market you know they just go up and down in a in a seemingly just go up and down in a in a seemingly just go up and down in a in a seemingly random pattern and in Usually that's random pattern and in Usually that's random pattern and in Usually that's what happens that that if you plug in a what happens that that if you plug in a what happens that that if you plug in a random number, you can actually prove at random number, you can actually prove at random number, you can actually prove at least initially that it would look like

least initially that it would look like least initially that it would look like um random walk. Um and that's actually a um random walk. Um and that's actually a um random walk. Um and that's actually a random walk with a downward drift. Um random walk with a downward drift. Um random walk with a downward drift. Um it's like if you're always gambling on it's like if you're always gambling on it's like if you're always gambling on on roulette at at the casino with odds on roulette at at the casino with odds on roulette at at the casino with odds slightly weighted against you. So slightly weighted against you. So slightly weighted against you. So sometimes you you win, sometimes you sometimes you you win, sometimes you sometimes you you win, sometimes you lose, but over in the long run you lose lose, but over in the long run you lose lose, but over in the long run you lose a bit more than you win. Um and so a bit more than you win. Um and so a bit more than you win. Um and so normally your wallet will hit will go to normally your wallet will hit will go to normally your wallet will hit will go to zero um if you just keep playing over zero um if you just keep playing over zero um if you just keep playing over and over again. So statistically it and over again. So statistically it and over again. So statistically it makes sense. Yes. So, so the result that makes sense. Yes. So, so the result that makes sense. Yes. So, so the result that I I proved roughly speaking such that I I proved roughly speaking such that I I proved roughly speaking such that that statistically like 99% of all that statistically like 99% of all that statistically like 99% of all inputs would would drift down to maybe inputs would would drift down to maybe inputs would would drift down to maybe not all the way to one, but to be much not all the way to one, but to be much not all the way to one, but to be much much smaller than what you started. So, much smaller than what you started. So, much smaller than what you started. So, it's it's like if I told you that if you it's it's like if I told you that if you it's it's like if I told you that if you go to a casino, most of the time you end go to a casino, most of the time you end go to a casino, most of the time you end up if you keep playing for long enough, up if you keep playing for long enough, up if you keep playing for long enough, you end up with a smaller amount in your you end up with a smaller amount in your you end up with a smaller amount in your wallet than when you started. That's wallet than when you started. That's wallet than when you started. That's kind of like the what the result that I kind of like the what the result that I kind of like the what the result that I proved. So why is that result like can proved. So why is that result like can proved. So why is that result like can you continue down that thread you continue down that thread you continue down that thread to prove the full conjecture? Well, the to prove the full conjecture? Well, the to prove the full conjecture? Well, the problem is that um my I I used arguments problem is that um my I I used arguments problem is that um my I I used arguments from probability theory um and there's from probability theory um and there's from probability theory um and there's always this exceptional event. So you always this exceptional event. So you always this exceptional event. So you know, so in probability we have this know, so in probability we have this know, so in probability we have this this law of large numbers um which tells this law of large numbers um which tells this law of large numbers um which tells you things like if you play a casino you things like if you play a casino you things like if you play a casino with a um a game at a casino with a with a um a game at a casino with a with a um a game at a casino with a losing um expectation over time you are losing um expectation over time you are losing um expectation over time you are guaranteed or almost surely with guaranteed or almost surely with guaranteed or almost surely with probably probability as close to 100% as probably probability as close to 100% as probably probability as close to 100% as you wish you're guaranteed to lose you wish you're guaranteed to lose you wish you're guaranteed to lose money. But there's always this money. But there's always this money. But there's always this exceptional outlier. Like it is exceptional outlier. Like it is exceptional outlier. Like it is mathematically possible that even in mathematically possible that even in mathematically possible that even in when the game is is the odds are not in when the game is is the odds are not in when the game is is the odds are not in your favor, you could just keep winning

your favor, you could just keep winning your favor, you could just keep winning slightly more often than you lose. Very slightly more often than you lose. Very slightly more often than you lose. Very much like how in Navia Stokes there much like how in Navia Stokes there much like how in Navia Stokes there could be, you know, um most of the time could be, you know, um most of the time could be, you know, um most of the time um your waves can disperse. There could um your waves can disperse. There could um your waves can disperse. There could be just one outlier choice of initial be just one outlier choice of initial be just one outlier choice of initial conditions that would lead you to blow conditions that would lead you to blow conditions that would lead you to blow up. And there could be one outlier up. And there could be one outlier up. And there could be one outlier choice of um um special number that they choice of um um special number that they choice of um um special number that they stick in that shoots off infinity while stick in that shoots off infinity while stick in that shoots off infinity while all other numbers crash to earth uh all other numbers crash to earth uh all other numbers crash to earth uh crash to one. Um in fact um there's some crash to one. Um in fact um there's some crash to one. Um in fact um there's some mathematicians um who Alex Kovvich for mathematicians um who Alex Kovvich for mathematicians um who Alex Kovvich for instance who've proposed that um that instance who've proposed that um that instance who've proposed that um that actually um these collat uh iterations actually um these collat uh iterations actually um these collat uh iterations are like the similar automator um are like the similar automator um are like the similar automator um actually if you look at what they happen actually if you look at what they happen actually if you look at what they happen on in binary they do actually look a on in binary they do actually look a on in binary they do actually look a little bit like like these game of life little bit like like these game of life little bit like like these game of life type patterns. Um and in an analogy to type patterns. Um and in an analogy to type patterns. Um and in an analogy to how the game of life can create these how the game of life can create these how the game of life can create these these massive like self-plicating these massive like self-plicating these massive like self-plicating objects and so forth possibly you could objects and so forth possibly you could objects and so forth possibly you could create some sort of heavier than air create some sort of heavier than air create some sort of heavier than air flying machine a number which is flying machine a number which is flying machine a number which is actually encoding this machine which is actually encoding this machine which is actually encoding this machine which is just whose job it is is to encode is to just whose job it is is to encode is to just whose job it is is to encode is to create a version of itself which which create a version of itself which which create a version of itself which which is larger heavier than air machine is larger heavier than air machine is larger heavier than air machine encoded in a number that flies forever.

encoded in a number that flies forever. encoded in a number that flies forever. Yeah. So Conway in fact worked on worked Yeah. So Conway in fact worked on worked Yeah. So Conway in fact worked on worked on this problem as well. Oh wow. So on this problem as well. Oh wow. So on this problem as well. Oh wow. So Conway so similar in fact that was one Conway so similar in fact that was one Conway so similar in fact that was one of inspirations for the Nebby Stokes of inspirations for the Nebby Stokes of inspirations for the Nebby Stokes project that Conway studied project that Conway studied project that Conway studied generalizations of the collapse problem generalizations of the collapse problem generalizations of the collapse problem where instead of multiplying by three where instead of multiplying by three where instead of multiplying by three and adding one or dividing by two you and adding one or dividing by two you and adding one or dividing by two you have a more complicated branch but but have a more complicated branch but but have a more complicated branch but but instead of having two cases maybe you instead of having two cases maybe you instead of having two cases maybe you have 17 cases and then you go up and have 17 cases and then you go up and have 17 cases and then you go up and down and he showed that once your down and he showed that once your down and he showed that once your iteration gets complicated enough you iteration gets complicated enough you iteration gets complicated enough you can actually encode touring machines and can actually encode touring machines and can actually encode touring machines and you can actually make these problems you can actually make these problems you can actually make these problems undecidable and and do things like this. undecidable and and do things like this. undecidable and and do things like this. In fact, he invented a programming In fact, he invented a programming In fact, he invented a programming language for uh these kind of fractional language for uh these kind of fractional language for uh these kind of fractional linear transformations. He called a linear transformations. He called a linear transformations. He called a factrat as a play on forrat. Uh and he factrat as a play on forrat. Uh and he factrat as a play on forrat. Uh and he showed that that you could um you can showed that that you could um you can showed that that you could um you can program it was too incomplete. You could program it was too incomplete. You could program it was too incomplete. You could you could you could uh um you could make you could you could uh um you could make you could you could uh um you could make a program that if if your number you a program that if if your number you a program that if if your number you insert in was encoded as a prime, it insert in was encoded as a prime, it insert in was encoded as a prime, it would sync to zero. It would go down would sync to zero. It would go down would sync to zero. It would go down otherwise it would go up uh and things otherwise it would go up uh and things otherwise it would go up uh and things like that. Um so the general class of like that. Um so the general class of like that. Um so the general class of problems is is really uh as complicated problems is is really uh as complicated problems is is really uh as complicated as all of mathematics. some of the as all of mathematics. some of the as all of mathematics. some of the mystery of the cellular automa that we mystery of the cellular automa that we mystery of the cellular automa that we talked about uh having a fra talked about uh having a fra talked about uh having a fra mathematical framework to say anything mathematical framework to say anything mathematical framework to say anything about cellular automa maybe the same about cellular automa maybe the same about cellular automa maybe the same kind of framework is required yeah kind of framework is required yeah kind of framework is required yeah injecture yeah if you want to do it not injecture yeah if you want to do it not injecture yeah if you want to do it not statistically but you really want 100% statistically but you really want 100% statistically but you really want 100% of all inputs to to fall to earth yeah of all inputs to to fall to earth yeah of all inputs to to fall to earth yeah so what might be feasible is is so what might be feasible is is so what might be feasible is is statistically 99% you know go to one but statistically 99% you know go to one but statistically 99% you know go to one but like everything yeah that looks hard like everything yeah that looks hard like everything yeah that looks hard what would you say is out of these

