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

The following is a conversation with Terrence Tao. Widely considered to be one of the greatest mathematicians in history. Often referred to as the Mozart of math, he won the Fields Medal and the Breakthrough Prize in mathematics and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. This was a huge honor for me for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions. It means the world. This is the Lex Freedman podcast. To support it, please check out our sponsors in the description or at lexfreedman.com/sponsors.

And now, dear friends, here's Terrence Tao.

What was the first really difficult research level math problem that you encountered? One that gave you pause maybe?

Well, I mean in your undergraduate, um, education, you learn about the really hard impossible problems like the Riemann hypothesis, the twin primes conjecture. You can make problems arbitrarily difficult. That's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting are the problems just at the on the boundary between what we can do relatively easily and what are hopeless. Um, but what are problems where like existing techniques can do like 90% of the job and then you just need that remaining 10%? Um, I think as a PhD student the Kakeya problem certainly caught my eye and it just got solved actually. It's a problem I've worked on a lot in my early research. Historically it came from a little puzzle by the Japanese mathematician Soji Kakeya, uh, in like 1918 or so.

Um, so the puzzle is that you you you have um a needle um in on the plane. Um, think like like a like driving like on on on a road something and you you want it to execute a U-turn. You want to turn the needle around. Um, but you want to do it in as little space as possible. So you want to use as little area in order to turn it around. So um, but the needle is infinitely maneuverable. So you can imagine just spinning it around its um as a unit needle. You can spin it around its center. Um, and I think, um, that gives you a disc of of area, I think pi over four. Um, or you can do a three-point U-turn, which is what they we teach people in in the driving schools to do. Uh, and that actually takes area pi over 8. So, it's it's a little bit more efficient than um a rotation. And so, for a while, people thought that was the most efficient uh way to turn things around. But, Maz'ya showed that in fact, you could actually uh turn the needle around using as little area as you wanted. So 0.0001 there was some really fancy multi- um u back and forth U-turn thing that you could you could do that that you could turn a needle around and in so doing it would pass through every intermediate direction.

Is this in the two-dimensional plane?

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

So this seems like a puzzle. Why is it interesting?

So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, comics. For example, in in wave propagation, you splash some some water around um you create water waves and they they travel in various directions. Um, but waves exhibit both both particle and wave type behavior. So you can have what's called a wave packet, which is like a a very localized wave that is localized in space and moving a certain direction in time. And so if you plot it in both space and time, it occupies a region which looks like a tube. And so what can happen is that you can have a wave which initially is very dispersed but it all comes it all focuses at a single point later in time. Like you can imagine dropping a pebble into a pond and ripples spread out. But then if you time reverse that that um that scenario and the equations of wave motion are time reversible. You can imagine ripples that are converging um to a single point and then a big splash occurs um maybe even a singularity. Um and so it's possible to do that. Uh and geometrically what's going on is that there's always s of light rays. Um so like if if if this wave represents light for example um you can imagine this wave as a superposition of photons um all traveling at the speed of light. They all travel on these light rays and they're all focusing at this one point. So you can have a very dispersed wave focus into a very concentrated wave at one point in space and time, but then it defocuses again and it separates. But potentially if the conjecture had a negative solution. So what that meant is that there's there's a very efficient way to pack um tubes pointing different directions into a very very narrow region of of of very narrow volume. Then you would also be able to create waves that start out some there'll be some arrangement of waves that start out very very dispersed but they would concentrate not just at a single point but um um there'll be a large um there'll be a lot of concentrations in space and time and uh um and you could create what's called a blowup where these waves their amplitude becomes so great that the laws of physics that they're governed by are no longer wave equations but something more complicated and nonlinear. Um and so in mathematical physics we care a lot about whether certain equations in in wave equations are stable or not whether they can create um these singularities.

There's a famous unsolved problem called the Navier-Stokes regularity problem. So the Navier-Stokes equations equations that govern the fluid flow for incompressible fluids like water. The question asks if you start with a smooth velocity field of water can it ever concentrate so much that like the velocity becomes infinite at some point that's called a singularity. We don't see that um in real life. You know, if you splash around water on the bathtub, it won't explode on you. Um or or have have water leaving at the speed of light, I think. But potentially, it is possible. Um and in fact, in recent years, the the consensus has has drifted towards the uh the belief that uh that in fact for certain very special initial configurations of of say water that singularities can form. But people have not yet been able to uh to actually establish this. The Clay foundation has these seven millennium prize problems has a million dollar prize for solving one of these problems that this is one of them. Of these seven only one of them has been solved the Poincaré conjecture by Perelman. So the Kakeya conjecture is not directly directly related to the Navier-Stokes problem but understanding it would help us understand some aspects of things like wave concentration which would indirectly probably help us understand the Navier-Stokes problem better.

Can you speak to the Navier-Stokes problem?

So the existence and smoothness like you said millennial prize problem right you've made a lot of progress on this one in 2016 you published a paper finite time blow up for an averaged three-dimensional Navier-Stokes equation right so we're trying to figure out if this thing usually doesn't blow up right but can we say for sure it never blows up right yeah so yeah that is literally the the million-dollar question yeah so this is what distinguishes mathematicians from pretty much everybody else like it If something holds 99.99% of the time, um that's good enough for most, you know, uh for for most things, but mathematicians are one of the few people who really care about whether every like 100% really 100% of all um situations are covered by by um yeah, so most fluid most of the time um water that does not blow up. But could you design a very special initial state that does this? And maybe we should say that this is a this is a set of equations that govern in the field of fluid dynamics. Trying to understand how fluid behaves and it's actually turns out to be a really comp you know fluid is yeah extremely complicated thing to try to model. Yeah. So it has practical importance. So this Clay prize problem concerns what's called the incompressible Navier-Stokes which governs things like water. There's something called the compressible Navier-Stokes which governs things like air. And that's particularly important for weather prediction. Weather prediction it does a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Navier-Stokes equations as best they can. Um also gathering a lot of data so that they can get they can in initialize the equation. There's a lot of moving parts. So it's very important practically.

Why is it difficult to prove general things about this set of equations like it not not blowing up?

Short answer is Maxwell's demon. Um so Maxwell's demon is a concept in thermodynamics like if you have a box of two gases and oxygen and hydrogen uh and maybe you start with all the oxygen one side and nitrogen the other side but there's no barrier between them right then they will mix um and they should stay mixed right there there's no reason why they should unmix but in principle because of all the collisions between them there could be some sort of weird conspiracy that that um like maybe there's a microscopic demon called Maxwell's demon that will um every time a oxygen and nitrogen atom collide they will bounce off in such a way that the oxygen sort of drifts onto one side and then goes to the other and uh you could have an extremely improbable configuration emerge. Uh which we never see. Um and and we statistically it's extremely unlikely but mathematically it's possible that this can happen and we can't rule it out. Um and this is a situation that shows up a lot in mathematics. Um a basic example is the digits of pi 3.14159 and so forth. The digits look like they have no pattern and we believe they have no pattern. On the long term, you should see as many ones and twos and threes as fours and fives and sixes. There should be no preference in the digits of pi to favor let's say 7 over 8. Um, but maybe there's some demon in the digits of pi that that like every time you compute more digits, it sort of biases one digit to another. Um and this is a conspiracy that should not happen. There's no reason it should happen, but um there's there's there's no way to prove it. uh with our current technology.

Okay. So getting back to Navier-Stokes, a fluid has a certain amount of energy and because a fluid is in motion, the energy gets transported around and water is also viscous. So if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and will it will go to zero. Um and this is what happens um in um uh when we actually experiment with water like you splash around there. there's some turbulence and waves and so forth. But eventually it it settles down and and and the the lower the amplitude, the smaller the velocity, the the more calm it gets. Um but potentially there is some sort of a demon that keeps pushing the uh the energy of the fluid into a smaller and smaller scale and it will move faster and faster and at faster speeds the effective viscosity is relatively less. And so it could happen that that it it creates a some sort of um um what's called a self-similar blowup scenario where you know um the energy of fluid starts off at some um large scale and then it all sort of um transfers it energy into a smaller um region of of of the fluid which then at a much faster rate um moves into um an even smaller region and so forth. Um and and each time it does this uh it takes maybe half as as long as as the previous one and then you you could you could actually uh converge to all the energy concentrating in one point in a finite amount of time. Um and that that's uh that scenario is called finite blow up. Um so in practice this doesn't happen. Um so water is what's called turbulent. Um so it is true that um if you have a big eddy of water it will tend to break up into smaller eddies but it won't transfer all the the energy from one big eddy into one smaller eddy. It will transfer into maybe three or four and then those must split up into maybe three or four small eddies of their own and so the energy gets dispersed to the point where the viscosity can can then keep that thing under control. Um but if it can somehow um concentrate um all the energy keep it all together um and do it fast enough that the viscous effects don't have enough time to calm everything down then this blob can occur. So there were papers who had claimed that oh you just need to take into account conservation energy and just carefully use the viscosity and you can keep everything under control for not just Navier-Stokes but for many many types of equations like this and so in the past there have been many attempts to try to obtain what's called global regularity for Navier-Stokes which is the opposite of final time blow up that velocity say smooth and it all failed there was always some sign error or some subtle mistake and and it couldn't be salvaged.

Um so what I was interested in doing was trying to explain why we were not able to disprove um finite time blow up. I couldn't do it for the actual equations of fluids which were too complicated. But if I could average the equations of motion of Navier-Stokes basically if if um if I could turn off certain types of of ways in which water interacts and only keep the ones that I want. Um, so in particular, um, if, um, if there's a fluid and it could transfer energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this this one and and direct it only into um, this smaller eddy while still preserving the law of conservation of energy.

So you're trying to make it blow up.

Yeah. Yeah. So I I I basically engineer um, a blow up by changing the laws of physics, which is one thing that mathematicians are allowed to do. We can change the equation.

How does that help you get closer to the proof of something?

Right? So, it provides what's called an obstruction in mathematics. Um, so, so what I did was that uh basically if I turned off the um certain parts of the equation, so which usually when you turn off certain interactions make it less nonlinear, it makes it more regular and less likely to blow up. But I found that by turning off a very well-designed set of of of of interactions, I could force all the energy to blow in finite time. So what that means is that if you wanted to prove um global regularity for Navier-Stokes um for the actual equation you had you must use some feature of the true equation which which my artificial equation um does not satisfy. So it it rules out certain um certain approaches. So um the thing about math is is it's not just about finding you know taking a technique that is going to work and applying it but you you need to not take the techniques that don't work. Um and for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem. But uh it's only after a lot of experience that you realize there's no way that these methods are going to work. So having these counter examples for nearby problems um kind of rules out um uh it saves you a lot of time because you you're not wasting um energy on on things that you now know cannot possibly ever work.

How deeply connected is it to that specific problem of fluid dynamics or just some more general intuition you build up about mathematics?

Right. Yeah. So the key phenomenon that uh my my technique exploits is what's called supercriticality. So in partial differential equations often these equations are like a tug-of-war between different forces. So in Navier-Stokes there's the dissipation um force coming from viscosity and it's very well understood. It's linear. It calms things down. If if viscosity was all there was, then then nothing bad would ever happen. Um but there's also transport um that that energy from in one location of space can get transported because the fluid is in motion to to other locations. Um and that's a nonlinear effect and that causes all the all the problems. Um so there are these two competing terms in the Navier-Stokes equation the dissipation term and the transport term. If the dissipation term dominates, if it's if it's large, then basically you get regularity. And if um if the transport term dominates, then uh then we don't know what's going on. It's a very nonlinear situation. It's unpredictable. It's turbulent. So sometimes these forces are in balance at small scales, but not in balance at large scales or or vice versa. Um so Navier-Stokes is what's called supercritical. So at at smaller and smaller scales, the transport terms are much stronger than the viscosity terms. So the viscosity are the things that calm things down. Um and so this is um um this is why the problem is hard in two dimensions. So the Soviet mathematician Ladyzhenskaya she in the 60s shows in two dimensions there is no blow up and in two dimensions the Navier-Stokes equations is what's called critical the effect of transport and the effect of viscosity about the same strength even at very very small scales and we have a lot of technology to handle critical and also subcritical equations and proof um regularity but for supercritical equations it was not clear what was going on and I did a lot of work and then there's been a lot of follow-up showing that for many other types of supercritical equations you create all kinds of blow up examples. Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen. So this is sort of one of the main insights of this this line of work is that supercriticality versus criticality and subcriticality. This this makes a big difference. I mean that's a key qualitative feature that distinguishes some equations for being sort of nice and predictable and you know like like planetary motion and I mean there are certain equations that that you can predict for millions of years and or thousands at least. Again, it's not really a problem, but but there's a reason why we can't predict the weather past 2 weeks into the future because it's a supercritical equation. Lots of really strange things are going on at very fine scales. So, whenever there is some huge source of nonlinearity, yeah, that can create a huge problem for predicting what's going to happen. Yeah. And if the nonlinearity is somehow more and more featured and interesting at at small scales. Um I mean there's there's many equations that are nonlinear but um in in many equations you can approximate things by the bulk. Um so for example planetary motion you know if you want to understand the orbit of the moon or Mars or something you don't really need the micro structure of like the seismology of the moon or or like exactly how the mass is distributed. um you just basically you can almost approximate these planets by point masses and just the aggregate behavior is important um but if you want to model a fluid um like like the weather you can't just say in Los Angeles the temperature is this the wind speed is this for supercritical equations the fine-grained information is is really important if we can just linger on the Navier-Stokes uh equations a little bit so you've suggested maybe you can describe it that one of the ways to uh solve it or to negatively resolve it would

Be to sort of to construct a liquid, a kind of liquid computer, right? And then show that the halting problem from computation theory has consequences for fluid dynamics. So, uh, show it in that way.