what would you say is out of these what would you say is out of these within reach famous problems is the within reach famous problems is the within reach famous problems is the hardest problem we have today. Is there hardest problem we have today. Is there hardest problem we have today. Is there a reman hypothesis? We want is up there. a reman hypothesis? We want is up there. a reman hypothesis? We want is up there. Um POS MP is a good one because like uh Um POS MP is a good one because like uh Um POS MP is a good one because like uh that's that's a meta problem like if you that's that's a meta problem like if you that's that's a meta problem like if you solve that in the um in the positive solve that in the um in the positive solve that in the um in the positive sense that you can find a PMP algorithm sense that you can find a PMP algorithm sense that you can find a PMP algorithm that potentially this solves a lot of that potentially this solves a lot of that potentially this solves a lot of other problems as well and we should other problems as well and we should other problems as well and we should mention some of the conjectures we've mention some of the conjectures we've mention some of the conjectures we've been talking about. You know a lot of been talking about. You know a lot of been talking about. You know a lot of stuff is built on top of them. Now stuff is built on top of them. Now stuff is built on top of them. Now there's ripple effects. P equ= 1 P has there's ripple effects. P equ= 1 P has there's ripple effects. P equ= 1 P has more ripple effects than basically any more ripple effects than basically any more ripple effects than basically any other right if the reman hypothesis is other right if the reman hypothesis is other right if the reman hypothesis is disproven um that would be a big mental disproven um that would be a big mental disproven um that would be a big mental shock to the number theorist uh but it shock to the number theorist uh but it shock to the number theorist uh but it would have follow on effects for um would have follow on effects for um would have follow on effects for um cryptography cryptography cryptography um because a lot of cryptography uses um because a lot of cryptography uses um because a lot of cryptography uses number theory um it uses number theory number theory um it uses number theory number theory um it uses number theory constructions involving primes and so constructions involving primes and so constructions involving primes and so forth and um it relies very much on the forth and um it relies very much on the forth and um it relies very much on the intuition that number theories are built intuition that number theories are built intuition that number theories are built over many many years of what operations over many many years of what operations over many many years of what operations involving prime behave randomly and what involving prime behave randomly and what involving prime behave randomly and what ones don't. Um, and in particular, our ones don't. Um, and in particular, our ones don't. Um, and in particular, our encryption encryption encryption um methods are designed to turn text um methods are designed to turn text um methods are designed to turn text with information on it into text which with information on it into text which with information on it into text which is indistinguishable from um from random is indistinguishable from um from random is indistinguishable from um from random noise. So um and hence we believe to be noise. So um and hence we believe to be noise. So um and hence we believe to be almost impossible to crack um at least almost impossible to crack um at least almost impossible to crack um at least mathematically. Um but uh if something mathematically. Um but uh if something mathematically. Um but uh if something has core to our belief as human has core to our belief as human has core to our belief as human hypothesis is is wrong it means that hypothesis is is wrong it means that hypothesis is is wrong it means that there are there are actual patterns of there are there are actual patterns of there are there are actual patterns of the primes that we not aware of and if the primes that we not aware of and if the primes that we not aware of and if there's one there's probably going to be there's one there's probably going to be there's one there's probably going to be more. Um and suddenly a lot of our

more. Um and suddenly a lot of our more. Um and suddenly a lot of our crypto systems are in doubt. Yeah. crypto systems are in doubt. Yeah. crypto systems are in doubt. Yeah. But then how do you then say stuff about But then how do you then say stuff about But then how do you then say stuff about the the primes? Yeah. That you're going the the primes? Yeah. That you're going the the primes? Yeah. That you're going towards the collect conjecture again. Um towards the collect conjecture again. Um towards the collect conjecture again. Um because if I I you you want it to be because if I I you you want it to be because if I I you you want it to be random, right? You want it to be random, right? You want it to be random, right? You want it to be randomly. Yeah. So more broadly, I'm randomly. Yeah. So more broadly, I'm randomly. Yeah. So more broadly, I'm just looking for more tools, more ways just looking for more tools, more ways just looking for more tools, more ways to show that that that things are to show that that that things are to show that that that things are random. How do you prove a conspiracy random. How do you prove a conspiracy random. How do you prove a conspiracy doesn't happen, right? Is there any doesn't happen, right? Is there any doesn't happen, right? Is there any chance to you that P equals NP? Is there chance to you that P equals NP? Is there chance to you that P equals NP? Is there some Can you imagine a possible some Can you imagine a possible some Can you imagine a possible universe? It is possible. I mean there's universe? It is possible. I mean there's universe? It is possible. I mean there's there's various uh scenarios. I mean there's various uh scenarios. I mean there's various uh scenarios. I mean there there's one where it is there there's one where it is there there's one where it is technically possible but in practice is technically possible but in practice is technically possible but in practice is never actually implementable. The never actually implementable. The never actually implementable. The evidence is sort of slightly pushing in evidence is sort of slightly pushing in evidence is sort of slightly pushing in favor of no that we probably is not favor of no that we probably is not favor of no that we probably is not equal to NP. I mean it seems like it's equal to NP. I mean it seems like it's equal to NP. I mean it seems like it's one of those cases similar similar to one of those cases similar similar to one of those cases similar similar to reman hypothesis that I think the reman hypothesis that I think the reman hypothesis that I think the evidence is le leaning pretty heavily on evidence is le leaning pretty heavily on evidence is le leaning pretty heavily on the no. Certainly more on the no than on the no. Certainly more on the no than on the no. Certainly more on the no than on on the yes. The funny thing about on the yes. The funny thing about on the yes. The funny thing about picompy is that we have also a lot more picompy is that we have also a lot more picompy is that we have also a lot more obstructions than we do for almost any obstructions than we do for almost any obstructions than we do for almost any other problem. Um so while there's other problem. Um so while there's other problem. Um so while there's evidence we also have a lot of results evidence we also have a lot of results evidence we also have a lot of results ruling out many many types of approaches ruling out many many types of approaches ruling out many many types of approaches to the problem. Uh this is the one thing to the problem. Uh this is the one thing to the problem. Uh this is the one thing that the computer scientists have that the computer scientists have that the computer scientists have actually been very good at. It's actually been very good at. It's actually been very good at. It's actually saying that that certain actually saying that that certain actually saying that that certain approaches cannot work. No go theorems.

approaches cannot work. No go theorems. approaches cannot work. No go theorems. It could be undecidable. We don't Yeah, It could be undecidable. We don't Yeah, It could be undecidable. We don't Yeah, we don't know. There's a funny story I we don't know. There's a funny story I we don't know. There's a funny story I read that when you won the Fields Medal, read that when you won the Fields Medal, read that when you won the Fields Medal, somebody from the internet wrote you somebody from the internet wrote you somebody from the internet wrote you and asked uh you know what are you going and asked uh you know what are you going and asked uh you know what are you going to do now that you've won this to do now that you've won this to do now that you've won this prestigious award? and and then you just prestigious award? and and then you just prestigious award? and and then you just quickly very humbly said that, you know, quickly very humbly said that, you know, quickly very humbly said that, you know, this a shiny metal is not going to solve this a shiny metal is not going to solve this a shiny metal is not going to solve any of the problems I'm currently any of the problems I'm currently any of the problems I'm currently working on. So, I'm just I'm going to working on. So, I'm just I'm going to working on. So, I'm just I'm going to keep I'm going to keep working on them. keep I'm going to keep working on them. keep I'm going to keep working on them. It's just first of all, it's funny to me It's just first of all, it's funny to me It's just first of all, it's funny to me that you would answer an email in that that you would answer an email in that that you would answer an email in that context, and second of all, it um it context, and second of all, it um it context, and second of all, it um it just shows your humility. But anyway, uh just shows your humility. But anyway, uh just shows your humility. But anyway, uh maybe you could speak to the Fields maybe you could speak to the Fields maybe you could speak to the Fields Medal, but it's another way for me to Medal, but it's another way for me to Medal, but it's another way for me to ask uh ask uh ask uh about Gregoria Pearlman. What do you about Gregoria Pearlman. What do you about Gregoria Pearlman. What do you think about him famously declining the think about him famously declining the think about him famously declining the Fields Medal and the Millennial Prize, Fields Medal and the Millennial Prize, Fields Medal and the Millennial Prize, which came with a $1 million of prize which came with a $1 million of prize which came with a $1 million of prize money? He stated that I'm not interested money? He stated that I'm not interested money? He stated that I'm not interested in money or fame. The prize is in money or fame. The prize is in money or fame. The prize is completely irrelevant for me. If the completely irrelevant for me. If the completely irrelevant for me. If the proof is correct, then no other proof is correct, then no other proof is correct, then no other recognition is needed. Yeah. No, he's recognition is needed. Yeah. No, he's recognition is needed. Yeah. No, he's he's somewhat of an outlier. Um even he's somewhat of an outlier. Um even he's somewhat of an outlier. Um even among mathematicians who tend to uh to among mathematicians who tend to uh to among mathematicians who tend to uh to have uh somewhat idealistic views. I've have uh somewhat idealistic views. I've have uh somewhat idealistic views. I've never met him. I think I'd be interested never met him. I think I'd be interested never met him. I think I'd be interested to meet him one day, but I I never had to meet him one day, but I I never had to meet him one day, but I I never had the chance. I know people who met him, the chance. I know people who met him, the chance. I know people who met him, but he's always had strong views about but he's always had strong views about but he's always had strong views about certain things. Um, you know, I mean, certain things. Um, you know, I mean, certain things. Um, you know, I mean, it's it's not like he was completely it's it's not like he was completely it's it's not like he was completely isolated from the math community. I isolated from the math community. I isolated from the math community. I mean, he would he would give talks and mean, he would he would give talks and mean, he would he would give talks and write papers and so forth. Um, but at write papers and so forth. Um, but at write papers and so forth. Um, but at some point he just decided not to engage some point he just decided not to engage some point he just decided not to engage with the rest of the community. He was with the rest of the community. He was with the rest of the community. He was he was disillusioned or something. Um, I he was disillusioned or something. Um, I he was disillusioned or something. Um, I don't know. Um, and he decided to to uh

don't know. Um, and he decided to to uh don't know. Um, and he decided to to uh uh to peace out uh and you know, collect uh to peace out uh and you know, collect uh to peace out uh and you know, collect mushrooms in St. Petersburg or mushrooms in St. Petersburg or mushrooms in St. Petersburg or something. And then that's that's fine. something. And then that's that's fine. something. And then that's that's fine. you know and you can you can do that. Um you know and you can you can do that. Um you know and you can you can do that. Um I mean that's another sort of flip side. I mean that's another sort of flip side. I mean that's another sort of flip side. I mean we are not a lot of our problems I mean we are not a lot of our problems I mean we are not a lot of our problems that we solve you know they some of them that we solve you know they some of them that we solve you know they some of them do have practical application and that's do have practical application and that's do have practical application and that's that's great but uh like if you stop that's great but uh like if you stop that's great but uh like if you stop thinking about a problem you know so thinking about a problem you know so thinking about a problem you know so he's he hasn't published since in in he's he hasn't published since in in he's he hasn't published since in in this field but that's fine there's many this field but that's fine there's many this field but that's fine there's many many other people who've done so as many other people who've done so as many other people who've done so as well. Um yeah so I guess one thing I well. Um yeah so I guess one thing I well. Um yeah so I guess one thing I didn't realize initially with the fields didn't realize initially with the fields didn't realize initially with the fields medal is that it it sort of makes you medal is that it it sort of makes you medal is that it it sort of makes you part of the establishment. Um you know part of the establishment. Um you know part of the establishment. Um you know so you know most mathematicians you so you know most mathematicians you so you know most mathematicians you there's uh just career mathematicians there's uh just career mathematicians there's uh just career mathematicians you know you just focus on publishing you know you just focus on publishing you know you just focus on publishing the next paper maybe getting one to the next paper maybe getting one to the next paper maybe getting one to promote one one rank you know and and promote one one rank you know and and promote one one rank you know and and starting a few projects maybe taking starting a few projects maybe taking starting a few projects maybe taking some students or something. Yeah. But some students or something. Yeah. But some students or something. Yeah. But then suddenly people want your opinion then suddenly people want your opinion then suddenly people want your opinion on things and uh you have to think a on things and uh you have to think a on things and uh you have to think a little bit about you know things that little bit about you know things that little bit about you know things that you might just so foolishly say because you might just so foolishly say because you might just so foolishly say because you know no one's going to listen to you know no one's going to listen to you know no one's going to listen to you. Uh it's more important now. Is it you. Uh it's more important now. Is it you. Uh it's more important now. Is it constraining to you? Are you able to constraining to you? Are you able to constraining to you? Are you able to still have fun and be a rebel and try still have fun and be a rebel and try still have fun and be a rebel and try crazy stuff and well play with ideas? I crazy stuff and well play with ideas? I crazy stuff and well play with ideas? I have a lot less free time than I had have a lot less free time than I had have a lot less free time than I had previously. Um I mean mostly by choice.