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

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

So what I realized is that if you could pull the same thing off for the actual equations. So if the equations of water support a computation so, um, like if you can imagine kind of a steampunk but really water punk, uh, type of thing where, um, you know so modern computers are electronic you know they they they're powered by by electrons passing through very tiny wires and interacting with other electrons and so forth. But instead of electrons, you can imagine these pulses of of water moving at certain velocity and maybe it's they're two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide, it would come out with some new configuration which is which would be something like an ANDgate or orgate. You know that if the the the output would depend in a very predictable way on on the inputs and like you could chain these together and maybe create a touring machine and and then you could you have computers which are made completely out of water, um, and if you have computers then maybe you can do robotics so I you know hydraulics and so forth, um, and so you could create some machine which is basically a fluid analog what's called a vonomian machine so vonomian proposed if you want to colonize Mars. The sheer cost of transporting people machines to Mars is just ridiculous. But if you could transport one machine to Mars and this machine had the ability to mine the planet, create some more materials to smelt them and build more copies of the same machine. Um, then you could colonize a whole planet, um, over time. Um, so uh if you could build a fluid machine, which, uh, yeah, so it's it's it's a it's a robot.

Okay. And what it would do it its purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state. It wouldn't start just yet. Once it's ready, the big robot configuration water would transfer all his energy into the smaller configuration and then power down. Okay? And then like I clean itself up. And then what's left is this newest state which would then turn on and do the same thing but smaller and faster. And then the equation has a certain scaling symmetry. Once you do that, it can just keep iterating. So this in principle would create a blow up, uh, for the actual Navia Stokes and this is what I managed to accomplish for this average Navia Stokes. So it provided the sort of road map to solve the problem.

Now this is, uh, a pipe dream because, uh, there are so many things that are missing for this to actually be a reality. Um, so, um, I I I can't create these basic logic gates. Um, I I don't I don't have these in these special configurations of water. Um, I mean there's candidates there things called vortex rings that might possibly work but, um, um, but also you know analog computing is really nasty, um, compared to digital computing. I mean because there's always errors, um, you you have to you have to do a lot of error correction along the way. I don't know how to completely power down the big machine so that it doesn't interfere with the the running of the smaller machine but everything in principle can happen like it doesn't contradict any of the laws of physics. Um, so it's sort of evidence that this thing is possible. Um, there are other groups who are now pursuing ways to make navis blow up which are nowhere near as ridiculously complicated as this. Um, um, they they actually are pursuing much closer to the the direct self similar model which can it doesn't quite work as is but there could be some simpler scheme than what I just described to make this work.

There is a real leap of genius here to go from Navia Stokes to this touring machine. So it goes from what the self similar blob scenario that you're trying to get the smaller and smaller blob to now having a liquid toying machine gets smaller and smaller and smaller and somehow seeing how that could be used to say something about a blowup. I mean that's a big leap.

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

Mhm. So they could build something like that. And once you could build and, um, these basic gates then just from software engineering you can build almost anything. Um, you can build a touring machine. I mean it's like an enormous steampunk type things. They look ridiculous. But then people also generated self-replicating objects in the game of life. A massive machine a bon machine which over a huge period of time and it always look like glider guns inside doing these very steampunk calculations. It would create another version of itself which could replicate. It's so incredible. A lot of this was like community crowdsourced by like amateur mathematicians actually. Um, so I knew about that that that work and so that is part of what inspired me to propose the same thing with Navia Stokes. Um, which is a much as I said analog is much worse than digital like it's going to be, um, you can't just directly take the constructions in the game of life and plunk them in. But again it just it shows it's possible. You know, there's a kind of emergence that happens with these cellular automa. Local rules. Maybe it's similar to fluids. I don't know. But local rules operating at scale can create these incredibly complex dynamic structures.

Do you think any of that is amendable to mathematical analysis? Do we have the tools to say something profound about that? The thing is you can get this emerg in very complicated structures but only with very carefully prepared initial conditions. Yeah. So so these these these glider guns and and gates and and so forth machines if you just plunk down randomly some cells and you and you will not see any of these. Um, and that's the analogous situation with Navia Stokes again you know that that with with typical initial conditions you you will not have any of this weird computation going on. Um, but basically through engineering you know by by by specially designing things in a very special way you can make clever constructions. I wonder if it's possible to prove the sort of the negative of like basically prove that only through engineering can you ever create something interesting.

This this is a recurring challenge in mathematics that, um, I call it the dichotomy between structure and randomness. That most objects that you can generate in mathematics are random. They look like rand like the digits of pi. Well, we believe is a good example. Um, but there's a very small number of things that have patterns. Um, but, um, now you can prove something has a pattern by just constructing, you know, like if something has a simple pattern and you have a proof that it it does something like repeat itself every so often. You can do that. But, um, and you you can prove that that for example, you can you can prove that most sequences of of digits have no pattern. Um, so like if you just pick digits randomly, there's something called low large numbers. It tells you you're going to get as many ones as as twos in the long run. Um, but, um, we have a lot fewer tools to to to if I give you a specific pattern like the digits of pi how can I show that this doesn't have some weird pattern to it.

Some other work that I spend a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is is very structured. So some functions are what's called additive like if you have a function that maps natural numbers with natural numbers. So maybe, um, you know two maps to four three maps to six and so forth. um, some functions what's called additive which means that if you add if you add two inputs together the output gets gets added as well, uh, for example multiplying by a constant if you multiply a number by 10, um, if you if you multiply a plus b by 10 that's the same as multiplying a by 10 and b by 10 and then adding them together so some, um, functions are additive some are kind of additive but not completely additive, um, so for example if I take a number n I multiply by the square root of two and I take the integer part of that So 10 by square of two is like 14 point something. So 10 up to 14. Um, 20 up to 28. Um, so in that case additively is true then. So 10 + 10 is 20 and 14 + 14 is 28. But because of this rounding sometimes there's roundoff errors and and sometimes when you, um, add a plus b this function doesn't quite give you the sum of of the two individual outputs but the sum plus minus one. Um, so it's almost additive but not quite additive. Um, so there's a lot of useful results in mathematics and I've worked a lot on developing things like this to the effect that if if a function exhibits some structure like this then, um, it's basically there's a reason for why it's true and the reason is because there's there's some other nearby function which is actually, um, completely structured which is explaining this sort of partial pattern that you have. Um, and so if you have these so inverse theorems it, um, it creates this sort of dichotomy that they either the objects that you study are either have no structure at all or they are somehow related to something that is structured. Um, and in either way in either, um, in either case you can make progress.

Um, a good example of this is that there's this old theorem in mathematics called sim theorem proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers. the patterns that have make progression things like 3 five and seven or or or 10 15 and 20 andreli proved that, um, any set of of numbers that are sufficiently big, um, what's called positive density has, um, arithmetic progressions in it of of any length you wish, um, so for example, um, the odd numbers have a set of density 1/2, um, and they contain arithmetic progressions of any length, um, so in that case it's obvious because the the odd numbers are really really structured I can just take 11 13 15 17 I just I can I can easily find arithmetic progressions in in in that set. Um, but, um, zerminism also applies to random sets. If I take the set of odd numbers and I flip a coin, um, and for each number and I only keep the numbers which for which I got a heads okay so I just flip coins. I just randomly take out half the numbers I keep one half. So that's a set that has no no patterns at all. But just from random fluctuations, you will still get a lot of, um, um, of arithmetic progressions in that set.

Can you prove that there's arithmetic progressions of arbitrary length within a random? Yes. Um, have you heard of the infinite monkey theorem? Usually mathematicians give boring names to theorists, but occasionally they they give colorful names. Yes. The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each with a typewriter they type out, uh, text randomly almost surely one of them is going to generate the entire screw of Hamlet or any other finite string of text. Uh, it will just take some time quite a lot of time actually but if you have an infinite number then it happens. Um, so, um, basically the the if you take an infinite string of of digits or whatever, um, eventually any finite pattern you wish will emerge. Um, it may take a long time but it will eventually happen. Um, in particular arithmetic progressions of any length will eventually happen. Okay. But you need that but you need an extremely long random sequence for this to happen. I suppose that's intuitive. It's just infinity. Yeah. Infinity absorbs a lot of sins. Yeah. How are we humans supposed to deal with infinity?

Well, you can think of infinity as as as just an abstraction of, um, a finite number for which you you do not have a bound for, um, that, uh, you know I mean so nothing in real life is truly infinite. Um, but you know you can, um, you know you can ask yourself questions like you know what if I had as much money as I wanted you know or what if I could go as fast as I wanted and a way in which mathematicians formalize that is mathematics has found a formalism to idealize instead of something being extremely large or extremely small to actually be exactly infinite or zero. Um, and often the the mathematics becomes a lot cleaner when you do that. I mean in physics we we joke about, uh, assuming spherical cows. um, you know like real world problems have got all kinds of real world effects but you can idealize send certain things to infinity send certain things to zero, um, and, um, and the mathematics becomes a lot simpler to work with there. I wonder how often using infinity, uh, forces us to deviate from, um, the physics of reality. Yeah. So there's a lot of pitfalls. Um, so you know we we spend a lot of time in undergraduate math classes teaching analysis. Um, and analysis is often about how to take limits and and and and whether you you know so for example a plus b is always b plus a. Um, so when you have a finite number of terms you add them you can swap them and there there's no problem. But when you have infinite number of terms there these sort of shell games you can play where you can have a series which converges to one value but you rearrange it and it suddenly converges to another value. And so you can make mistakes. You have to know what you're doing when you allow infinity. Um, you have to introduce these epsilons and deltas and and this there's a certain type of way of reasoning that helps you avoid mistakes. Um, in more recent years, um, people have started taking results that are true in infinite limits and what's called finetizing them. Um, so you know that something's true eventually but, um, you don't know when. Now give me a rate. Okay. Okay, so it's such a if I have don't have an infinite number of monkeys but but a large finite number of monkeys, how long do I have to wait for H to come out? Um, and that's a more quantitative question. Um, and this is something that you can you can, um, attack by purely finite methods and you can use your finite intuition. Um, and in this case it turns out to be exponential in the length of the text that you're you're trying to generate. Um, so, um, and so this is why you never see the monkeys create Hamilton. You can maybe see them create a four-letter word, but nothing that big. And so I personally find once you finitize an infinite statement, it's it does become

Much more intuitive, and it's no longer so weird. Um, so even if you're working with infinity, it's good to finitize so that you can have some intuition. Yeah. The downside is that the finite groups are just much, much messier, and uh, yeah. So the infinite ones are found first, usually like decades earlier, and then later on people finitize them.

So since we mentioned a lot of math and a lot of physics, uh, what is the difference between mathematics and physics as disciplines, as ways of understanding, of seeing the world? Maybe we can throw in engineering in there. You mentioned your wife is an engineer, give it a new perspective on circuits, right? So this different way of looking at the world, given that you've done mathematical physics, so you've worn all the hats, right?

So I think science in general is an interaction between three things: um, there's the real world, um, there's what we observe of the real world, our observations, and then our mental models as to how we think the world works. Um, so we can't directly access reality. Okay. Uh, all we have are the observations, which are incomplete, and they have errors. Um, and there are many, many cases where we would um, uh, we want to know, for example, what is the weather like tomorrow, and we don't yet have the observation; we'd like a prediction. Um, and then we have these simplified models, sometimes making unrealistic assumptions—you know, spherical cow type things. Those are the mathematical models. Mathematics is concerned with the models. Science collects the observations and it proposes the models that might explain these observations.

What mathematics does: we stay within the model and we ask what are the consequences of that model? What observations would, what predictions would the model make of future observations, um, or past observations? Does it fit observed data? Um, so there's definitely a symbiosis; it's, I guess mathematics is unusual among other disciplines is that we start from hypotheses, like the axioms of a model, and ask what conclusions come up from that model. Um, in almost any other discipline, uh, you start with the conclusions—you know, I want to do this, I want to build a bridge, you know, I want to make money, I want to do this. Okay. And then you find the path to get there. Um, a lot, there's a lot less sort of speculation about, suppose I did this, what would happen? Um, you know, planning and modeling, um, speculative fiction maybe is one other place. Uh, but uh, that's about it, actually. Most of the things we do in life is conclusions driven, including physics and science. You, I mean, they want to know, you know, where is this asteroid going to go? What was, what is the weather going to be tomorrow? Um, but mathematics also has this other direction of going from the axioms.

What do you think? There is this tension in physics between theory and experiment. Mhm. What do you think is the more powerful way of discovering truly novel ideas about reality?

Well, you need both top-down and bottom-up. Um, yeah, it's a real interaction between all these things. So over time, the observations and the theory and the modeling should both get closer to reality. But initially, and it is, I mean, this is always the case. You know, they're always far apart to begin with. Um, but you need one to figure out where to push the other, you know. So, um, if your model is predicting anomalies um that are not picked up by experiment, that tells experimenters where to look, you know, um, to find more data to refine the models. Um, yeah, so it goes back and forth.

Within mathematics itself, there's also a theory and experimental component. It's just that until very recently, theory has dominated almost completely—like 99% of mathematics is theoretical mathematics, and there's a very tiny amount of experimental mathematics. Um, I mean, people do do it, you know, like if they want to study prime numbers or whatever, they can just generate large data sets, and with a, so once we had computers, um, we were able to do it a little bit. Um, although even before, well, like Gauss, for example, he discovered, he conjectured the most basic theorem in number theory, to call the prime number theorem, which predicts how many primes there are up to a million, up to a trillion. It's not an obvious question, and basically what he did was that he computed, I mean mostly by himself, but also hired human computers—people whose professional job it was to do arithmetic—to compute the first 100,000 primes or something and made tables and made a prediction. Um, that was an early example of experimental mathematics, um, but until very recently it was not, yeah, I mean, theoretical mathematics was just much more successful, I mean, because doing complicated mathematical computations was just not feasible until very recently.

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

And I do hope that mathematics will have a larger experimental component in the future, perhaps powered by AI. We'll of course talk about that, but in the case of chess, and there's a similar thing in mathematics, that I don't believe it's providing a kind of formal explanation of the different positions. It's just saying which position is better or not, that you can intuit it as a human being, and then from that we humans can construct a theory of the matter.

You've mentioned Plato's cave allegory. Mhm. So, in case people don't know, it's where people are observing shadows of reality, not reality itself, and they believe what they're observing to be reality. Is that, in some sense, what mathematicians and maybe all humans are doing—is looking at shadows of reality? Is it possible for us to truly access reality?

Well, there are these three ontological things: there's actual reality, there's our observations, and our models. Um, and technically they are distinct, and I think they will always be distinct. Um, but they can get closer over time. Um, you know, so, and the process of getting closer often means that you have to discard your initial intuitions. Um, so, like astronomy provides great examples, you know, like, you know, like your initial model of the world is flat because it looks flat, you know, and, and it's big, you know, and the rest of the universe, the skies is not, you know, like the sun, for example, looks really tiny, um, and so you start off with a model which is actually really far from reality, um, but it fits kind of the observations that you have, um, you know, so, you know, so things look good, you know, but but over time as you make more and more observations, bring it closer to reality. Okay. Um, the model gets dragged along with it, and so over time we had to realize that the Earth was round, that it spins, it goes around the solar system, the solar system goes around the galaxy, and so on and so forth. And the universe is expanding, the expansion itself is expanding, accelerating, and in fact, very recently, in this year. So this, even the acceleration of the universe itself is this evidence that this is non-constant, and the explanation behind why that is, it's catching up.

It's catching up. I mean, it's still, you know, the dark matter or dark energy, this kind of thing. We have, we have a model that sort of explains that fits the data really well. It just has a few parameters that um you have to specify. Um, but so, you know, people say that's fudge factors, you know, with enough fudge factors you can explain anything. Um, but the mathematical point of the model is that um you want to have fewer parameters in your model than data points in your observational set. So if you have a model with 10 parameters that explains 10 observations, that is a completely useless model. It's what's called overfitted. But like if you have a model with, you know, two parameters and it explains a trillion observations, which is basically, so yeah, the dark matter model, I think has like 14 parameters and it explains petabytes of data um that the astronomers have.

Um, you can think of a theory, like one way to think about physical, math theory, theory, is it's a compression of the universe, um, and data compression. So you know, you have these petabytes of observations, you'd like to compress it to a model which you can describe in five pages and specify a certain number of parameters, and if it can fit to reasonable accuracy, you know, almost all of your observations. I mean, the more compression that you make, the better your theory. In fact, one of the great surprises of our universe and of everything in it is that it's compressible at all. It's the unreasonable effectiveness of mathematics. Yeah, Einstein had a quote like that. The most incomprehensible thing about the universe is that it is comprehensible, right? And not just comprehensible. You can do an equation like E=MC². There is actually some mathematical possible explanation for that.

Um, so there's this phenomenon in mathematics called universality. So many complex systems at the macro scale are coming out of lots of tiny interactions at the micro scale, and normally, because of the combinatorial explosion, you would think that the macro scale equations must be like infinitely, exponentially more complicated than the micro scale ones, and they are, if you want to solve them completely exactly—like if you want to model all the atoms in a box of air, that's like Avogadro's number is humongous, right? There's a huge number of particles; if you actually have to track each one, it'll be ridiculous. This, but certain laws emerge at the macroscopic scale that almost don't depend on what's going on at the micro scale, or only depend on a very small number of parameters. So if you want to model a gas of, you know, quintillion particles in a box, you just need to know its temperature and pressure and volume and a few parameters, like five or six, and it models almost everything you need to know about these 10 to the 23 or whatever particles.

Um, so we, we don't understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs. Um, the most basic one is the central limit theorem, that explains why the bell curve shows up everywhere in nature, that so many things are distributed by what's called a Gaussian distribution—the famous bell curve. There's now even a meme with this curve, and even the meme applies broadly universality to the meme.

Yeah. Yes, you can go meta if you like. But there are many, many processes, for example, you can take lots and lots of independent random variables and average them together, um, in various ways. You take a simple average or more complicated average, and we can prove in various cases that these bell curves, these Gaussians emerge, and it is a satisfying explanation. Um, sometimes they don't. Um, so so if you have many different inputs and they're all correlated in some systemic way, then you can get something very far from a bell curve show up. Uh, and this is also important to know when this system fails. So universality is not a 100% reliable thing to rely on, that um, the global financial crisis was a famous example of this. Uh, people thought that um mortgage defaults um had this sort of Gaussian type behavior, that that if you, if you ask, if a population of 100,000 Americans with mortgages, ask what proportion of them would default on the mortgages. Um, if everything was decorrelated, it would be a nice bell curve, and and like you can manage risk with options and derivatives and so forth, and um, and it there's a very beautiful theory, um, but if there are systemic shocks in the economy uh that can push everybody to default at the same time, that's very non-Gaussian behavior, um, and uh this wasn't fully accounted for in 2008. Now I think there's some more awareness that this systemic risk is actually a much bigger issue, and just because the model is pretty and nice, uh it may not match reality.

Right. So the mathematics of working out what models do is really important. Um, but also the science of validating when the models fit reality and when they don't. Um, I mean, that you need both. Um, and but mathematics can help because it, it can, for example, these central limit theorems, it tells you that if you have certain axioms, like like non-correlation, that if all the inputs were not correlated to each other, um, then you have this kind of behavior—things are fine. It tells you where to look for weaknesses in the model. So if you have a mathematical understanding of the central limit theorem and someone proposes to use these Gaussian copies or whatever to model um default risk, um if you're mathematically trained, you would say, okay, but what if this systemic correlation between all your inputs? And so then then you can ask the economists, you know, how much of a risk is that, um, and then you can, you can, you can go look for that. So there's always this, this synergy between science and mathematics.

A little bit on the topic of universality. Mhm. You're known and celebrated for working across an incredible breadth of mathematics, reminiscent of Hilbert a century ago. In fact, the great Fields Medal-winning mathematician Tim Gowers has said that you are the closest thing we get to Hilbert. He's a colleague of yours.

Oh, yeah. Good friend. But anyway, so you are known for this ability to go both deep and broad in mathematics. So you're the perfect person to ask, do you think there are threads that connect all the disparate areas of mathematics? Is there a kind of deep underlying structure to all of mathematics?

There's certainly a lot of connecting threads. Um, and a lot of the progress of mathematics has, can be represented by taking, by stories of two fields of mathematics that were previously not connected and finding connections. Um, an ancient example is geometry and number theory, you know. So, so in the times of the ancient Greeks, these were considered different subjects. Um, I mean, mathematicians worked on both. You know, you could work both on geometry, most famously, but also on numbers. Um, but they were not really considered related. Um, I mean, a little bit like, you know, you could say that that this length was five times this length because you could take five copies of this length and so forth. But it wasn't until Descartes who really realized that, who developed analytic geometry, that you can parameterize the plane, a geometric object, by two real numbers. Every point can be, and so geometric problems can be turned into problems about numbers. Um, and today this feels almost trivial, like, like there's no content to this, like, of course, you, you know, a plane is x, x and y, because that's what we teach and it's internalized. Um, but it was an important development that these two fields were unified. Um, and this process has just gone on throughout mathematics over and over again. Algebra and geometry were separated, and now we have a subject, algebraic geometry, that connects them, and over and over again, and that's certainly the type of mathematics that I enjoy the most.

So I think there's sort of different styles to being a mathematician. I think hedgehogs and foxes—a fox knows many things a little bit, but a hedgehog knows one thing very, very well. Um, and in mathematics, there's definitely both hedgehogs and foxes. Um, and then there's people who are kind of, who can play both roles. Um, and I think like ideal collaboration between mathematicians involves a very, you need some diversity, like a fox working with many hedgehogs, or or vice versa. So yeah, but I identify mostly as a fox—certainly, I like arbitrage, somehow, you like, like learning how one field works, learning the tricks of that field, and then going to another field which people don't think is related, but I can adapt the tricks. So see the connections between the fields. Yeah.

So there are other mathematicians who are far deeper than I am, like who really, they're really hedgehogs. They know everything about one field, and they're much faster and and more effective in that field. But I can give them these extra tools.

I mean, you said that you can be both the hedgehog and the fox depending on the context, depending on the collaboration. So what can you, if it's at all possible, speak to the difference between those two ways of thinking about a problem? Say you're encountering a new problem, you know, searching for the connections versus like very singular focus.

I'm much more comfortable with the fox paradigm. Yeah. So, um, yeah, I like looking for analogies, narratives. Um, I spend a lot of time, if there's a result I see in one field and I like the result, it's a cool result, but I don't like the proof—like it uses types of mathematics that I'm not super familiar with. Um, I often try to reprove it myself using the tools that I favor. Um, often my proof is worse. Um, but by the exercise of doing so, um, I can say, oh, now I can see what the other proof was trying to do. Um, and from that I can get some understanding of the tools that are used in that field. So it's very exploratory, very doing crazy things in crazy fields and like reinventing the wheel a lot. Yeah.

Whereas the hedgehog style is, I think much more scholarly, you know, you're very knowledge-based. You, you stay up to speed on like all the developments in this field. You know all the history. Um, you have a very good understanding of exactly the strengths and weaknesses of each particular technique. Um, yeah, I think you rely a lot more on sort of calculation than sort of trying to find narratives. Um, so yeah, I mean, I can do that too, but there are other people who are extremely good at that.