previously. Um I mean mostly by choice. previously. Um I mean mostly by choice. I mean I I I obviously I have the option I mean I I I obviously I have the option I mean I I I obviously I have the option to sort of uh decline. So I decline a to sort of uh decline. So I decline a to sort of uh decline. So I decline a lot of things. I I could decline even lot of things. I I could decline even lot of things. I I could decline even more. Um or I could acquire a reputation more. Um or I could acquire a reputation more. Um or I could acquire a reputation for being so unreliable that people for being so unreliable that people for being so unreliable that people don't even ask anymore. Uh this is I don't even ask anymore. Uh this is I don't even ask anymore. Uh this is I love the different algorithms here. This love the different algorithms here. This love the different algorithms here. This is great. This is it's always an option. is great. This is it's always an option. is great. This is it's always an option. Um but you know um there are things that Um but you know um there are things that Um but you know um there are things that are like are like are like I mean so I mean I I I don't spend as I mean so I mean I I I don't spend as I mean so I mean I I I don't spend as much time as I do as a postto you know much time as I do as a postto you know much time as I do as a postto you know just just working on one problem at a just just working on one problem at a just just working on one problem at a time or um fooling around. I still do time or um fooling around. I still do time or um fooling around. I still do that a little bit but yeah as you that a little bit but yeah as you that a little bit but yeah as you advance in your career somehow the more advance in your career somehow the more advance in your career somehow the more soft skills so math somehow frontloads soft skills so math somehow frontloads soft skills so math somehow frontloads all the technical skills to the early all the technical skills to the early all the technical skills to the early stages of your career. So um yeah, so stages of your career. So um yeah, so stages of your career. So um yeah, so it's as a post office publisher or it's as a post office publisher or it's as a post office publisher or parish you're you're incent you're parish you're you're incent you're parish you're you're incent you're incentivized to basically focus on on incentivized to basically focus on on incentivized to basically focus on on proving very technical themsel proving very technical themsel proving very technical themsel um as well as proof the theorems. Um but um as well as proof the theorems. Um but um as well as proof the theorems. Um but then as as you get more senior you have then as as you get more senior you have then as as you get more senior you have to start you know mentoring and and and to start you know mentoring and and and to start you know mentoring and and and and giving interviews uh and uh and and giving interviews uh and uh and and giving interviews uh and uh and trying to shape um direction of the trying to shape um direction of the trying to shape um direction of the field both research wise and and you field both research wise and and you field both research wise and and you know uh sometimes you have to uh u you know uh sometimes you have to uh u you know uh sometimes you have to uh u you know do various administrative things know do various administrative things know do various administrative things and it's kind of the right social and it's kind of the right social and it's kind of the right social contract because you you need to to work contract because you you need to to work contract because you you need to to work in the trenches to see what can help in the trenches to see what can help in the trenches to see what can help mathematicians. the other side of the mathematicians. the other side of the mathematicians. the other side of the establishment sort of the the really establishment sort of the the really establishment sort of the the really positive thing is that um you get to be positive thing is that um you get to be positive thing is that um you get to be a light that's an inspiration to a lot a light that's an inspiration to a lot a light that's an inspiration to a lot of young mathematicians or young people of young mathematicians or young people of young mathematicians or young people that are just interested in mathematics.

that are just interested in mathematics. that are just interested in mathematics. It's like it's just how the human mind It's like it's just how the human mind It's like it's just how the human mind works. This is where I would probably uh works. This is where I would probably uh works. This is where I would probably uh say that I like the fields metal say that I like the fields metal say that I like the fields metal that it does inspire a lot of young that it does inspire a lot of young that it does inspire a lot of young people somehow. I don't this just how people somehow. I don't this just how people somehow. I don't this just how human brains work. Yeah. At the same human brains work. Yeah. At the same human brains work. Yeah. At the same time, I also want to give sort of time, I also want to give sort of time, I also want to give sort of respect to somebody like Gregoria respect to somebody like Gregoria respect to somebody like Gregoria Pearlman who Pearlman who Pearlman who is critical of awards in his mind. Those is critical of awards in his mind. Those is critical of awards in his mind. Those are his principles and any human that's are his principles and any human that's are his principles and any human that's able for their principles to like do the able for their principles to like do the able for their principles to like do the thing that most humans would not be able thing that most humans would not be able thing that most humans would not be able to do. It's beautiful to see. Some to do. It's beautiful to see. Some to do. It's beautiful to see. Some recognition is is necessarily important. recognition is is necessarily important. recognition is is necessarily important. Uh but yeah, it's it's also important to Uh but yeah, it's it's also important to Uh but yeah, it's it's also important to not let these things take over your not let these things take over your not let these things take over your life. um and like only be concerned life. um and like only be concerned life. um and like only be concerned about uh getting the next big award or about uh getting the next big award or about uh getting the next big award or whatever. Um I mean yeah so again you whatever. Um I mean yeah so again you whatever. Um I mean yeah so again you see these people try to only solve like see these people try to only solve like see these people try to only solve like a really big math problems and not work a really big math problems and not work a really big math problems and not work on on on things that are less uh sexy if on on on things that are less uh sexy if on on on things that are less uh sexy if you wish but but but actually still you wish but but but actually still you wish but but but actually still interesting and instructive as you say interesting and instructive as you say interesting and instructive as you say like the way the human mind works it's like the way the human mind works it's like the way the human mind works it's um we understand things better when um we understand things better when um we understand things better when they're attached to humans um and also they're attached to humans um and also they're attached to humans um and also uh if they're attached to a small number uh if they're attached to a small number uh if they're attached to a small number of humans like this this way our human of humans like this this way our human of humans like this this way our human mind is is wired we can comprehend and mind is is wired we can comprehend and mind is is wired we can comprehend and the relationships between you know 10 or the relationships between you know 10 or the relationships between you know 10 or 20 people you know but once you get 20 people you know but once you get 20 people you know but once you get beyond like 100 people like there beyond like 100 people like there beyond like 100 people like there there's a there's a limit I think there's a there's a limit I think there's a there's a limit I think there's a name for it um beyond which uh there's a name for it um beyond which uh there's a name for it um beyond which uh it just becomes the other um and so we it just becomes the other um and so we it just becomes the other um and so we have you have to simplify the pole have you have to simplify the pole have you have to simplify the pole master you know 99.9% of humanity

master you know 99.9% of humanity master you know 99.9% of humanity becomes the other um and uh often these becomes the other um and uh often these becomes the other um and uh often these models are are incorrect and this causes models are are incorrect and this causes models are are incorrect and this causes all kinds of problems but um so yeah so all kinds of problems but um so yeah so all kinds of problems but um so yeah so to humanize a subject you know if you to humanize a subject you know if you to humanize a subject you know if you identify a small number of people and identify a small number of people and identify a small number of people and say you know these say you know these say you know these representative people of the subject representative people of the subject representative people of the subject role models for example um that has some role models for example um that has some role models for example um that has some role um but it can also be um uh yeah role um but it can also be um uh yeah role um but it can also be um uh yeah too much of it can be harmful because too much of it can be harmful because too much of it can be harmful because it's it's it's I'll be the first to say that my own I'll be the first to say that my own I'll be the first to say that my own career path is not that of a typical career path is not that of a typical career path is not that of a typical mathematician um I the very accelerated mathematician um I the very accelerated mathematician um I the very accelerated education I skipped a lot of classes um education I skipped a lot of classes um education I skipped a lot of classes um I think I was had very fortunate I think I was had very fortunate I think I was had very fortunate mentoring opportunities um and I think I mentoring opportunities um and I think I mentoring opportunities um and I think I was at the right place at the right time was at the right place at the right time was at the right place at the right time just because someone does doesn't have just because someone does doesn't have just because someone does doesn't have my um trajectory, you it doesn't mean my um trajectory, you it doesn't mean my um trajectory, you it doesn't mean that they can't be good mathematicians. that they can't be good mathematicians. that they can't be good mathematicians. I mean they be ma good mathematician in I mean they be ma good mathematician in I mean they be ma good mathematician in a very different style. Uh and we need a very different style. Uh and we need a very different style. Uh and we need people of a different style. Um and you people of a different style. Um and you people of a different style. Um and you know even if and sometimes too much know even if and sometimes too much know even if and sometimes too much focus is given on the on the person who focus is given on the on the person who focus is given on the on the person who does the last step to complete um a does the last step to complete um a does the last step to complete um a project in mathematics or elsewhere project in mathematics or elsewhere project in mathematics or elsewhere that's that's really taken you know that's that's really taken you know that's that's really taken you know centuries or decades with lots and lots centuries or decades with lots and lots centuries or decades with lots and lots of building lots of previous work. Um, of building lots of previous work. Um, of building lots of previous work. Um, but that's a a story that's difficult to but that's a a story that's difficult to but that's a a story that's difficult to tell um if you're not an expert because, tell um if you're not an expert because, tell um if you're not an expert because, you know, it's easier to just say one you know, it's easier to just say one you know, it's easier to just say one person did this one thing. You know, it person did this one thing. You know, it person did this one thing. You know, it makes for a much simpler history. I makes for a much simpler history. I makes for a much simpler history. I think on the whole it um is a hugely think on the whole it um is a hugely think on the whole it um is a hugely positive thing to to talk about Steve positive thing to to talk about Steve positive thing to to talk about Steve Jobs as a representative of Apple when I Jobs as a representative of Apple when I Jobs as a representative of Apple when I personally know and of course everybody personally know and of course everybody personally know and of course everybody knows the incredible design, the

knows the incredible design, the knows the incredible design, the incredible engineering teams, just the incredible engineering teams, just the incredible engineering teams, just the individual humans on those teams. individual humans on those teams. individual humans on those teams. They're not a team. They're individual They're not a team. They're individual They're not a team. They're individual humans on a team. And there's a lot of humans on a team. And there's a lot of humans on a team. And there's a lot of brilliance there. But it's just a nice brilliance there. But it's just a nice brilliance there. But it's just a nice shorthand like a very like pi. Yeah. shorthand like a very like pi. Yeah. shorthand like a very like pi. Yeah. Steve Jobs. Yeah. Yeah. As as a starting Steve Jobs. Yeah. Yeah. As as a starting Steve Jobs. Yeah. Yeah. As as a starting point, you know, as a first point, you know, as a first point, you know, as a first approximation that's how you and then approximation that's how you and then approximation that's how you and then read some biographies and then look into read some biographies and then look into read some biographies and then look into much deeper. First approximation. Yeah. much deeper. First approximation. Yeah. much deeper. First approximation. Yeah. That's right. Uh so you mentioned you That's right. Uh so you mentioned you That's right. Uh so you mentioned you were a Princeton to um Andrew Wilds at were a Princeton to um Andrew Wilds at were a Princeton to um Andrew Wilds at that time. He's a professor there. It's that time. He's a professor there. It's that time. He's a professor there. It's a funny moment how history is just all a funny moment how history is just all a funny moment how history is just all interconnected. And at that time he interconnected. And at that time he interconnected. And at that time he announced that he proved the form last announced that he proved the form last announced that he proved the form last theorem. What did you think maybe theorem. What did you think maybe theorem. What did you think maybe looking back now with more context about looking back now with more context about looking back now with more context about that moment in math history? Yes. So I that moment in math history? Yes. So I that moment in math history? Yes. So I was a graduate student at the time. I was a graduate student at the time. I was a graduate student at the time. I mean I I vaguely remember you know there mean I I vaguely remember you know there mean I I vaguely remember you know there was press attention and uh um we all had was press attention and uh um we all had was press attention and uh um we all had the same um we had pigeon holes in the the same um we had pigeon holes in the the same um we had pigeon holes in the same mail room you know. So we all same mail room you know. So we all same mail room you know. So we all picked our mail and like suddenly Andrew picked our mail and like suddenly Andrew picked our mail and like suddenly Andrew W's mailbox exploded to be overflowing. W's mailbox exploded to be overflowing. W's mailbox exploded to be overflowing. That's a good that's a good metric.

That's a good that's a good metric. That's a good that's a good metric. Yeah. um you know so yeah we we all Yeah. um you know so yeah we we all Yeah. um you know so yeah we we all talked about it at at tea and so forth I talked about it at at tea and so forth I talked about it at at tea and so forth I mean we we didn't understand most of us mean we we didn't understand most of us mean we we didn't understand most of us didn't understand the proof um we didn't understand the proof um we didn't understand the proof um we understand sort of high level details um understand sort of high level details um understand sort of high level details um fact there's an ongoing project to fact there's an ongoing project to fact there's an ongoing project to formalize it in lean right Kevin puzzly formalize it in lean right Kevin puzzly formalize it in lean right Kevin puzzly yeah can can we take that small tangent yeah can can we take that small tangent yeah can can we take that small tangent is it is it how difficult does that cuz is it is it how difficult does that cuz is it is it how difficult does that cuz as as I understand the for last the as as I understand the for last the as as I understand the for last the proof for uh for last theorem has like proof for uh for last theorem has like proof for uh for last theorem has like super complicated objects yeah really super complicated objects yeah really super complicated objects yeah really difficult to formalize now yeah I guess difficult to formalize now yeah I guess difficult to formalize now yeah I guess yeah you're right the objects that they yeah you're right the objects that they yeah you're right the objects that they use um you can define them. Uh so use um you can define them. Uh so use um you can define them. Uh so they've been defined in lean. Okay. So they've been defined in lean. Okay. So they've been defined in lean. Okay. So so just defining what they are can be so just defining what they are can be so just defining what they are can be done. Uh that's really not trivial but done. Uh that's really not trivial but done. Uh that's really not trivial but it's been done. But there's a lot of it's been done. But there's a lot of it's been done. But there's a lot of really basic facts about um these really basic facts about um these really basic facts about um these objects that have taken decades to prove objects that have taken decades to prove objects that have taken decades to prove and that they're in all these different and that they're in all these different and that they're in all these different math papers and so lots of these have to math papers and so lots of these have to math papers and so lots of these have to be formalized as well. Um Kevin's uh be formalized as well. Um Kevin's uh be formalized as well. Um Kevin's uh Kevin Buzzard's goal actually he has a Kevin Buzzard's goal actually he has a Kevin Buzzard's goal actually he has a five-year grraft to formalize fossil five-year grraft to formalize fossil five-year grraft to formalize fossil theorem and his aim is that he doesn't theorem and his aim is that he doesn't theorem and his aim is that he doesn't think he will be able to get all the way think he will be able to get all the way think he will be able to get all the way down to the basic axioms but he wants to down to the basic axioms but he wants to down to the basic axioms but he wants to formalize it to the point where the only formalize it to the point where the only formalize it to the point where the only things that he needs to rely on as black things that he needs to rely on as black things that he needs to rely on as black boxes are things that were known by 1980 boxes are things that were known by 1980 boxes are things that were known by 1980 to um to number theorist at the time. Um to um to number theorist at the time. Um to um to number theorist at the time. Um and then some other person some other and then some other person some other and then some other person some other work would have to done to to to get work would have to done to to to get work would have to done to to to get from there. Um so it's it's a different from there. Um so it's it's a different from there. Um so it's it's a different area of mathematics than um the type of area of mathematics than um the type of area of mathematics than um the type of mathematics I'm used to. Um um in mathematics I'm used to. Um um in mathematics I'm used to. Um um in analysis, which is kind of my area, um analysis, which is kind of my area, um analysis, which is kind of my area, um the objects we study are kind of much the objects we study are kind of much the objects we study are kind of much closer to the ground. We study I study

closer to the ground. We study I study closer to the ground. We study I study things like prime numbers and and things like prime numbers and and things like prime numbers and and functions and things that are within functions and things that are within functions and things that are within scope of a high school um math education scope of a high school um math education scope of a high school um math education to at least uh define. Um yeah, but then to at least uh define. Um yeah, but then to at least uh define. Um yeah, but then there's this very advanced algebraic there's this very advanced algebraic there's this very advanced algebraic side of number theory where people have side of number theory where people have side of number theory where people have been building structures upon structures been building structures upon structures been building structures upon structures for quite a while. Um and it's it's a for quite a while. Um and it's it's a for quite a while. Um and it's it's a very sturdy structure. It's it's been very sturdy structure. It's it's been very sturdy structure. It's it's been it's been very um at the base at least it's been very um at the base at least it's been very um at the base at least is extremely well developed in the is extremely well developed in the is extremely well developed in the textbooks and so forth. But um um it textbooks and so forth. But um um it textbooks and so forth. But um um it does get to the point where um if you if does get to the point where um if you if does get to the point where um if you if you haven't taken these years of study you haven't taken these years of study you haven't taken these years of study and you want to ask about what what is and you want to ask about what what is and you want to ask about what what is going on at um like level six of of this going on at um like level six of of this going on at um like level six of of this tower, you have to spend quite a bit of tower, you have to spend quite a bit of tower, you have to spend quite a bit of time before they can even get to the time before they can even get to the time before they can even get to the point where you can see you see point where you can see you see point where you can see you see something you recognize. What uh something you recognize. What uh something you recognize. What uh inspires you about his journey that we inspires you about his journey that we inspires you about his journey that we similar as we talked about seven years similar as we talked about seven years similar as we talked about seven years mostly working in secret? Yeah. Uh that mostly working in secret? Yeah. Uh that mostly working in secret? Yeah. Uh that is a romantic uh Yeah. So it kind of is a romantic uh Yeah. So it kind of is a romantic uh Yeah. So it kind of fits with sort of the the romantic image fits with sort of the the romantic image fits with sort of the the romantic image I think people have of mathematicians to I think people have of mathematicians to I think people have of mathematicians to the extent they think of them at all as the extent they think of them at all as the extent they think of them at all as these kind of eccentric uh you know these kind of eccentric uh you know these kind of eccentric uh you know wizards or something. Um so that wizards or something. Um so that wizards or something. Um so that certainly kind of uh uh accentuated that certainly kind of uh uh accentuated that certainly kind of uh uh accentuated that perspective you know I mean it's it is a perspective you know I mean it's it is a perspective you know I mean it's it is a great achievement his style of solving great achievement his style of solving great achievement his style of solving problems is so different from my own um problems is so different from my own um problems is so different from my own um but which but which is great. I mean we but which but which is great. I mean we but which but which is great. I mean we we need people speak to it like what uh we need people speak to it like what uh we need people speak to it like what uh in in terms of like the you like the in in terms of like the you like the in in terms of like the you like the collaborative I like moving on from a collaborative I like moving on from a collaborative I like moving on from a problem if it's giving too much problem if it's giving too much problem if it's giving too much everybody. Um got it. But you need the