Let's step back and maybe look at a bit of a romanticized version of mathematics. Mhm. So I think you've said that early on in your life, uh, math was more like a puzzle-solving activity when you were young. When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it?

That's a good question. Um, when I came to graduate school in Princeton, um, so John Conway was there at the time. He passed away a few years ago. But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof. So Conway had just had this amazing way of thinking about all kinds of things in a way that you would normally think of. So he thought of proofs themselves as occupying some sort of space, you know. So, so if you want to prove something, let's say that there's infinitely many primes, okay, you have different proofs, but you could rank them in different axes, like some proofs

Are elegant, some are long, some proofs are are um elementary and so forth. Um and so there's this cloud. So the space of all proofs itself has some sort of shape. Um and so he was interested in in extreme points of this shape like out of all all these proofs what is one that is the shortest at the the extent of every everything else or or the most elementary or or whatever. Um and so he gave some examples of well-known theorems and then he would give what he thought was was the extreme proof um in these different aspects. Um and I I just found that really eye-opening um that that um you know it's not just getting a proof for a result was interesting but but once you have that proof you know trying to to uh to optimize it in various ways. Um that that proof um uh proofing itself had some craftsmanship to it.

Um it it certainly informed my writing style. Um but you know like when you do your your math assignments and as undergraduate your homework and so forth, you you're sort of encouraged to just write down any proof that works, okay, and hand it in and get a get as long as it gets a tick mark, you you move on. Um but if you want your your results to actually be influential and be read by people, um it can't just be correct. It should also um be a pleasure to read, you know, um motivated um be adaptable to to generalize to other um things.

Um it's the same in many other disciplines like like coding. It's a there's a lot of analogies between math and coding. I like analogies if you haven't noticed. Um but um you know like you can code something spaghetti code that works for a certain task and it's quick and dirty and it works. But uh there's lots of good principles for for um writing code well so that other people can use it build upon it and so on and has fewer bugs and whatever. Um and there's similar things with mathemat mathematics. So yeah the first of all there's so many beautiful things there and and is one of the great minds uh in mathematics ever and computer science. Uh just even considering the space of proofs. Yeah. and saying, "Okay, what does this space look like and what are the extremes?"

Uh, like you mentioned, coding as an analogy is interesting because there's also this activity called the code golf. Oh, yeah. Yeah. Yeah. Which I also find beautiful and fun where people use different programming languages to try to write the shortest possible program that accomplishes a particular tasks. Then I believe there's even competitions on this. Yeah. And uh it's also a nice way to stress test not just the sort of the programs or in this case the proofs but also the different languages maybe that's the different notation or whatever to use to to accomplish a different task. Yeah, you learn a lot. I mean it may seem like a frivolous exercise but it can generate all these insights which if you didn't have this artificial um objective to to to pursue you might not see.

What to you is the most beautiful or elegant equation in mathematics? I mean one of the things that people often look to in in beauty is the simplicity. So if you look at E=MC², so when when a few concepts come together that's why the oiler identity is often considered uh the most beautiful equation in mathematics. Do you do you find beauty in that one and the oil identity? Yeah. Well, as I said, I mean, what I find most appealing is is connections between different things that um so the if ei= minus one um so yeah people oh uses all the fundamental constants okay that that's I mean that's cute um but but to me so the exponential function was interested by oil to measure exponential growth you know so compound interest or decay anything which is continuously growing continuously decreasing growth and decay or dilation or contraction is modeled by the exponential function Um whereas pi uh comes around from circles and rotation right if you want to rotate a needle for example 180° you need to rotate by pi radians and i complex numbers represents the swing between imagine axis of a 90° rotation so a change in direction so the x function represents growth and decay in the direction where you really are um when you stick an i in the exponential it now it's it's instead of motion in the same direction as your current position it's the motion has right angles to composition. So rotation um and then so e e pi equ= minus 1 tells you that if you rotate for time pi you end up at the other direction. So it unifies geometry through dilation and exponential growth or dynamics through this act of of complexification rotation by by i. So it connects together all these tools mathematics. Yeah. Yeah. dynamic structure and complex and complex and um the complex numbers they all considered almost yeah they were all next door neighbors in mathematics because of this identity.

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

Um and this uh helped actually uh this change of perspective really helped when quantum mechanics came along. Uh because um the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking because everything was a particle and so forth to to to quantum mechanics, you know, because I think because it was a wave. It just looked really really weird. Um like you ask what is the quantum version of F equals MA? And it's really really hard to to give an answer to that. Um but it turns out that the Hamiltonian which was so um secretly behind the scenes in classical mechanics also is the key uh object in um um in quantum mechanics that there's there's also an object called Hamiltonian. It's a different type of object. It's what's called an operator rather than than a function. But um and um but again once you specify it you specify the entire dynamics. So there's something called Shingers equation that tells you exactly how quantum systems evolve once you have a Hamiltonian. So side by side they look completely different objects you know like so one involves particles one involves waves and so forth but with this centrality you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.

For example, in classical mechanics, there's this thing called ner's theorem. Every time there's a symmetry in a physical system, there is a conservation law. So the laws of physics are translation invariant. Like if I move 10 steps to the left, I experience the same laws of physics as if I was here. And that corresponds to conservation momentum. Um if I turn around by by some angle again, I experience the same laws of physics. This corresponds to conservation angular momentum. If I wait for 10 minutes, um I still have the same laws of physics. Um so this time translation variance. this corresponds to the low conservation of energy. Um, so there's this fundamental connection between symmetry and conservation. Um, and that's also true in quantum mechanics. Even though the equations are completely different, but because they're both coming from the Hamiltonian, the Hamiltonian controls everything. Um, every time the Hamiltonian has a symmetry, the equations will will have a conservation law. Um, so it's it's it's it's once you have the right language, it actually makes things um a lot a lot cleaner.

One of the problems why we can't unify quantum mechanics and general relativity yet we haven't figured out what the fundamental objects are like for example we have to give up the notion of space and time being these almost uklidian type spaces and there has to be um you know and you know we kind of know that at very tiny scales um there's going to be quite fluctuations of space space-time foam um and trying to to use cartigian coord xyz is going to be it's it's just it's it's a non-starter but we don't know how to what to replace it with um We don't actually have the mathematical um um concepts the analog Hamiltonian that sort of organized everything.

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

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

This is why analogies are so important, you know. I mean, so yeah, the round earth is not intuitive because we're stuck on it, but you know, but you know, but round objects in general, we have pretty good intuition over uh and we have intuition about light works and so forth. And like it's it's actually a good exercise to actually work out how eclipses and phases of of the sun and the moon and so forth can be really easily explained by by by by round earth and round moon, you know, um and models. Um and and you can just take you know a basketball and a golf ball and and and a light source and actually do these things yourself. Um so the intuition is there. Um but yeah you have to transfer it. That is a big leap intellectually for us to go from flat to round earth because you know our life is mostly lived in flat land. Yeah. To load that information and we all like take it for granted. We take so many things for granted because science has established a lot of evidence for this kind of thing. But you know, we're on a round rock. Yeah. Flying through space. Yeah. Yeah. And it's a big leap and you have to take a chain of those leaps the more and more and more we progress. Right. Yeah. So modern science is maybe again a victim of its own success is that you know in order to be more accurate it has to to move further and further away from your initial intuition. And so um for someone who hasn't gone through the whole process of science education it looks more more suspicious because of that. So, you know, we we need we need more grounding. I mean, I I think um I mean, you know, there are there are scientists who do excellent outreach. Um but there's this there this there's there there's lots of science things that you can do at home. There's lots of YouTube videos. I did a YouTube video recent of Grant Sanderson. We talked about this earlier that uh you know how the ancient Greeks were able to measure things like the distance to the moon, distance to the earth, and you know, using techniques that you you could also replicate yourself. Um it doesn't all have to be like fancy space telescopes and and very intimidating mathematics. Yeah, that's uh I highly recommend that. I believe you give a lecture and you also did an incredible video with Grant. It's a beautiful experience to try to put yourself in the mind of a person from that time. Mhm. Shrouded in mystery, right? You know, you're like on this planet, you don't know the shape of it, the size of it. You see some stars, you see some you see some things and you try to like localize yourself in this world. Yeah. Yeah. And try to make some kind of general statements about distance to places. Change your perspective is really important. You say travel bordens the mind. This is intellectual travel. You know put yourself in the mind of the ancient Greeks or or some other person some other time period. Make hypothesis spherical cows whatever you know speculate. Um and you know this is this is what mathematicians do and some what artists do actually. It's just incredible that given the extreme constraints, you could still say very powerful things. That's why it's inspiring looking back in history. How much can be figured out right when you don't have much to figure out stuff like if you propose axioms then the mathematics lets you follow those a to their conclusions and sometimes you can get quite a quite a long way from you know initial hypothesis.

If we can stay in the land of the weird, you mentioned general relativity. You've uh you've contributed uh to the mathematical understanding of Einstein's field equations. Can you explain this work and from a sort of mathematical standpoint uh what aspects of general relativity are intriguing to you, challenging to you? I have worked on some equations. There's something called the the wave maps equation or the sigma field model which is not quite the equation of space-time gravity itself but of certain fields that might exist on top of spaceime. Um so Einstein's equations of relativity just describes space and time itself. Um but then there's other fields that live on top of that. There's the electromagnetic field. Um there's control fields and there's this whole hierarchy of different equations of which Einstein is considered one of the most nonlinear and difficult. But relatively low in the hierarchy was this thing called the wave maps equation. So it's a wave which at any given point uh is fixed to be like on a sphere. Um so uh I can think of a bunch of arrows in space and time and and the arrows pointing in in different directions. Um but they propagate like waves. If you wiggle an arrow it was it will propagate and make all the arrows move kind of like sheets of wheat in the wheat field. And I was interested in the global regularity problem again for this question like is it possible for for all the energy here to collect at a point. So the equation I considered was actually what's called a critical equation where it's actually the behavior at all scales is roughly the same. Um and I was able barely to show that um that you couldn't actually force a scenario where all the energy concentrated at one point that the energy had to disperse a little bit and the moment it dis little bit it it would it would stay regular. Yeah. This was back in 2000. That was part of why I got interested in narrows afterwards actually. Yeah. So I developed some techniques to um solve that problem. So part of it is it was um this problem is really nonlinear uh because of the curvature of the sphere. Um this there was a certain nonlinear effect which was a non-perturbative effect. It was when you sort of looked at it normally it looked larger than the linear effects of the wave equation. Um and so it was hard to to keep things under control even when the energy was small. But I developed what's called a gauge transformation. So the equation is kind of like an evolution of of of heaves of wheat and and they're all bending back and forth and so there's a lot of motion. Um but like if you imagine like stabilizing the flow by attaching little cameras at different points in space which are trying to move in a way that captures most of the motion and under this stabilized flow the flow becomes a lot more linear. I discovered a way to transform the the equation to reduce the amount of of nonlinear effects. Um and then I was able to to to to solve the equation. I found this transformation while visiting my aunt in Australia and I was trying to understand the dynamics of all these fields and I I couldn't do it with pen and paper. Um and I had not enough facility of computers to do any computer simulations. So I ended up closing my eyes being on on the floor and just imagining myself to actually be this vector field and rolling around to try to to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion. And yeah, my aunt walked in on me while I was doing that and she was asking what do I what am I doing doing this? It's complicated is the answer. Yeah. Yeah. And you know, okay, fine. You know, you're a young man. I don't ask questions. I I I have to ask about the you know um how do you approach solving difficult problems? What if it's possible to go inside your mind when you're thinking? Are you visualizing in your mind the mathematical objects symbols maybe what are you visualizing in your mind usually when you're thinking um a lot of pen and paper one thing you pick up as a mathematician is sort of uh I call it cheating strategically um so u the the beauty of mathematics is that is that you get to change the rule change the problem change the rules as you

Wish this you don't get to do this for any other field. Like, you know, if if you're an engineer and someone says, "Build a bridge over this," you can't say, "I want to build this up bridge over here instead," or "I want to build it out of paper instead of steel." Um, but a mathematician, you can you can do whatever you want. Um, it's it's like trying to solve a computer game where you can, there's unlimited cheat codes available. Uh, and so you know, you you can you can set this. So there's a dimension that's too large. I'll set it to one. I'd solve the one-dimension problem first. So there's a main term and an error term. I'm going to make a spherical cow assumption. I'll assume the error term is zero.