everybody. Um got it. But you need the everybody. Um got it. But you need the people who have the tenacity and the people who have the tenacity and the people who have the tenacity and the fearlessness. Um you I've collaborated fearlessness. Um you I've collaborated fearlessness. Um you I've collaborated with with people like that where where I with with people like that where where I with with people like that where where I want to give up uh cuz the first want to give up uh cuz the first want to give up uh cuz the first approach that we tried didn't work and approach that we tried didn't work and approach that we tried didn't work and the second one didn't approach but the second one didn't approach but the second one didn't approach but they're convinced and they have the they're convinced and they have the they're convinced and they have the third fourth and the fifth approach third fourth and the fifth approach third fourth and the fifth approach works. Um and I have to eat my words. works. Um and I have to eat my words. works. Um and I have to eat my words. Okay. I didn't think this was going to Okay. I didn't think this was going to Okay. I didn't think this was going to work, but yes, you were right all along. work, but yes, you were right all along. work, but yes, you were right all along. And we should say for people who don't And we should say for people who don't And we should say for people who don't know, not only are you known for the know, not only are you known for the know, not only are you known for the brilliance of your work, but the brilliance of your work, but the brilliance of your work, but the incredible productivity, just the number incredible productivity, just the number incredible productivity, just the number of papers, which are all of very high of papers, which are all of very high of papers, which are all of very high quality. So there's something to be said quality. So there's something to be said quality. So there's something to be said about being able to jump from topic to about being able to jump from topic to about being able to jump from topic to topic. Yeah, it works for me. Yeah, I topic. Yeah, it works for me. Yeah, I topic. Yeah, it works for me. Yeah, I mean there also people who are very mean there also people who are very mean there also people who are very productive and they focus very deeply on productive and they focus very deeply on productive and they focus very deeply on Yeah. I think everyone has to find their Yeah. I think everyone has to find their Yeah. I think everyone has to find their own workflow. Um like one thing which is own workflow. Um like one thing which is own workflow. Um like one thing which is a shame in mathematics is that we have a shame in mathematics is that we have a shame in mathematics is that we have mathematics there's sort of a one size mathematics there's sort of a one size mathematics there's sort of a one size fits all approach to teach teaching fits all approach to teach teaching fits all approach to teach teaching mathematics um and you know so we have a mathematics um and you know so we have a mathematics um and you know so we have a certain curriculum and so forth I mean certain curriculum and so forth I mean certain curriculum and so forth I mean you know maybe like if you do math you know maybe like if you do math you know maybe like if you do math competitions or something you get a competitions or something you get a competitions or something you get a slightly different experience but um I slightly different experience but um I slightly different experience but um I think many people um they don't find think many people um they don't find think many people um they don't find their their native math language uh their their native math language uh their their native math language uh until very late or usually too late so until very late or usually too late so until very late or usually too late so they they stop doing mathematics and they they stop doing mathematics and they they stop doing mathematics and they have a bad experience with a they have a bad experience with a they have a bad experience with a teacher who's trying to teach them one teacher who's trying to teach them one teacher who's trying to teach them one way to do mathematics. They don't like way to do mathematics. They don't like way to do mathematics. They don't like it. Um my theory is that um humans don't it. Um my theory is that um humans don't it. Um my theory is that um humans don't come evolution has not given us a math come evolution has not given us a math come evolution has not given us a math center of a brain directly. We have a center of a brain directly. We have a center of a brain directly. We have a vision center and a language center and vision center and a language center and vision center and a language center and some other centers um which have some other centers um which have some other centers um which have evolution has honed but we it doesn't we

evolution has honed but we it doesn't we evolution has honed but we it doesn't we don't have innate sense of mathematics. don't have innate sense of mathematics. don't have innate sense of mathematics. Um but our other centers are Um but our other centers are Um but our other centers are sophisticated enough that different sophisticated enough that different sophisticated enough that different people we we we can repurpose other people we we we can repurpose other people we we we can repurpose other areas of our brain to do mathematics. So areas of our brain to do mathematics. So areas of our brain to do mathematics. So some people have figured out how to use some people have figured out how to use some people have figured out how to use the visual center to do mathematics and the visual center to do mathematics and the visual center to do mathematics and so they think very visually when they do so they think very visually when they do so they think very visually when they do mathematics. Some people have repurposed mathematics. Some people have repurposed mathematics. Some people have repurposed their their language center and they their their language center and they their their language center and they think very symbolically. Um, you know, think very symbolically. Um, you know, think very symbolically. Um, you know, um, some people like if they are very um, some people like if they are very um, some people like if they are very competitive and they they like gaming, competitive and they they like gaming, competitive and they they like gaming, there's a type there's this part of your there's a type there's this part of your there's a type there's this part of your brain that's very good at at at uh at brain that's very good at at at uh at brain that's very good at at at uh at solving puzzles and games and and and solving puzzles and games and and and solving puzzles and games and and and that can be repurposed. But like when I that can be repurposed. But like when I that can be repurposed. But like when I talked about the mathematicians, you talked about the mathematicians, you talked about the mathematicians, you know, they don't quite think they I can know, they don't quite think they I can know, they don't quite think they I can tell that they're using some different tell that they're using some different tell that they're using some different styles of of thinking than I am. I mean, styles of of thinking than I am. I mean, styles of of thinking than I am. I mean, not not disjoint, but they they may not not disjoint, but they they may not not disjoint, but they they may prefer visual. Like I I don't actually prefer visual. Like I I don't actually prefer visual. Like I I don't actually prefer visual so much. I need lots of prefer visual so much. I need lots of prefer visual so much. I need lots of visual aids myself. Um, you know, visual aids myself. Um, you know, visual aids myself. Um, you know, mathematics provides a common language. mathematics provides a common language. mathematics provides a common language. So, we can still talk to each other even So, we can still talk to each other even So, we can still talk to each other even if we are thinking in in different ways.

if we are thinking in in different ways. if we are thinking in in different ways. But you can tell there's a different But you can tell there's a different But you can tell there's a different set of subsystems being used in the set of subsystems being used in the set of subsystems being used in the thinking process like they take thinking process like they take thinking process like they take different paths. They're very quick at different paths. They're very quick at different paths. They're very quick at things that I struggle with and vice things that I struggle with and vice things that I struggle with and vice versa. Um, and yet they still get to the versa. Um, and yet they still get to the versa. Um, and yet they still get to the same goal. Um, that's beautiful. And same goal. Um, that's beautiful. And same goal. Um, that's beautiful. And yeah, but I mean the way we educate yeah, but I mean the way we educate yeah, but I mean the way we educate unless you have like a personalized unless you have like a personalized unless you have like a personalized tutor or something. I mean education tutor or something. I mean education tutor or something. I mean education sort of just by natural scale has to be sort of just by natural scale has to be sort of just by natural scale has to be mass-produced you know you have to teach mass-produced you know you have to teach mass-produced you know you have to teach to 30 kids and you know if they have 30 to 30 kids and you know if they have 30 to 30 kids and you know if they have 30 different styles you can't you can't different styles you can't you can't different styles you can't you can't teach 30 different ways. On that topic teach 30 different ways. On that topic teach 30 different ways. On that topic what advice would you give to students what advice would you give to students what advice would you give to students uh young students who are struggling uh young students who are struggling uh young students who are struggling with math and but are interested in it with math and but are interested in it with math and but are interested in it and would like to get better. Is there and would like to get better. Is there and would like to get better. Is there something in this Yeah. um in this something in this Yeah. um in this something in this Yeah. um in this complicated educational context, what complicated educational context, what complicated educational context, what what would you Yeah, it's a tricky what would you Yeah, it's a tricky what would you Yeah, it's a tricky problem. One nice thing is that there problem. One nice thing is that there problem. One nice thing is that there are now lots of sources for mathematical are now lots of sources for mathematical are now lots of sources for mathematical enrichment outside the classroom. Um so enrichment outside the classroom. Um so enrichment outside the classroom. Um so in in in my day there already there are in in in my day there already there are in in in my day there already there are math competitions. Um and you know there math competitions. Um and you know there math competitions. Um and you know there also like popular math books in the also like popular math books in the also like popular math books in the library. Um yeah but but now you have library. Um yeah but but now you have library. Um yeah but but now you have you know YouTube uh there there are you know YouTube uh there there are you know YouTube uh there there are forums just devoted to solving you know forums just devoted to solving you know forums just devoted to solving you know math puzzles and um and math shows up in math puzzles and um and math shows up in math puzzles and um and math shows up in other places you know like um for other places you know like um for other places you know like um for example there there are hobbyists who example there there are hobbyists who example there there are hobbyists who play poker for fun uh and um they they play poker for fun uh and um they they play poker for fun uh and um they they you know they for very specific reasons you know they for very specific reasons you know they for very specific reasons are interested in very specific are interested in very specific are interested in very specific probability questions um and and they probability questions um and and they probability questions um and and they actually know there's a community of actually know there's a community of actually know there's a community of amateur proists in in in poker um in amateur proists in in in poker um in amateur proists in in in poker um in chess, in baseball. I mean, there's chess, in baseball. I mean, there's chess, in baseball. I mean, there's there's there's uh yeah um there's math

there's there's uh yeah um there's math there's there's uh yeah um there's math all over the place. Um and I'm I'm I'm all over the place. Um and I'm I'm I'm all over the place. Um and I'm I'm I'm hoping actually with with these new sort hoping actually with with these new sort hoping actually with with these new sort of tools for lean and so forth that of tools for lean and so forth that of tools for lean and so forth that actually we can incorporate the broader actually we can incorporate the broader actually we can incorporate the broader public into math research projects um public into math research projects um public into math research projects um like this is almost is doesn't happen at like this is almost is doesn't happen at like this is almost is doesn't happen at all currently. So in the sciences all currently. So in the sciences all currently. So in the sciences there's some scope for citizen science there's some scope for citizen science there's some scope for citizen science like astronomers uh they amateurs who like astronomers uh they amateurs who like astronomers uh they amateurs who discover comets and there's biologists discover comets and there's biologists discover comets and there's biologists there people who could identify there people who could identify there people who could identify butterflies and so forth. Um and in butterflies and so forth. Um and in butterflies and so forth. Um and in mathematics mathematics mathematics whereum amateur mathematicians can like whereum amateur mathematicians can like whereum amateur mathematicians can like discover new primes and so forth but but discover new primes and so forth but but discover new primes and so forth but but previously because we have to verify previously because we have to verify previously because we have to verify every single contribution um like most every single contribution um like most every single contribution um like most mathematical research projects it would mathematical research projects it would mathematical research projects it would not help to have input from the general not help to have input from the general not help to have input from the general public. In fact, it would it would just public. In fact, it would it would just public. In fact, it would it would just be be timeconuming because just error be be timeconuming because just error be be timeconuming because just error checking and everything. Um but you know checking and everything. Um but you know checking and everything. Um but you know one thing about these formalization one thing about these formalization one thing about these formalization projects is that they are bringing projects is that they are bringing projects is that they are bringing together more bringing in more people. together more bringing in more people. together more bringing in more people. So I'm sure there are high school So I'm sure there are high school So I'm sure there are high school students who've already contributed to students who've already contributed to students who've already contributed to some of these these formalizing projects some of these these formalizing projects some of these these formalizing projects who contributed into math liib. Um you who contributed into math liib. Um you who contributed into math liib. Um you know you don't need to be a PhD holder know you don't need to be a PhD holder know you don't need to be a PhD holder to just work on one atomic thing.

to just work on one atomic thing. to just work on one atomic thing. There's something about the There's something about the There's something about the formalization here that also at as a formalization here that also at as a formalization here that also at as a very first step opens it up to the very first step opens it up to the very first step opens it up to the programming community too. The people programming community too. The people programming community too. The people who are already comfortable with who are already comfortable with who are already comfortable with programming. It seems like programming programming. It seems like programming programming. It seems like programming is somehow maybe just the feeling but it is somehow maybe just the feeling but it is somehow maybe just the feeling but it feels more accessible to folks than feels more accessible to folks than feels more accessible to folks than math. Math is seen as this like extreme math. Math is seen as this like extreme math. Math is seen as this like extreme especially modern mathematics seen as especially modern mathematics seen as especially modern mathematics seen as this extremely difficult to enter area this extremely difficult to enter area this extremely difficult to enter area and programming is not. So that could be and programming is not. So that could be and programming is not. So that could be just an entry point. you can execute just an entry point. you can execute just an entry point. you can execute code and you can get results. You know, code and you can get results. You know, code and you can get results. You know, you can print a hello world pretty you can print a hello world pretty you can print a hello world pretty quickly. Um, you know, like if uh if quickly. Um, you know, like if uh if quickly. Um, you know, like if uh if programming was taught as almost programming was taught as almost programming was taught as almost entirely theoretical subject where you entirely theoretical subject where you entirely theoretical subject where you just taught the the computer science, just taught the the computer science, just taught the the computer science, the theory of functions and and and the theory of functions and and and the theory of functions and and and routines and so forth and and outside of routines and so forth and and outside of routines and so forth and and outside of some some very specialized homework some some very specialized homework some some very specialized homework assignments, you're not actually program assignments, you're not actually program assignments, you're not actually program like on the weekend for fun. Yeah. Or like on the weekend for fun. Yeah. Or like on the weekend for fun. Yeah. Or Yeah. They would be as considered as Yeah. They would be as considered as Yeah. They would be as considered as hard as math. Mhm. Um Yeah. Yeah. So, as hard as math. Mhm. Um Yeah. Yeah. So, as hard as math. Mhm. Um Yeah. Yeah. So, as I said, you know, there are communities I said, you know, there are communities I said, you know, there are communities of non- mathematicians where they're of non- mathematicians where they're of non- mathematicians where they're deploying math for some very specific deploying math for some very specific deploying math for some very specific purpose, you know, like like optimizing purpose, you know, like like optimizing purpose, you know, like like optimizing their poker game and and for them then their poker game and and for them then their poker game and and for them then math becomes fun for them. Uh what math becomes fun for them. Uh what math becomes fun for them. Uh what advice would you give in general to advice would you give in general to advice would you give in general to young people how to pick a career, how young people how to pick a career, how young people how to pick a career, how to find themselves like that's a tough to find themselves like that's a tough to find themselves like that's a tough tough tough question. Yeah. So um tough tough question. Yeah. So um tough tough question. Yeah. So um there's a lot less certainty now in the there's a lot less certainty now in the there's a lot less certainty now in the world you know I mean I there was this world you know I mean I there was this world you know I mean I there was this period after the war where uh at least period after the war where uh at least period after the war where uh at least in the west you know if you came from a in the west you know if you came from a in the west you know if you came from a good demographic you uh you know like good demographic you uh you know like good demographic you uh you know like you there was a very stable path to to a

you there was a very stable path to to a you there was a very stable path to to a good career you go to college you get an good career you go to college you get an good career you go to college you get an education you pick one profession and education you pick one profession and education you pick one profession and you stick to it becoming much more a you stick to it becoming much more a you stick to it becoming much more a thing of the past so I think you just thing of the past so I think you just thing of the past so I think you just have to be adaptable and flexible I have to be adaptable and flexible I have to be adaptable and flexible I think people have to get skills that are think people have to get skills that are think people have to get skills that are transferable you know like like learning transferable you know like like learning transferable you know like like learning one specific programming language or one one specific programming language or one one specific programming language or one specific subject of mathematics or specific subject of mathematics or specific subject of mathematics or something. It's it's it's that itself is something. It's it's it's that itself is something. It's it's it's that itself is not a super transferable skill but sort not a super transferable skill but sort not a super transferable skill but sort of knowing how to um reason with with of knowing how to um reason with with of knowing how to um reason with with abstract concepts or how to problem abstract concepts or how to problem abstract concepts or how to problem solve when things go wrong. So these are solve when things go wrong. So these are solve when things go wrong. So these are things which I think we will still need things which I think we will still need things which I think we will still need even as our tools get get better and you even as our tools get get better and you even as our tools get get better and you know you you would be working with AI know you you would be working with AI know you you would be working with AI sport and so forth. But actually you're sport and so forth. But actually you're sport and so forth. But actually you're an interesting case study. I mean you're an interesting case study. I mean you're an interesting case study. I mean you're like a like a like a one of the great living mathematicians one of the great living mathematicians one of the great living mathematicians right and then you had a way of doing right and then you had a way of doing right and then you had a way of doing things and then all of a sudden you things and then all of a sudden you things and then all of a sudden you start learning I mean first of all you start learning I mean first of all you start learning I mean first of all you kept learning new fields but you learn kept learning new fields but you learn kept learning new fields but you learn lean that's not that's a non-trivial lean that's not that's a non-trivial lean that's not that's a non-trivial thing to learn like that's a that's a thing to learn like that's a that's a thing to learn like that's a that's a for a lot of people that's an extremely for a lot of people that's an extremely for a lot of people that's an extremely uncomfortable leap to take right yeah uncomfortable leap to take right yeah uncomfortable leap to take right yeah mathematicians um first of all I've mathematicians um first of all I've mathematicians um first of all I've always been interested in new ways to do always been interested in new ways to do always been interested in new ways to do mathematics I I I feel like a lot of the mathematics I I I feel like a lot of the mathematics I I I feel like a lot of the ways we do things right now are ways we do things right now are ways we do things right now are inefficient. Um I I I I spend me my inefficient. Um I I I I spend me my inefficient. Um I I I I spend me my colleagues, we spend a lot of time doing colleagues, we spend a lot of time doing colleagues, we spend a lot of time doing very routine computations or doing very routine computations or doing very routine computations or doing things that other mathematicians would things that other mathematicians would things that other mathematicians would instantly know how to do and we don't instantly know how to do and we don't instantly know how to do and we don't know how to do them. Uh and why can't we know how to do them. Uh and why can't we know how to do them. Uh and why can't we search and get a quick response and so search and get a quick response and so search and get a quick response and so that's why I've always been interested that's why I've always been interested that's why I've always been interested in exploring new workflows.

in exploring new workflows. in exploring new workflows. About four or five years ago, I was on a About four or five years ago, I was on a About four or five years ago, I was on a committee where we had to ask for ideas committee where we had to ask for ideas committee where we had to ask for ideas for interesting workshops to run at a for interesting workshops to run at a for interesting workshops to run at a math institute. And at the time, Peter math institute. And at the time, Peter math institute. And at the time, Peter Schulzer had just formalized one of his Schulzer had just formalized one of his Schulzer had just formalized one of his his um new theorems. And um there are his um new theorems. And um there are his um new theorems. And um there are some other developments in computer some other developments in computer some other developments in computer assisted proof that look quite assisted proof that look quite assisted proof that look quite interesting. And I said, "Oh, we should interesting. And I said, "Oh, we should interesting. And I said, "Oh, we should we should uh um we should run a workshop we should uh um we should run a workshop we should uh um we should run a workshop on this. This be a good idea." Um and on this. This be a good idea." Um and on this. This be a good idea." Um and then I was a bit too enthusiastic about then I was a bit too enthusiastic about then I was a bit too enthusiastic about this idea. So I I got volunte. this idea. So I I got volunte. this idea. So I I got volunte. Um, so I did with a bunch of other Um, so I did with a bunch of other Um, so I did with a bunch of other people, Kevin Bisard and Jordan people, Kevin Bisard and Jordan people, Kevin Bisard and Jordan Ellenburg and and a bunch of other Ellenburg and and a bunch of other Ellenburg and and a bunch of other people. Um, and it was it was a a nice people. Um, and it was it was a a nice people. Um, and it was it was a a nice success. We brought together a bunch of success. We brought together a bunch of success. We brought together a bunch of mathematicians and computer scientists mathematicians and computer scientists mathematicians and computer scientists and other people and and we got up to and other people and and we got up to and other people and and we got up to speed and state um and it was really speed and state um and it was really speed and state um and it was really interesting um developments that that interesting um developments that that interesting um developments that that most mathematicians didn't know was most mathematicians didn't know was most mathematicians didn't know was going on. Um that lots of nice proofs of going on. Um that lots of nice proofs of going on. Um that lots of nice proofs of concept, you know, just sort of hints of concept, you know, just sort of hints of concept, you know, just sort of hints of of what was going to happen. this was of what was going to happen. this was of what was going to happen. this was just before chat GBD but there was even just before chat GBD but there was even just before chat GBD but there was even then there was one talk about language then there was one talk about language then there was one talk about language models and the potential um capability models and the potential um capability models and the potential um capability of those in the future. So that got me of those in the future. So that got me of those in the future. So that got me excited about the subject. So I started excited about the subject. So I started excited about the subject. So I started giving talks um about this is something giving talks um about this is something giving talks um about this is something we should more of us should start we should more of us should start we should more of us should start looking at um now that I' arranged to looking at um now that I' arranged to looking at um now that I' arranged to run this conference and then chat GPT run this conference and then chat GPT run this conference and then chat GPT came out and like suddenly AI was came out and like suddenly AI was came out and like suddenly AI was everywhere and so uh I got interviewed a everywhere and so uh I got interviewed a everywhere and so uh I got interviewed a lot um about about this topic um and in lot um about about this topic um and in lot um about about this topic um and in particular um the interaction between AI particular um the interaction between AI particular um the interaction between AI and formal proof assistance and I said and formal proof assistance and I said and formal proof assistance and I said yeah they should be combined this this yeah they should be combined this this yeah they should be combined this this is this is um this perfect synergy to is this is um this perfect synergy to is this is um this perfect synergy to happen here and at some point I realized