And so the way you should solve these problems is is not in sort of this Iron Man mode where you make things maximally difficult. Um, but actually the way you should you should approach any reasonable math problem is that you if if there are 10 things that are making your life difficult, find a version of the problem that turns off nine of the difficulties but only keeps one of them. Um, and so that um and then that just so you you you install nine cheats. Okay. You install 10 cheats, then then the game is trivial. You solve nine cheats, you solve one problem that that that teaches you how how to deal with that particular difficulty, and then you turn that one off and you turn someone else something else else on, and then you solve that one. And after you you know how to solve the 10 problems 10 difficulties separately, then you have to start merging them a few at a time.

Um, I I as a kid I watched a lot of these Hong Kong action movies. Um, it's from a culture. Um, and uh one thing is that every time there was a fight scene, you know, so maybe the the hero will get swarmed by a hundred bad guy goons or whatever. But it would always be choreographed so that he'd always be only fighting one person at a time, and then he would defeat that person and move on. And and because of that, he could he could defeat all of them, right? But whereas if they had fought a bit more intelligently and just swarmed the guy at once, uh it would make for much much worse um cinema, but uh but they would win.

Are you usually uh pen and paper? Are you working uh with computer and latte? I'm mostly pen and paper actually. So in my office, I have four giant blackboards. Um, and sometimes I just have to write everything I know about the problem on the four blackboards and then sit my couch and just sort of see the whole thing.

Is it all symbols like notation or is there some drawings? Oh, there's a lot of drawing and a lot of bespoke doodles that that only make sense to me. Um, I mean and and the beauty of blackboard is you erase and it's it's a very organic thing. Um, I'm beginning to use more and more computers. Um, partly because AI makes it much easier to do simple coding things that you know if I wanted to plot a function before, which is moderately complicated as some iteration or something, you know, I'd have to to remember how to set up a Python program and and and and and how does a for loop work and and and debug it and it would take two hours and so forth. And and now I can do it in 10 15 minutes. It's much um yeah, I'm using more and more uh computers to do simple explorations.

Let's talk about AI a little bit if we could. So um maybe a good entry point is just talking about computer-assisted proofs in general. Can you describe the Lean formal proof programming language and how it can help as a proof assistant and maybe how you started using it and how uh it has helped you?

So um Lean is a computer language um much like sort of standard languages like Python and C and so forth, except that in most languages the focus is on producing executable code. Lines of code do things, you know, they they flip bits or or they make a robot move or or they they deliver you text on the internet or something. Um, so Lean is a language that can also do that. Uh, it can also be run as a standard traditional language, but it can also produce certificates. So a software like like Python might do a computation and give you that the answer is seven. Okay, that does a sum of 3+4 is equal to 7, but uh Lean can produce not just the answer but but a proof that how it got the the answer of seven as 3+4 and all the steps involved in in so it creates these more complicated objects, not just statements but statements with proofs attached to them. um and um every line of code is just a way of p piecing together previous statements to to create new ones. So the idea is not new. These things are are called proof assistants, and so they provide languages for which you you can create quite complicated um intricate mathematical proofs and um they produce these certificates that that give a 100% um guarantee that your arguments are correct if you trust the compiler of but they made the compiler really small, and you can there are several different compilers available for the same for

Can you give people some intuition about the the difference between writing on pen and paper versus using Lean programming language? How hard is it to formalize a statement? So Lean, a lot of mathematicians were involved in the design of Lean. So it's it's designed so that individual lines of code resemble individual lines of mathematical argument. Like you might want to introduce a variable. You want want to prove a contradiction. You you um there are various standard things that you can do and and it's it's written so ideally it should like a one correspondence. In practice, it isn't because Lean is like explaining a proof to an extremely pedantic colleague who will will point out, "Okay, did you really mean this? Like what what happens if this is zero?" Okay. Um, "Did you how do you justify this?" Um, so Lean has a lot of automation in it um to try to to uh to be less annoying. Um, so for example um every mathematical object has to come with a type. Like if I if I talk about X, is X a real number or um a natural number or or a function or something. Um, if you write things informally um it's up in terms of context. You say, you know, um clearly x is equal to let x be the sum of y and z and y and z were already real numbers so x should also be a real number. Um, so Lean can do a lot of that um but every so often it it says, "Wait a minute, can you tell me more about what this object is uh what type of object it is." You see, you have to think more um at a philosophical level. Well, not just sort of computations you're doing, but sort of what each object actually um is in some sense.

Is he using something like LLMs to do uh the type inference or like you mention the real number? It's it's using much more traditional what's called good old fashioned AI. Yeah, you can represent all these things as trees, and there's always an algorithm to match one tree to another tree. So it's actually doable to figure out if something is a a real number or a natural number. Yeah. Every object sort of comes with a history of where it came from, and you can you can kind of trace. Oh, I see. Um, yeah, so it's it's designed for reliability. So uh modern AIs are not used in it's a disjoint technology. People are beginning to use AIs on top of Lean. So when a mathematician tries to program um a proof in Lean um often there's a step, okay, now I want to use um the fundamental thing of calculus, say okay to do the next step. So the Lean developers have built this this massive project called mathlib, a collection of tens of thousands of useful facts about mathematical objects, and somewhere in there is the fundamental theorem of calculus, but you need to find it. So a lot the bottleneck now is actually lemma search, you know, there's a tool that that you know is in there somewhere and you need to find it um and so you can there are various search engines specialized for mathlib that you can do um but there's now these large language models that you can say um I need the fundamental theorem of calculus at this point and it say okay uh um uh for example um when I code I have GitHub Copilot installed as a plugin to my IDE and it scans my text and it sees what I need says you know I might even type here okay now I need to use the fundamental theorem of calculus okay and then it might suggest okay try this and like maybe 25% of the time it works exactly and then another 10 15% of the time it doesn't quite work but it it's close enough that I can say oh if I just change it here and here it it will work and then like half the time it gives me complete rubbish um so but people are beginning to use AI a little bit on top um mostly on the level of basically fancy autocomplete um but uh you can type half of one line of a proof and it will find it will tell you

Yeah, but a fancy especially fancy with the sort of capital letter F is uh uh removes some of the friction a mathematician might feel when they move from pen and paper to formalizing. Yes. Yeah. So, right now I estimate that the effort time and effort taken to formalize a proof is about 10 times the amount taken to to write it out. Yeah. So, it's doable, but uh you don't it's it's annoying.

But doesn't it like kill the whole vibe of being a mathematician? Yeah. So, I mean having a pedantic coworker, right? Yeah. If if that was the only aspect of it. Okay. But um Okay. there there are some there's some case it was actually more pleasant to do things formally. So there was there was a theorem I formalized, and there was a certain constant 12 um that that came out at um in the final statement and so this 12 had to be carried all through the proof um and like everything had to be checked that it goes all the all these other numbers had to be consistent with this final number 12 and so we wrote a paper through this theorem with this number 12 and then a few weeks later someone said, "Oh, we can actually improve this 12 to an 11 by reworking some of these steps." And when this happens with pen and paper um like every time you change a parameter you have to check line by line that every single line of your proof still works and there can be subtle things that you didn't quite realize. Some properties on the number 12 that you didn't even realize that you were taking advantage of. So a proof can break down at a subtle place. Um, so we had formalized the proof with this constant 12 and then when this this new paper came out uh we said okay let's so that took like 3 weeks to formalize and and like 20 people to formalize this this this original proof. I said, "Oh, but now now let's let's um uh uh let's update the 12 to 11." And what you can do with Lean is that you just in your headline theorem you you change a 12 to 11. You run the compiler and like of the thousands of lines of code you have 90% of them still work and there's a couple that are lined in red. Now I can't justify this these steps, but it it immediately isolates which steps you need to change, but you can skip over everything which which works just fine. Um, and if you program things correctly, um with sort of good programming practices, most of your lines will not be read. Um, and there'll just be a few places where you, I mean, if if you don't hard code your constants, but you sort of, uh, um, um, you use smart tactics and so forth. Yeah, you can localize um the things you need to change to to a very small um period of time. So like within a day or two, we had updated our proof to this is a very quick process. You um you make a change, there are 10 things now that don't work. For each one you make a change and now there's five more things that don't work, but but the process converges much more smoothly than with pen and paper. So that's for writing.

Are you able to read it like if somebody else sends a proof are you able to like how what's what's the uh versus paper and yeah? So the proofs are longer, but each individual piece is easier to read. So, um, if you take a math paper and you jump to page 27 and you look at paragraph 6 and you have a line of of of text of math, I often can't read it immediately because it assumes various definitions which I have to to go back and and maybe 10 pages earlier this was defined and this um the proof is scattered all over the place and you basically are forced to read fairly sequentially. Um, it's it's not like say a novel where like you know in theory you could you open up a novel halfway through and start reading. There's a lot of context. But when a proven Lean, if you put your cursor on a line of code, every single object there, you can hover over it and it would it would say what it is, where it came from, where stuff is justified. You can trace things back much easier than sort of flipping through a math paper. So, one thing that Lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn't do in the past. So traditionally with pen and paper um when you want to collaborate with another mathematician um either you do it as a blackboard where you um you can really interact, but if you're doing it sort of by email or something um basically yeah you have to segment it say I'm going to I'm going to finish section three you do section four, but uh you can't really sort of work on the same thing collaboratively at the same time. But with Lean you can be trying to formalize some portion of the proof and say I got stuck at line 67 here I need to prove this thing but it it doesn't quite work here is like the three lines of code I'm having trouble with. Um, but because all the context is there, someone else can say, "Oh, okay. I recognize what you need to do. You need to to apply this trick or this tool," and you can do extremely atomic level conversations. So, because of Lean, I can collaborate, you know, with dozens of people across the world, most of whom I don't have never met in person. Um, and I may not know actually even whether they're um how reliable they are in in in their um um in in the process, but Lean gives me a certificate of of of trust. Um, so I can do I can do trustless mathematics. So there's so many interesting questions there's.

So one, you're you're known for being a great collaborator. So what is the right way to approach solving a difficult problem in mathematics? When you're collaborating, are you doing a divide and conquer type of thing or are you brains are you focusing on a particular part and you're brainstorming? There's always a brainstorming process first. Yeah. So math research projects sort of by their nature when you start you don't really know how to do the problem. Um, it's not like an engineering project where somehow the theory has been established for decades and it's it's implementation is the main difficulty. You have to figure out even what is the right path. So so this is what I said about about cheating first, you know um it's like um to go back to the bridge building analogy, you know so first assume you have infinite budget and and like unlimited amounts of of of workforce and so forth. Now can you can you build this bridge? Okay. Okay. now have infinite budget but only finite workforce, right? Now can you do that and so forth um so uh I mean of course you know no engineer can actually do this like I say they have fixed requirements. Yes, there's this sort of jam sessions always at the beginning where you try all kinds of crazy things and you you make all these assumptions that are unrealistic but you plan to fix later um and you try to see if there's even some skeleton of an approach that might work um and then hopefully that breaks up the problem into smaller sub problems which you don't know how to do but then you uh you focus on on sub ones and sometimes different collaborators are better at at working on on certain things. Um, so one of my themes I'm known for is a theorem of Ben Green which called the Green-Tao theorem. Um, it's a statement that the primes contain arithmetic progressions of any length. So it was a modification of this theorem, and the way we collaborated was that Ben had already proven a similar result for progressions of length three. Um, he showed that sets like the primes contain lots and lots of progressions of length three. Um, even and even um subsets of the prime certain subsets do um but his techniques only worked for um for length three progressions. They didn't work for longer progressions. Um, but I had these techniques coming from ergodic theory which is something that I had been playing with and and uh I knew better than Ben at the time. Um, and so um if I could justify certain randomness properties of some set relating to primes like there there's a certain technical condition which if I could have it if if Ben could supply me this fact I could I could conclude the theorem, but I what I asked was a really difficult question in number theory which um he said there's no way we can prove this, can so he said, "Can you prove your part of the theorem using a weaker hypothesis that I have a chance to prove it?" And he proposed something which he could prove, but it was too weak for me; I can't use this. Um, so there's this there was this conversation going back and forth. Um, so different cheats to Yeah. Yeah. I want to cheat more, he 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 a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a 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 could prove and b I could use um and then we we could prove our view and um yeah so there's there's a there all kinds of dynamics you know I mean it's every every um collaboration has a has a has some story, no two are the same.

And then on on the flip side of that like you mentioned with Lean programming, now that's almost like a different story because you can do you can create I think you've mentioned a kind of a blueprint right for a problem and then you can really do a divide and conquer with Lean where you're working on separate parts right and they're using the computer system proof checker essentially to make sure that everything is correct along the way. Yeah. So it makes everything compatible and uh yeah and trustable. Um, yeah so currently only a few mathematical projects can be cut up in this way at the current state of the art. Most of the Lean activity is on formalizing proofs that have already been proven by humans. A math paper basically is a proof; a blueprint in a sense it is taking a a difficult statement like a big theorem and breaking up into 100 little lemmas um but often not all written with enough detail that each one can be sort of directly formalized. A blueprint is like a really pedantically written version of a paper where every step is explained as to as much detail as as as possible and trying to make each step kind of self-contained um and or

Depending on only a very specific number of previous statements that have been proven, so that each node of this blueprint graph that gets generated can be tackled independently of the others, and you don't even need to know how the whole thing works.

Um, so it's like a modern supply chain; you know, like if you want to create an iPhone or some other complicated object, um, no one person can build a single object, but you can have a specialist who, if they're given some widgets from some other company, they can combine them together to form a slightly bigger widget. I think that's a really exciting possibility because you can have, if you can find problems that could be broken down this way, then you can have, you know, thousands of contributors, right, distributed.

So I told you before about the split between theoretical and experimental mathematics, and right now most mathematics is theoretical, and when you type it, it's experimental. I think the platform that Lean and other software tools—so, um, GitHub and things like that—allow, they will allow experimental mathematics to scale up to a much greater degree than we can do now. So right now, if you want to do any mathematical exploration of some mathematical pattern or something, you need some code to write out the pattern, and I mean sometimes there are some computer algebra packages that help, but often it's just one mathematician coding lots and lots of Python or whatever, and because coding is such an error-prone activity, it's not practical to allow other people to collaborate with you on writing modules for your code because if one of the modules has a bug in it, the whole thing is unreliable.

Um, so it's these are, so you get these bespoke spaghetti code that's written by not professional programmers but by mathematicians, you know, and they're clunky and and slow, and um, and so because of that, it's hard to really mass-produce experimental results, um, but, yeah, but I think with Lean—I mean, so I'm already starting some projects where we are not just experimenting with data but experimenting with proofs. So I have this project called the equation theories project. Basically, we generated about 22 million little problems in abstract algebra. Maybe I should back up and tell you what the project is.

Okay. So abstract algebra studies operations like multiplication and addition and the abstract properties. Okay. So multiplication, for example, is commutative. X * Y is always Y * X, at least for numbers. Um, and it's also associative. X * Y * Z is the same as X * Y * Z. Um, so these operations obey some laws that don't obey others. For example, X * X is not always equal to X. So that law is not always true. So given any operation, it obeys some laws and not others.

Um, and so we generated about 4,000 of these possible laws of algebra that certain operations can satisfy. And our question is which laws imply which other ones? Um, so for example, does commutativity imply associativity? And the answer is no because it turns out you can describe an operation which obeys the commutative law but doesn't obey the associative law. So by producing an example, you can show that commutativity does not imply associativity, but some other laws do imply other laws by substitution and so forth, and you can write down some algebraic proof. So we look at all the pairs between these 4,000 laws and this 22 million of these pairs, and for each pair we ask, does this law imply this law? If so, give a give u give a proof. If not, give a counter-example.

Mhm. Um, so 22 million problems, each one of which you could give to like an undergraduate algebra student, and they had a decent chance of solving the problem. Although there are a few of these 22 million—there are like 100 or so—that are really quite hard. Okay. But a lot are easy, and the project was just to work out to determine the entire graph, like which ones imply which other ones.

That's an incredible project, by the way. Such a good idea. Such a good test of the very thing we've been talking about at a scale that's remarkable.

Yeah. So it would not have been feasible.

Yeah, I mean the state-of-the-art in the literature was like, you know, 15 equations and sort of how they apply. That's sort of at the limit of what a human repentant paper can do. So, so you need to scale it up. So, you need to crowdsource, but you also need to trust all the—I mean, no one person can check 22 million of these proofs. You needed it to be computerized, and so it only became possible with with Lean.

Um, we were hoping to use a lot of AI as well. Um, so the project is almost complete. Um, so of these 22 million, all but two had been settled.

Um, wow. And, well, actually, and of those two, we have a pen and paper proof of the two, and we're formalizing it. In fact, this morning I was working on finishing it. Um, so we're almost done on this—incredible—is yeah, fantastic.

How many people were able to get about 50, um, which in mathematics is considered a huge number.

It's a huge number. That's crazy.

Yeah. So we kind of have a paper with 50 authors and a big appendix of who contributed to what.

Here's an interesting question. Now, to maybe speak even more generally about it. When you have this pool of people, is there a way to organize the contributions by level of expertise of the people, of the contributors? Now, okay, um, I'm asking you a lot of pthead questions here, but I'm imagining a bunch of humans and maybe in the future some AIs. Can there be like an ELO rating type of situation where like a gamification of this?

The beauty of these Lean projects is that automatically you get all this data, you know, so like everything has to be uploaded for this GitHub, and GitHub tracks who contributed what. Um, so you could generate statistics from at any at any later point in time. You can say, oh, this person contributed this many this many lines of code or whatever. I mean these are very crude metrics. Um, I would definitely not want this to become, you know, part of your tenure review or something. Uh, um, but, I mean, I think already in in in enterprise computing, right, people do use some of these metrics as part of the assessment of the performance of an employee. Um, again, this is a direction which is a bit scary for academics to go down. We don't like metrics so much, and yet academics use metrics; they just use old ones.

Number of papers.

Yeah.

Yeah. It's true. It's true that—Yeah. I mean, um, it feels like this is a metric, while flawed, is going in the more in the right direction.

Right.

Yeah. It's an interesting—at least it's a very interesting metric.

Yeah. I think it's interesting to study. I mean, I think you can you can do studies of whether these are better predictors. Um, there's this problem called Goodhart's Law. If a statistic is actually used to incentivize performance, it becomes gamed. Um, and then it is no longer a useful measure.

Oh, humans always.

Yeah.

Yeah. I know. It's rational.

So what we've done for this project is self-report. So, um, there are actually standard categories from the sciences of what types of contributions people give. So there's there's concept and validation and resources and and and coding and so forth. Um, so we we we there's a standard list of, you know, categories. Um, and we just ask each contributor to—there's a big matrix of all the of all the authors in all the categories—just to tick the boxes where they think that they contributed. Um, and just give a rough idea, you know, like, oh, so you did some coding and and and you provided some compute, but you didn't do any of the pen and paper verification or whatever. And I think that that works out. Traditionally, mathematicians just order alphabetically by surname. So we don't have this tradition as in the sciences of, you know, lead author and second author and so forth, like which we're proud of, you know, we make all the authors equal status, but it doesn't quite scale to this size. So a decade ago, I was involved in these things called Polymath projects; it was the crowdsourcing mathematics but without the Lean component, so it was limited by you needed a human moderator to actually check that all the contributions coming in were actually valid, and and this was a huge bottleneck actually, um, but still we had projects that were, you know, 10 author or so. But we had decided at the time, um, not to try to decide who did what, um, but to have a single pseudonym. So we created this fictional character called DHJ Polymath in the spirit of Bourbaki. Bourbaki is the pseudonym for a famous group of mathematicians in the 20th century. But, um, and so the paper was authored under the pseudonym. So none of us got the author credit.

Um, this actually turned out to be not so great for a couple of reasons. So, so one is that if you actually wanted to be considered for tenure or whatever, you could not use this paper in your, uh, as your submitted as one of your publications because it wasn't you didn't have the formal author credit. Um, um, but the other thing that we've recognized much later is that when people referred to these projects, they naturally refer to the most famous person who was involved in the project. Oh, so this was Tim Gowers' project. This was Terence Tao's project, and not mention the the other 19 or whatever people that were involved.

Yeah. So we're trying something different this time around where we have everyone as an author. Um, but we will have an appendix with this matrix, and we'll see how that works. I mean, uh, so both projects are incredible, just the fact that you're involved in such huge collaborations. But I think I saw a talk from Kevin Buzzard about the Lean programming language just a few years ago, and he was saying that this might be the future of mathematics. And so it's also exciting that you're embracing—one of the greatest mathematicians in in the world embracing this—what seems like the paving of the future of mathematics.

Um, so I have to ask you here about the integration of AI into this whole process. So DeepMind's AlphaProof was trained using reinforcement learning on both failed and successful formal Lean proofs of IMO problems. So this is sort of high-level high school—oh, very high level—yes, very high-level high school level mathematics problems. What do you think about the system and maybe what is the gap between this system that is able to prove the high school level problems versus graduate level problems?

Yeah, the difficulty increases exponentially with the number of steps involved in the proof. It's a combinatorial explosion, right? So the thing with large language models is that they make mistakes. And so if a proof has got 20 steps and your model has a 10% failure rate at each step of going in the wrong direction, like, uh, it's just extremely unlikely to actually reach the end. Actually, just to take a small tangent here, is how hard is the problem of mapping from natural language to the formal program?

Oh, yeah, it's extremely hard actually. Um, natural language, you know, it's very fault-tolerant. Um, like you can make a few minor grammatical errors, and a speaker in the second language can get some idea of what you're saying. Um, yeah, but but formal language, yeah, you if you get one little thing wrong, um, like the whole thing is is nonsense, um, even formal to formal is is very hard. There are different incompatible, um, proof assistant languages. Uh, there's Lean, but also Coq and Isabelle and so forth, and actually even converting from a formal language to a formal language, um, is is an unsolved—basically unsolved—problem.

That is fascinating. Okay. So, but once you have an informal language, they're using their RL-trained model. So something akin to AlphaZero that they used to go to then try to come up with proofs; they also have a model—I believe it's a separate model—for geometric problems. So what impresses you about the system, and um, what do you think is the gap? Yeah, we talked earlier about things that are amazing; over time they become kind of normalized. Um, so yeah, now somehow it's, oh, of course, geometry is a silver problem, right?

That's true. That's true. I mean, it's still beautiful. Yeah, these are great works; it shows what's possible. I mean, it's, it, the approach doesn't scale currently. Is—yeah, 3 days of Google's server time to solve one high school math problem. This is not a scalable prospect. Um, especially with the exponential increase in, as as the complexity increases. We should mention that they got a silver medal performance—the equivalent of—I mean, yeah, equivalent of a silver—so first of all, they took way more time than was allotted, um, and they had this assistance where where the humans started helped by by formalizing, um, but, also, they they're giving us those full marks for the solution, which I guess is formally verified. So I guess that that's that's fair.

Um, yeah, there there are efforts; there was there will be a proposal at some point to actually have an AI math olympiad where at the same time as the human contestants get the the actual olympiad problems, AIs will also be given the same problems with the same time period, um, and the outputs will have to be graded by the same judges, um, um, and which means that will have to be written in natural language rather than formal language.

Oh, I hope that happens. I hope that this IMO it happens. I hope I hope next one it won't happen. This IMO, the performance is not good enough in in the time period and and uh, um, but there are smaller competitions, um, there are competitions where the the answer is a is a number rather than a long-form proof, um, and that's that's um, AI are actually a lot better at problems where there's a specific numerical answer, um, because it's it's easy to to to reinforce do reinforcement learning on it. Yeah, you got the right answer, you got the wrong answer. It's a very clear signal. But a long-form proof either has to be formal, and then the Lean can give it a thumbs up, thumbs down, or it's informal. Um, but then you need a human to grade it to tell, and if you're trying to do billions of of reinforcement learning, um, you know, um, um, runs, you're not you can't hire enough humans to to grade those. Um, it's already hard enough for for the large language to do reinforcement learning on on just the regular text that that people get. But now if you actually hire people not just give thumbs up, thumbs down, but actually check the the output mathematically.

Yeah, that's too expensive.

So if we just explore this possible future, what what what is the thing that humans do that's most special in in mathematics? So that you could see AI not cracking for a while.

So inventing new theories. So coming up with new conjectures versus proving the conjectures, right? Building new abstractions, new representations, maybe an AI turning style with seeing new connections between disparate fields.