happen here and at some point I realized happen here and at some point I realized that I have to actually do not just talk that I have to actually do not just talk that I have to actually do not just talk the talk but walk the book you know like the talk but walk the book you know like the talk but walk the book you know like you know I don't work in machine you know I don't work in machine you know I don't work in machine learning I and I don't work in proof learning I and I don't work in proof learning I and I don't work in proof formalization and there's a limit to how formalization and there's a limit to how formalization and there's a limit to how much I can just rely on authority and much I can just rely on authority and much I can just rely on authority and saying you know I I'm a I'm a warn saying you know I I'm a I'm a warn saying you know I I'm a I'm a warn mathematician just trust me you know mathematician just trust me you know mathematician just trust me you know when I say that this is going to change when I say that this is going to change when I say that this is going to change athletics and I'm not doing it any when athletics and I'm not doing it any when athletics and I'm not doing it any when I don't do any of it myself so I felt I don't do any of it myself so I felt I don't do any of it myself so I felt like I had to actually uh uh justify it like I had to actually uh uh justify it like I had to actually uh uh justify it yeah a lot of what I get into actually I yeah a lot of what I get into actually I yeah a lot of what I get into actually I don't quite see in advance as how much don't quite see in advance as how much don't quite see in advance as how much time I'm going to spend on it and it's time I'm going to spend on it and it's time I'm going to spend on it and it's only after I'm sort of waste deep in in only after I'm sort of waste deep in in only after I'm sort of waste deep in in in in in a project that I I I realized in in in a project that I I I realized in in in a project that I I I realized by that point I'm committed. Well, by that point I'm committed. Well, by that point I'm committed. Well, that's deeply admirable that you're that's deeply admirable that you're that's deeply admirable that you're willing to go into the fray be in some willing to go into the fray be in some willing to go into the fray be in some small way a beginner, right? Or have small way a beginner, right? Or have small way a beginner, right? Or have some of the sort of challenges that a some of the sort of challenges that a some of the sort of challenges that a beginner would, right? beginner would, right? beginner would, right? new concepts, new ways of thinking also, new concepts, new ways of thinking also, new concepts, new ways of thinking also, you know, sucking at a thing that others you know, sucking at a thing that others you know, sucking at a thing that others I think I think in that talk you could I think I think in that talk you could I think I think in that talk you could be a fields med metal winning be a fields med metal winning be a fields med metal winning mathematician and undergrad knows mathematician and undergrad knows mathematician and undergrad knows something better than you. Yeah. Um I something better than you. Yeah. Um I something better than you. Yeah. Um I think mathematics inherently I mean think mathematics inherently I mean think mathematics inherently I mean mathematics is so huge these days that mathematics is so huge these days that mathematics is so huge these days that nobody knows all of modern mathematics.

nobody knows all of modern mathematics. nobody knows all of modern mathematics. Um and inevitably we make mistakes and Um and inevitably we make mistakes and Um and inevitably we make mistakes and um you know uh you can't cover up your um you know uh you can't cover up your um you know uh you can't cover up your mistakes with just sort of bravado and mistakes with just sort of bravado and mistakes with just sort of bravado and and uh I mean because people will ask and uh I mean because people will ask and uh I mean because people will ask for your proofs and if you don't have for your proofs and if you don't have for your proofs and if you don't have the proofs you don't have the proofs. Um the proofs you don't have the proofs. Um the proofs you don't have the proofs. Um I don't love math. Yeah. So it does keep I don't love math. Yeah. So it does keep I don't love math. Yeah. So it does keep us honest. I mean not not I mean you can us honest. I mean not not I mean you can us honest. I mean not not I mean you can still it's not a perfect uh panacea but still it's not a perfect uh panacea but still it's not a perfect uh panacea but I think uh we do have more of a culture I think uh we do have more of a culture I think uh we do have more of a culture of admitting error than because we're of admitting error than because we're of admitting error than because we're forced to all the time. Big ridiculous forced to all the time. Big ridiculous forced to all the time. Big ridiculous question. I'm sorry for it once again. question. I'm sorry for it once again. question. I'm sorry for it once again. Who is the greatest mathematician of all Who is the greatest mathematician of all Who is the greatest mathematician of all time? Maybe one who's no longer with us. time? Maybe one who's no longer with us. time? Maybe one who's no longer with us. Uh who are the candidates? Zyler, Gaus, Uh who are the candidates? Zyler, Gaus, Uh who are the candidates? Zyler, Gaus, Newton, Raman, Hilbert. So, first of Newton, Raman, Hilbert. So, first of Newton, Raman, Hilbert. So, first of all, as as mentioned before, like all, as as mentioned before, like all, as as mentioned before, like there's there's some time dependent there's there's some time dependent there's there's some time dependent on the day. Yeah. Like like if if you if on the day. Yeah. Like like if if you if on the day. Yeah. Like like if if you if you if you plot cumulatively over time, you if you plot cumulatively over time, you if you plot cumulatively over time, for example, Uklid like like sort of for example, Uklid like like sort of for example, Uklid like like sort of like is is one of the leading like is is one of the leading like is is one of the leading contenders. Um and then maybe some contenders. Um and then maybe some contenders. Um and then maybe some unnamed anonymous mathematicians before unnamed anonymous mathematicians before unnamed anonymous mathematicians before that um you know whoever came up with that um you know whoever came up with that um you know whoever came up with the concept of of numbers you know you the concept of of numbers you know you the concept of of numbers you know you know um do mathematicians today still know um do mathematicians today still know um do mathematicians today still feel the impact of Hilbert just oh yeah feel the impact of Hilbert just oh yeah feel the impact of Hilbert just oh yeah directly of everything that's happened directly of everything that's happened directly of everything that's happened in the 20th century yeah Hilbert spaces in the 20th century yeah Hilbert spaces in the 20th century yeah Hilbert spaces we have lots of things that are named we have lots of things that are named we have lots of things that are named after him of course just the arrangement after him of course just the arrangement after him of course just the arrangement of mathematics and just the introduction of mathematics and just the introduction of mathematics and just the introduction of certain concepts I mean 23 problems of certain concepts I mean 23 problems of certain concepts I mean 23 problems have been extremely influential have been extremely influential have been extremely influential there's some strange power to the there's some strange power to the there's some strange power to the declaring ing which problems are hard to declaring ing which problems are hard to declaring ing which problems are hard to solve. The statement of the open solve. The statement of the open solve. The statement of the open problems. Yeah. I mean this is bystander

problems. Yeah. I mean this is bystander problems. Yeah. I mean this is bystander effect in everywhere. Like if no one effect in everywhere. Like if no one effect in everywhere. Like if no one says you should do X, everyone just says you should do X, everyone just says you should do X, everyone just moves around waiting for somebody else moves around waiting for somebody else moves around waiting for somebody else to to uh to do something and and like to to uh to do something and and like to to uh to do something and and like nothing gets done. Um so and and like it nothing gets done. Um so and and like it nothing gets done. Um so and and like it like it's one one thing that actually uh like it's one one thing that actually uh like it's one one thing that actually uh you have to teach undergraduates in you have to teach undergraduates in you have to teach undergraduates in mathematics is that you should always mathematics is that you should always mathematics is that you should always try something. So um you see a lot of try something. So um you see a lot of try something. So um you see a lot of paralysis um in an undergraduate trying paralysis um in an undergraduate trying paralysis um in an undergraduate trying a math problem if they recognize that a math problem if they recognize that a math problem if they recognize that there's a certain technique that that there's a certain technique that that there's a certain technique that that can be applied they will try it but can be applied they will try it but can be applied they will try it but there are problems for which they see there are problems for which they see there are problems for which they see none of their standard techniques none of their standard techniques none of their standard techniques obviously applies and the common obviously applies and the common obviously applies and the common reaction is then just paralysis I don't reaction is then just paralysis I don't reaction is then just paralysis I don't know what to do or um or I think there's know what to do or um or I think there's know what to do or um or I think there's a quote from the Simpsons I've tried a quote from the Simpsons I've tried a quote from the Simpsons I've tried nothing and I'm all out of ideas um so nothing and I'm all out of ideas um so nothing and I'm all out of ideas um so you know like the next step then is to you know like the next step then is to you know like the next step then is to try anything like no matter how stupid try anything like no matter how stupid try anything like no matter how stupid um and in fact almost as stupid of the um and in fact almost as stupid of the um and in fact almost as stupid of the better um which you know and one a better um which you know and one a better um which you know and one a technique which is almost guaranteed to technique which is almost guaranteed to technique which is almost guaranteed to fail but the way it fails is going to be fail but the way it fails is going to be fail but the way it fails is going to be instructive um like it fails because you instructive um like it fails because you instructive um like it fails because you you you're not at all taking into you you're not at all taking into you you're not at all taking into account this hypothesis oh this account this hypothesis oh this account this hypothesis oh this hypothesis must be useful that's a clue hypothesis must be useful that's a clue hypothesis must be useful that's a clue I I think you also suggested somewhere I I think you also suggested somewhere I I think you also suggested somewhere this this fascinating approach which this this fascinating approach which this this fascinating approach which really stuck with me I started using it really stuck with me I started using it really stuck with me I started using it and really works I think you said it's and really works I think you said it's and really works I think you said it's called structured procrastination no yes called structured procrastination no yes called structured procrastination no yes it's when you really don't want to do a it's when you really don't want to do a it's when you really don't want to do a thing. Do you imagine a thing you don't thing. Do you imagine a thing you don't thing. Do you imagine a thing you don't want to do more? Yes. That's worse than want to do more? Yes. That's worse than want to do more? Yes. That's worse than that. And then in that way, you that. And then in that way, you that. And then in that way, you procrastinate by not doing the thing procrastinate by not doing the thing procrastinate by not doing the thing that's worse. Yeah. Yeah. It's a nice that's worse. Yeah. Yeah. It's a nice that's worse. Yeah. Yeah. It's a nice It's a nice hack. It actually works.

It's a nice hack. It actually works. It's a nice hack. It actually works. Yeah. Yeah. This um I mean with anything Yeah. Yeah. This um I mean with anything Yeah. Yeah. This um I mean with anything like you know I mean like you um like you know I mean like you um like you know I mean like you um psychology is really important like you psychology is really important like you psychology is really important like you you talk to athletes like marathon you talk to athletes like marathon you talk to athletes like marathon runners and so forth and and they talk runners and so forth and and they talk runners and so forth and and they talk about what's the most important thing is about what's the most important thing is about what's the most important thing is it their training regimen or the diet it their training regimen or the diet it their training regimen or the diet and so forth. Actually so much of it is and so forth. Actually so much of it is and so forth. Actually so much of it is actually psychology. Um you know just actually psychology. Um you know just actually psychology. Um you know just tricking yourself to to think that the tricking yourself to to think that the tricking yourself to to think that the problem is feasible um so that you can problem is feasible um so that you can problem is feasible um so that you can you're motivated to do it. Is there you're motivated to do it. Is there you're motivated to do it. Is there something our human mind will never be something our human mind will never be something our human mind will never be able to comprehend? able to comprehend? able to comprehend? Well I sort of as a mathematician I mean Well I sort of as a mathematician I mean Well I sort of as a mathematician I mean you you you there must be some suffer that you can't there must be some suffer that you can't there must be some suffer that you can't understand. That was the first thing understand. That was the first thing understand. That was the first thing that came to mind. So that but even that came to mind. So that but even that came to mind. So that but even broadly is there are we li is there broadly is there are we li is there broadly is there are we li is there something about our mind that's we're something about our mind that's we're something about our mind that's we're going to be limited even with the help going to be limited even with the help going to be limited even with the help of mathematics well okay I mean like how of mathematics well okay I mean like how of mathematics well okay I mean like how much augmentation are you willing like much augmentation are you willing like much augmentation are you willing like like for example if if I didn't even like for example if if I didn't even like for example if if I didn't even have pen and paper um like if I had no have pen and paper um like if I had no have pen and paper um like if I had no technology whatsoever okay so I'm not technology whatsoever okay so I'm not technology whatsoever okay so I'm not allowed blackboard pen and paper right allowed blackboard pen and paper right allowed blackboard pen and paper right you're already much more limited than you're already much more limited than you're already much more limited than you would be incredibly limited even you would be incredibly limited even you would be incredibly limited even language the English language is a language the English language is a language the English language is a technology technology technology It's a It's one that's been very It's a It's one that's been very It's a It's one that's been very internalized. So, you're right. There internalized. So, you're right. There internalized. So, you're right. There really the the the formulation of the really the the the formulation of the really the the the formulation of the problem is incorrect because there problem is incorrect because there problem is incorrect because there really is no longer a just a solo human.