It's a good question. Um, I think the nature of what mathematicians do over time has changed a lot. Um, you know, um, so a thousand years ago, mathematicians had to compute the date of Easter, uh, and there was really complicated calculations, you know, but it's all automated—been automated for centuries; we don't need that anymore, you know. They used to navigate—to do spherical navigation, spherical trigonometry—to navigate how to get from from the old world to the new—or very complicated calculations; again, we've been automated. Um, you know, even a lot of undergraduate mathematics, even before AI, like Wolfram Alpha, for example, it's not a language model, but it can solve a lot of undergraduate level math tasks. So on the computational side, verifying routine things like having a a problem and and say, here's a problem in partial equations. Could you solve it using any of the 20 standard techniques? Um, and they say yes, I've tried all 20, and here are the 100 different permutations and and here's my results. Um, and that type of thing I think it will work very well, um, type of scaling to once you solve one problem to to make the AI attack 100 adjacent problems. Um, the things that humans do still—Yeah. So so where the AI really struggles right now, um, is knowing when it's made a wrong turn. Um, that it can say, "Oh, I'm going to solve this problem. I'm going to split up this problem into into these two cases. I'm going to try this technique." And um, sometimes if you're lucky and it's a simple problem, it's the right technique and you solve the problem, and sometimes it it will get it will have a problem; it would propose an approach which is just complete nonsense. Um, and but like it looks like a proof. Um, so this is one annoying thing about LM-generated mathematics. So, um, yeah, we we we've had human-generated mathematics as very low quality, um, like, you know, submissions, people who don't have the formal training and so forth. But if a human proof is bad, you can tell it's bad pretty quickly. It makes really basic mistakes. But the AI-generated proofs, they can look superficially flawless. Uh, and that's partly because that's what the reinforcement learning has actually trained them to do, right? To to make things to to produce text that looks like um what is correct, which for many applications is good enough. Um, uh, so the errors are often really subtle, and then when you spot them, they're really stupid. Um, like, you know, like no human would have actually made that mistake.

Yeah, it's actually really frustrating in the programming context because I I program a lot, and yeah, when a human makes—when low-quality code, there's something called code smell, right? You can you can tell you can tell immediately like, okay, there's signs. But with with a generated code, and then you're right, eventually you find an obvious dumb thing that just looks like good code.

Yeah. So, um, it's very tricky to and frustrating for some reason to—Yeah—to work.

Yeah. So the sense of smell. Okay, there you go. This is this is one thing that humans have. Um, and there's a metaphorical mathematical smell that uh this we it's not clear how to get the AI to duplicate that eventually. Um, I mean, so the way AlphaZero and so forth make progress on Go and and chess and so forth is is in some sense they have developed a sense of smell for Go and chess positions, you know, that that this position is good for white, is good for black. Um, they can't initiate why. Um, but just having that that sense of smell lets them strategize. So if AIs gain that ability to sort of a sense of viability of certain proof strategies, say, so so you can say, I'm going to try to break up this problem into two small subtasks, and they can say, well, this looks good; two tasks look like they're simpler tasks than than your main task, and they still got a good chance of being true. Um, so this is good to try, or no, you've you've made the problem worse because each of the two subproblems is actually harder than your original problem, which is actually what normally happens if you try a random thing to try; normally, actually, it's very easy to transform a problem into an even harder problem.

Mhm. Very rarely do you problem transform into a simpler problem.

Um, yeah, so if they can pick up a sense of smell, then they could maybe start competing with human-level mathematicians.

So, this is a hard question, but not competing, but collaborating. Yeah. If—Okay, hypothetical. If I gave you an oracle that was able to do some aspect of what you do, and you could just collaborate with it.

Yeah.

Yeah. What would that oracle—What would you like that oracle to be able to do? Would you like it to, maybe, be a verifier? Like check—

Mhm.

Do the codes like you're—Yes—professor to this is the correct—this is a good—this is a promising, fruitful direction.

Yeah.

Yeah. Or or would you like it to generate possible proofs and then you see which one is the right one? Um, or would you like it to maybe generate different representations, different totally different ways of seeing this problem?

Yeah, I think all of the above. Um, a lot of it is we don't know how to use these tools because it's a paradigm that is not—yeah, we have not had in the past systems.

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

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

Um, so in formalization I I mentioned before that it takes 10 times longer to formalize a proof than to write it by hand with these modern AI tools is and also just better tooling um the lean um um developers are doing a great job adding more and more features and making it user friendly. It's going up from 9 to 8 to 7. Okay, no big deal. But one day it will drop below one. Um, and that's a phase shift because suddenly um it makes sense when you write a paper to to write it in lean first or through a conversation with AI who is generally um on the fly with you and it becomes natural for journals to accept, you know, maybe they'll offer expedite refereeing you know if if a paper has already been formalized in in lean um they'll just ask the referee to comment on on the significance of the results and how it connects to literature and not worry so much about the correctness. Um, because that's been certified. Um, papers are getting longer and longer in mathematics and actually it's harder and harder to get good refereeing for um the really long ones unless they're really important. It is actually an issue which and the formalization is coming in at just the right time for this to be and the easier and easier to guess because of the tooling and all the other factors then you're going to see much more like math lib will grow potentially exponentially. It's a it's a it's a virtuous uh cycle. Okay.

I mean one facet of this type that happened in the past was the adoption of latte. So so latte is this type setting language that all mathematicians use now. So in the past people use all kinds of word processors and typewriters and whatever but at some point latte became easier to use than all other competitors and that people just switched you know within a few years like it was just a dramatic um pay shift. It's a wild out there question, but what what year how far away are we from a uh AI system being a collaborator on a proof that wins the Fields medal. So that level. Okay. Um, well, it depends on the level of collaboration. I mean, no, like it deserves to be to get the Fields Medal. Like so half and half already like I I can imagine if it was a winning paper having some AI systems in writing it you know uh just you know like the order complete alone is already I I use it like it speeds up my my own writing um um like you know you you can have a theorem you have a proof and the proof has three cases and I I write down the proof of the first case and the autocomplete just suggests all right now now here's how the proof of second case could work and like it was exactly correct that was great saved me like 5 10 minutes of uh of typing but in that case The AI system doesn't get the Fields medal. No. Uh are we talking 20 years, 50 years, 100 years? What do you think? Okay.

So I I gave a prediction in print. So by 2026, which is now next year, um there will be math collaborations, you know, where the AI, so not Fields Medal winning, but but like actual research level math like published ideas that in part generated by AI. Um maybe not the ideas but at least uh some of the computations um um the verifications. Yeah. I mean has that already happened? Has that already happened? Yeah. There are there are problems that were solved uh by a complicated process conversing with AI to propose things and the human goes and tries it and the contract doesn't work but it might propose a different idea. Um it it's it's hard to disentangle exactly. Um there are certainly math results which could only have been accomplished because there was a math method human mathematician and an AI involved. Um but it's hard to sort of disentangle credit. Um I mean these tools they they do not uh replicate all the skills needed to do mathematics but they can replicate sort of some non-trivial percentage of them you know 30 40%. They can fill in gaps. Um, you know, so, uh, coding is is is a is a good example, you know. So, I I um um it's annoying for me to code in Python. I'm not I'm not a native um I'm not a professional um programmer. Um, but um the with AI that the the friction cost of of doing it is is is much reduced. Uh so it it fills in that gap for me.

Um AI is getting quite good at literature review. Um I mean there's still a problem with um hallucinating you know the references that don't exist. Um but this I think is a civil war problem if you train in the right way and so forth you can you can and um and verify um you know using the internet um you know um you should in a few years get to the point where you you have a a lema that you need and uh we say has anyone proven this lema before and it will do basically a fancy web search AI assistant and say yeah yeah there are these six papers where something similar has happened and I mean it you can ask it right now and it'll give you six papers of which maybe one is is legitimate and relevant. One exists but is not relevant and four are hallucinated. Um it has a non-zero success rate right now, but uh it's there's so much garbage. Uh so much the signal to noise ratio is so poor that it's it's um it's most helpful when you already somewhat know the literature. Um and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory versus helping you discover new you were not even aware of but is the correct citation. Yeah, that's yeah, that it can sometimes do. But but when it does, it's it's buried in in a list of options for which the other that are bad. Yeah. I mean, being able to automatically generate a related work section that is correct. Yeah. That's actually a beautiful thing that might be another phase shift because it assigns credit correctly. Yeah. It does. It breaks you out of the silos of Yeah. Yeah. Yeah. thought, you know. Yeah. No, there's a big hump to overcome right now. I mean, it's it's like self-driving cars, you know. The the safety margin has to be really high for it to be um uh to be feasible. So yeah, so there's a last mile problem um with a lot of AI applications um that uh you know they can develop tools that work 20% 80% of the time but it's still not good enough um and in fact even worse than good some ways.

I mean another way of asking the Fields metal question is what year do you think you'll wake up and be like real surprised? You read the headline, the news of something happened that AI did like you know real breakthrough something it doesn't you know like feels metal even hypothesis it could be like really just this alpha zero moment with go that kind of thing right um yeah this this decade I can I can see it like making a conjecture between two unrelated two two things that people thought was unrelated oh interesting generating a conjecture that's a beautiful conjecture. Yeah. And and actually has a real chance of being correct and and and meaningful and um because that's actually kind of doable I suppose but the word of the data is Yeah. No, that would be truly amazing. Um the current models struggle a lot. I mean so um a version of this is um I mean the physicists have a dream of getting the AI to discover new new laws of physics. Um you know the the dream is you just feed it all this data. Okay. and and this is here's a new patent that we didn't see before but it actually even struggle the current state of the art even struggles to discover old laws of physics um from the data uh or if it does there's a big concern contamination that that it did it only because like somewhere in this training data it some new um you know boils law or whatever ball you're trying to to to reconstruct um part of it is that we don't have the right type of training data for this um yeah so for laws of physics like we we don't have like a million different universes with a million infant laws of nature. Um and um like a lot of what we're missing in math is actually the negative space of so we have published things of things that people have been able to prove um and conjectures that ended up being verified um or maybe counter examples produced but um we don't have data on on things that were proposed and they're kind of a good thing to try but then people quickly realized that it was the wrong conjecture and then they they said oh but we we should actually change um our claim to modify it in this way to actually make it more plausible. Um there's this there's a trial and error process which is a real integral part of human mathematical discovery which we don't record cuz it's embarrassing. Uh we make mistakes and and we only like to publish our wins. Um and uh the AI has no access to this data to train on. Um I sometimes joke that basically AI has to go through um grad school and actually you know go to grad courses, do the assignments, go to office hours, make mistakes, um get advice on how to correct the mistakes and learn from that.