really is no longer a just a solo human. really is no longer a just a solo human. We're already augmented in extremely We're already augmented in extremely We're already augmented in extremely complicated intricate ways, right? Yeah. complicated intricate ways, right? Yeah. complicated intricate ways, right? Yeah. Yeah. We're already like a collective Yeah. We're already like a collective Yeah. We're already like a collective intelligence. Yes. Yeah. Yes. So, intelligence. Yes. Yeah. Yes. So, intelligence. Yes. Yeah. Yes. So, humanity plural has much more humanity plural has much more humanity plural has much more intelligence in principle on it good intelligence in principle on it good intelligence in principle on it good days than than the individual humans put days than than the individual humans put days than than the individual humans put together. It can also have less. Okay. together. It can also have less. Okay. together. It can also have less. Okay. But uh um yeah, so yeah, mathemat But uh um yeah, so yeah, mathemat But uh um yeah, so yeah, mathemat mathematical community plural is is is mathematical community plural is is is mathematical community plural is is is incredibly super intelligent uh entity incredibly super intelligent uh entity incredibly super intelligent uh entity um that uh no single human mathematician um that uh no single human mathematician um that uh no single human mathematician can can come closer to to replicating. can can come closer to to replicating. can can come closer to to replicating. You see it a little bit on these like You see it a little bit on these like You see it a little bit on these like question analysis sites. Um so this math question analysis sites. Um so this math question analysis sites. Um so this math overflow which is the math version of overflow which is the math version of overflow which is the math version of stack overflow and like sometimes you stack overflow and like sometimes you stack overflow and like sometimes you get like this very quick responses to get like this very quick responses to get like this very quick responses to very difficult questions from the very difficult questions from the very difficult questions from the community. Um, and it's it's it's a community. Um, and it's it's it's a community. Um, and it's it's it's a pleasure to watch actually as a as an pleasure to watch actually as a as an pleasure to watch actually as a as an expert. I'm a fan spectator of that uh expert. I'm a fan spectator of that uh expert. I'm a fan spectator of that uh of that site, just seeing the brilliance of that site, just seeing the brilliance of that site, just seeing the brilliance of the different people, the um the of the different people, the um the of the different people, the um the depth of knowledge that people have and depth of knowledge that people have and depth of knowledge that people have and the the willingness to engage in the in the the willingness to engage in the in the the willingness to engage in the in the rigor and the nuance of the the rigor and the nuance of the the rigor and the nuance of the particular question. It's pretty cool to particular question. It's pretty cool to particular question. It's pretty cool to watch. It's fun. It's almost like just watch. It's fun. It's almost like just watch. It's fun. It's almost like just fun to watch. Uh what gives you hope fun to watch. Uh what gives you hope fun to watch. Uh what gives you hope about this whole thing we have going on, about this whole thing we have going on, about this whole thing we have going on, human civilization? I think uh yeah. Um human civilization? I think uh yeah. Um human civilization? I think uh yeah. Um the younger generation is always like the younger generation is always like the younger generation is always like like really creative and enthusiastic like really creative and enthusiastic like really creative and enthusiastic and and inventive. Um it's a pleasure and and inventive. Um it's a pleasure and and inventive. Um it's a pleasure working with with with uh with uh with working with with with uh with uh with working with with with uh with uh with young students. Um young students. Um young students. Um you know the uh the progress of science

you know the uh the progress of science you know the uh the progress of science tells us that the problems that used to tells us that the problems that used to tells us that the problems that used to be really difficult can become extremely be really difficult can become extremely be really difficult can become extremely you know can become like trivial to you know can become like trivial to you know can become like trivial to solve. you know, I mean, like it was solve. you know, I mean, like it was solve. you know, I mean, like it was like navigation, you know, just just like navigation, you know, just just like navigation, you know, just just knowing where you were on the planet was knowing where you were on the planet was knowing where you were on the planet was this horrendous problem. People died um this horrendous problem. People died um this horrendous problem. People died um you know, or or lost fortunes because you know, or or lost fortunes because you know, or or lost fortunes because they couldn't navigate, you know, and we they couldn't navigate, you know, and we they couldn't navigate, you know, and we have devices in our pockets that do this have devices in our pockets that do this have devices in our pockets that do this automatically for us, I guess, a automatically for us, I guess, a automatically for us, I guess, a completely solved problem, you know. So completely solved problem, you know. So completely solved problem, you know. So things that are seem unfeasible for us things that are seem unfeasible for us things that are seem unfeasible for us now could be maybe just sort of homework now could be maybe just sort of homework now could be maybe just sort of homework exercises for exercises for exercises for Yeah. But one of the things I find Yeah. But one of the things I find Yeah. But one of the things I find really sad about the finitness of life really sad about the finitness of life really sad about the finitness of life is that I won't get to see all the cool is that I won't get to see all the cool is that I won't get to see all the cool things we create as a civilization. You things we create as a civilization. You things we create as a civilization. You know that cuz in the next 100 years, 200 know that cuz in the next 100 years, 200 know that cuz in the next 100 years, 200 years, just imagine showing showing up years, just imagine showing showing up years, just imagine showing showing up in 200 years. Yeah. Well, already plenty in 200 years. Yeah. Well, already plenty in 200 years. Yeah. Well, already plenty has happened, you know, like if if you has happened, you know, like if if you has happened, you know, like if if you could go back in time and and talk to could go back in time and and talk to could go back in time and and talk to your teenage self or something, you know your teenage self or something, you know your teenage self or something, you know what I mean? Yeah. and just the internet what I mean? Yeah. and just the internet what I mean? Yeah. and just the internet and and our AI. I mean again they and and our AI. I mean again they and and our AI. I mean again they they've been in they're beginning to be they've been in they're beginning to be they've been in they're beginning to be internalized and say yeah of course an internalized and say yeah of course an internalized and say yeah of course an AI can understand our voice and and give AI can understand our voice and and give AI can understand our voice and and give reasonable you know slightly incorrect reasonable you know slightly incorrect reasonable you know slightly incorrect answers to to any question but yeah this answers to to any question but yeah this answers to to any question but yeah this was mind-blowing even 2 years ago and in was mind-blowing even 2 years ago and in was mind-blowing even 2 years ago and in the moment it's hilarious to watch on the moment it's hilarious to watch on the moment it's hilarious to watch on the internet and so on the the drama uh the internet and so on the the drama uh the internet and so on the the drama uh people take everything for granted very people take everything for granted very people take everything for granted very quickly and then they we humans seem to quickly and then they we humans seem to quickly and then they we humans seem to entertain ourselves with drama out of entertain ourselves with drama out of entertain ourselves with drama out of anything that's created somebody needs anything that's created somebody needs anything that's created somebody needs to take one opinion another person needs to take one opinion another person needs to take one opinion another person needs to take an opposite opinion, argue with to take an opposite opinion, argue with to take an opposite opinion, argue with each other about it. But when you look each other about it. But when you look each other about it. But when you look at the arc of things, I mean just even at the arc of things, I mean just even at the arc of things, I mean just even in progress of robotics. Yeah. Just to

in progress of robotics. Yeah. Just to in progress of robotics. Yeah. Just to take a step back and be like, "Wow, this take a step back and be like, "Wow, this take a step back and be like, "Wow, this is beautiful that we humans are able to is beautiful that we humans are able to is beautiful that we humans are able to create this." Yeah. When the create this." Yeah. When the create this." Yeah. When the infrastructure and the culture is is infrastructure and the culture is is infrastructure and the culture is is healthy, you know, the community of healthy, you know, the community of healthy, you know, the community of humans can be so much more intelligent humans can be so much more intelligent humans can be so much more intelligent and mature and and and rational than the and mature and and and rational than the and mature and and and rational than the individuals within it. Well, one place I individuals within it. Well, one place I individuals within it. Well, one place I can always count on rationality is the can always count on rationality is the can always count on rationality is the comment section of your blog, which I'm comment section of your blog, which I'm comment section of your blog, which I'm a big fan of. There's a lot of really a big fan of. There's a lot of really a big fan of. There's a lot of really smart people there. And thank you, of smart people there. And thank you, of smart people there. And thank you, of course, for uh for putting those ideas course, for uh for putting those ideas course, for uh for putting those ideas out on the blog, and it's I can't tell out on the blog, and it's I can't tell out on the blog, and it's I can't tell you how you how you how uh honored I am that you would spend uh honored I am that you would spend uh honored I am that you would spend your time with me today. I was looking your time with me today. I was looking your time with me today. I was looking forward this for a long time, Terry. I'm forward this for a long time, Terry. I'm forward this for a long time, Terry. I'm a huge fan. Um you inspire me. You a huge fan. Um you inspire me. You a huge fan. Um you inspire me. You inspire millions of people. Thank you so inspire millions of people. Thank you so inspire millions of people. Thank you so much for talking. Oh, thank you. It was much for talking. Oh, thank you. It was much for talking. Oh, thank you. It was a pleasure. a pleasure. a pleasure. Thanks for listening to this Thanks for listening to this Thanks for listening to this conversation with Terrence Tao. To conversation with Terrence Tao. To conversation with Terrence Tao. To support this podcast, please check out support this podcast, please check out support this podcast, please check out our sponsors in the description or at our sponsors in the description or at our sponsors in the description or at lexfreedman.com/sponsors. lexfreedman.com/sponsors. lexfreedman.com/sponsors. And now, let me leave you with some And now, let me leave you with some And now, let me leave you with some words from Galileo Galile.

words from Galileo Galile. words from Galileo Galile. Mathematics is a language with which God Mathematics is a language with which God Mathematics is a language with which God has written the universe. has written the universe. has written the universe. Thank you for listening and hope to see Thank you for listening and hope to see Thank you for listening and hope to see you next time.