Let me uh ask you if I may about uh Gregori Pearlman. Mhm. You mentioned that you try to be careful in your work and not let a problem completely consume you. Just you really fall in love with the problem and really cannot rest until you solve it. But you also hasted to add that sometimes this approach actually can be very successful. An example you gave is Gregoria Pearlman who proved the point conjecture and did so by working alone for 7 years with basically little contact with the outside world. Can you explain this one millennial prize problem that's been solved point conjecture and maybe speak to the journey that Gagora Pearlman's been on. All right. So it's it's a question about curved spaces. Earth is a good example. So you can think of a 2D surface in being round could maybe be a Taurus with a hole in it or it can have many holes and there there are many different topologies up priori that that a surface could have. Um even if you assume that it's it's bounded and and uh and smooth and so forth. So we have figured out how to classify surfaces as a first approximation everything is determined by something called the genus how many holes it has. So a sphere has genus 0 a donut has genus one and so forth and one way you can tell these surfaces apart probably the sphere has which is called simply connected if you take any closed loop on the sphere like a big closed little rope you can contract it to a point and while staying on the surface and the sphere has this property but a taurus doesn't if on a taurus and you take a rope that goes around say the the outer diameter taurus there's no way it can't get through the hole there's no way to to contract it to a point so it turns out that the this the sphere is the only surface with this property of contractability up to like continuous deformationations of the sphere. So um things that I want to call topologically um equivalent of the sphere. So point asked the same question in higher dimensions. Um so this it becomes hard to visualize because um surface you can think of as embedded in three dimensions but a curved free space we don't have good intuition of 4D space to to to live and and there are also 3D spaces that can't even fit into four dimensions. You need five or six or or higher. But anyway, uh mathematically you can still pose this question that if you have a bounded threedimensional space now which is also has this simply connected property that every loop can be contracted. Can you turn it into a threedimensional version of a sphere? And so this is the point conjecture. Weirdly in higher dimensions four and five it was actually easier. So uh it was solved first in higher dimensions. There's somehow more room to do the deformation. It's easier to to to move things around to a sphere. But three was really hard. So people tried many approaches. There sort of commentary approaches where you chop up the the surface into little triangles or or tetrahedra and you you just try to argue based on how the faces interact each other. Um there were um algebraic approaches. There's there's various algebraic objects like things called the fundamental group that you can attach to these homology and coology and and and all these very fancy tools. Um they also didn't quite work. Um but Richard Hamilton's proposed a um partial differential equations approach. So you take um you take so the problem is that you so you have this object which is so secretly is a sphere but it's given to you in a in a really um in in a weird way. So like like think of a ball that's been kind of crumpled up and twisted and it's not obvious that it's a ball. Um but um like if you if you have some sort of surface which is which is a deformed sphere, you could um u you could for example think of it as a surface of a balloon. You could try to inflate it. You you blow it up. Um and naturally as you fill it with air um the the wrinkles will sort of smooth out and it will turn into um um a nice round sphere. Um unless of course it was a Taurus or something in which case it would get stuck at some point like if you instead of Taurus it would there'll be a point in the middle when the inner ring shrinks to zero you get you get a singularity and you can't blow up any further. You can't flow any further. So he created this flow which is called Richie flow which is a way of taking an arbitrary surface or or space and smoothing it out to make it rounder and rounder to make it look like a sphere. And he wanted to show that that either uh this process would give you a sphere or it would create a singularity. Um actually very much like how PDS either they have global regularity or finite blow like basically it's almost exactly the same thing. It's all connected. Um and so and and he showed that for two dimensions two dimensional services surfaces um if you started simply connected no singularities ever formed um you never ran into trouble and you could flow and it would give you a sphere and it so he he got a new proof of the two dimensional result but by the way that's a beautiful explanation of reach flow and its application in this context how difficult is the mathematics here like for the 2D case is it yeah these are quite sophisticated equations on par with the Einstein equations slightly simpler but um Um yeah but but they were considered hard nonlinear equations to solve um and there's lots of special tricks in 2D that that that helped but in 3D the problem was that uh this equation was actually super critical the same problems as Nabia Stokes as you blow up um maybe the curvature could get constraint in finer smaller smaller regions and it um it looked more and more nonlinear and things just look worse and worse and there could be all kinds of singularities that showed up. um some singularities um like if there's these things called neck pinches where where the uh the surface sort of creates behaves like like a like a a barbell and it it pinches at a point. Some some singularities are simple enough that you can sort of see what to do next. You just make a snip and then you can turn one surface into two and evol them separately. But there was there was a the prospect that there's some really nasty like knotted singularities showed up that you you couldn't see how to um resolve in any way that you couldn't do any surgery to. Um so you need to classify all the singularities like what are all the possible ways that things can go wrong. Um so what Pearlman did was first of all he he made the problem he turned the problem a super critical problem to a critical problem. Um I said before about how um the invention of the of of energy the Hamiltonian like really clarified um Newtonian mechanics. Um uh so he introduced something which is now called permanence reduced volume and permanence entropy. He introduced new quantities kind of like energy that look the same at every single scale and turned the problem into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did before. Um and then he had to solve he still had to analyze the singularities of this critical problem. uh and that itself was a problem similar to this wake up thing I worked on actually um so on the on the level of difficulty of that. So he managed to classify all the singularities of this problem and show how to apply surgery to each of these and through that was able to to resolve the point Cray conjecture. um quite like a lot of really ambitious steps um and like like nothing that a large language model today for example could I mean um at best uh I could imagine model proposing this idea as one of hundreds of different things to try um but the other 99 would be complete dead ends but you'd only find out after months of work he must have had some sense that this was the right track to pursue because you know I it takes years to get them from A to B so you've done like you said Actually you see even strictly mathematically but more broadly in terms of the process he's done similarly difficult things what what can you infer from the process he was going through because he was doing it alone what are some low points

In a process like that, when you start to, like you've mentioned, hardship, like uh, AI doesn't know when it's failing, what happens to you? You're sitting in your office when you realize the thing you did for the last few days, maybe weeks, weeks. Yeah. Is a failure.

Well, for me, I switch to a different problem. Uh, so, uh, as said, I'm I'm a fox. I'm not a hedgehog. But you legitimately, that is a break that you can take, is is to step away and look at a different problem. Yeah, you can modify the problem too. Um, I mean, um, yeah, you can ask some cheat, if if there's a specific thing that's blocking you, that this um some bad case keeps showing up, that that that for which your tool doesn't work, you can just assume by fiat this this bad case doesn't occur. So you you do some magical thinking, um, for the, but but but strategically, okay, for the point to see if the rest of the argument goes through. Um, if there's multiple problems uh with with with your approach, then maybe you just give up, okay? But if this is the only problem that you know, but everything else checks out, then it's still worth fighting. Um, so yeah, you have to do some some sort of forward reconnaissance sometimes to uh, you know, and that is sometimes productive to assume like, okay, we'll figure it out, oh yeah, yeah, eventually.

Sometimes actually it's even productive to make mistakes. So um, one of the, I mean, um, there was a project which actually u we won some prizes for, actually, four other people. Um, we worked on this PD problem again, actually this blow of regularity type problem. Um, and it was considered very hard. Um, Sean Bain, who was another field methodist who worked on a special case of this, but he could not solve the general case. Um, and we worked on this problem for two months and we found, we thought we solved it. We we had this this cute argument that if everything fit, and we were excited, uh, we were planning celebrationally um to all get together and have champagne or something. Um, and we started writing it up. Um, and one of one of us, not me actually, but another co-author said, "Oh, um, in this in this lemma here, we um we have to estimate these 13 terms that that show up in this expansion." And we estimate 12 of them, but in our notes, I can't find the estimation of the 13th. Can you can someone supply that? And I said, "Sure, I'll look at this," and actually, yeah, we didn't cover, we completely omitted this term, and this term turned out to be worse than the other 12 terms put together. Um, in fact, we could not estimate this term, um, and we tried for a few more months and all different permutations, and there was always this one thing, one term that we could not control. Um, and so like, um, this was very frustrating, um, but because we had already invested months and months of effort into this already, um, we stuck at this, we we tried increasingly desperate things and and crazy things, um, and after two years we found an approach which was actually somewhat different, by quite a bit, from our initial um strategy, which did actually didn't generate these problematic terms and and and actually solve the problem. So we we solve a problem after 2 years, but if we hadn't had that initial false dawn of nearly solving a problem, we would have given up by month two or something and and worked on an easier problem. Um, yeah, if we had known it would take two years, not sure we would have started the project.

Yeah, sometimes actually having the incorrect, you know, it's like Columbus, New, incorrect version of measurement of the size of the earth. He thought he was going to find a new trade route to India, or at least that was how he sold it in his prospectus. I mean, it could be that he actually secretly knew, but just on the psychological element. Do you have like emotional or like self-doubt that just overwhelms you in moments like that? You know, because this stuff, it feels like math is is so engrossing that like it can break you when you like invest so much yourself in the problem and then it turns out wrong. You could start to, in a similar way, chess has broken some people. Yeah.

Um, I I think different mathematicians have different levels of emotional investment in what they do. I mean, I think for some people it's just a job. You know, you you have a problem, and if it doesn't work out, you you you go on to the next one. Um, yeah, so the fact that you can always move on to another problem um it reduces the emotional connection. I mean, there are cases, you know, so there are certain problems that are what I call back diseases, where where where you just latch on to that one problem and they spend years and years thinking about nothing but that one problem, and um, you know, maybe their career suffers and so forth, but okay, this big win, this will, you know, once I once I finish this problem, I will make up for all the years of of of lost opportunity, but that's that's I mean, occasionally, occasionally it works, but I I um I really don't recommend it for people without the the right fortitude. Yeah. So I I've never been super invested in any one problem.

Um, one thing that helps is that we don't need to call our problems in advance. Uh um, well, uh, when we do grant proposals, we say we we will study this set of problems. But even then we don't promise definitely by 5 years I will supply a proof of all these things. You know, or um you promise to make some progress or discover some interesting phenomena. Uh, and maybe you don't solve the problem, but you find some related problem that you you can say something new about, and that's that's a much more feasible task. But I'm sure for you there's problems like this. You have you have um made so much progress towards the hardest problems in the history of mathematics. So is there is there a problem that just haunts you? It sits there in the dark corners, you know, twin prime conjecture, Riemann hypothesis, Goldbach conjecture. Twin prime, that sounds again.

So, I mean, the problem is like a Riemann hypothesis, those are so far out of reach. Why do you think so? Yeah. There's no even viable strategy, like even if I activate all my all the cheats that I know of in this problem, like it there's just still no way to get me to be um like it's um I think it needs a breakthrough in another area of mathematics to happen first and for someone to recognize that it that would be a useful thing to transport into this problem. So we we should maybe step back for a little bit and just talk about prime numbers. Okay. So they're often referred to as the atoms of mathematics. Can you just speak to the structure that these uh atoms, the natural numbers, have? Two basic operations attached to them: addition and multiplication. Um, so if you want to generate the natural numbers, you can do one of two things. You can just start with one and add one to itself over and over again, and that generates you the natural numbers. So additively they're very easy to generate: 1, 2, 3, 4, 5. Or you can take the prime, if you want to generate multiplicatively, you can take all the prime numbers: 2, 3, 5, 7, and multiply them all together. um, and together that gives you all the the natural numbers, except maybe for one. So there these two separate ways of thinking about the natural numbers, from an additive point of view and a multiplicative point of view. Um, and separately they're not so bad. Um, so like any question about that only was addition is relatively easy to solve, and any question that only was multiplication is easy to solve. Um, but what has been frustrating is that you combine the two together. Um, and suddenly you get this extremely rich, I mean, we know that there are statements in number theory that are actually undecidable. There are certain polynomials in some number of variables. You know, is there a solution in the natural numbers, and the answer depends on on an undecidable statement um like like whether um the axioms of of mathematics are consistent or not. Um, but um, yeah, but even the simplest problems that combine something multiplicative such as the primes with something additive such as shifting by two, uh, separately we understand both of them well, but if you ask, when you shift the prime by two, do you can you get a, how often can you get another prime? We it's been amazingly hard to relate the two, and we should say that the twin prime conjecture is just that it posits that there are infinitely many pairs of prime numbers that differ by two. Yes.

Now, the interesting thing is that you have been very successful at pushing forward the field in answering these complicated questions uh of this variety, like you mentioned the Green-Tao theorem. It proves that prime numbers contain arithmetic progressions of any length, right? Which is mind-blowing that you can prove something like that, right?

Yeah. So, what we've realized because of this this this type of of research is that different patterns have different levels of uh indestructibility. Um, so, so what makes the twin prime problem hard is that if you take all the primes in the world, you know, 3, 5, 7, 11, so forth, there are some twins in there. 11 and 13 is a twin prime pair of twin primes, and so forth. But you could easily, if you wanted to, um redact the primes to get rid of to get rid of the um these twins, like the twins, they show up, and they're infinitely many of them, but they're actually reasonably sparse. Um, not there there's not, I mean, initially there's quite a few, but once you got to the millions, the trillions, they become rarer and rarer, and you could actually just, you know, if if someone was given access to the database of primes, you just edit out a a few primes here and there, they could make the twin prime conjecture false by just removing like 0.1% of the primes. Or something, um, just well well chosen to to um to do this. And so you could present a censored database of the primes which passes all of the statistical tests of the primes. You know, that it it obeys things like the prime number theorem and and other tests about the primes, but doesn't contain any twin primes anymore. Um, and this is a real obstacle for the twin prime conjecture. It means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes. And so it must be some very um subtle, delicate feature of the primes that you can't just get from like like aggregate statistical analysis.

Okay. So that's all, yeah. On the other hand, progressions have turned out to be much more robust. um, like you can take the primes and you can eliminate 99% of the primes, actually, you know, and you can take take any 99% you want, and uh, it turns out, and another thing we prove is that you still get arithmetic progressions, um, arithmetic progressions are much, you know, they're like cockroaches of arbitrary length. Yes, that's crazy. I mean, so so this for for people who don't know, arithmetic progressions is a sequence of numbers that differ by some fixed amount. Yeah, but it's again like it's an infinite monkey type phenomenon, for any fixed length of your set. You don't get arbitrary long progressions. You only get quite short progressions. But you're saying twin prime is not an infinite monkey phenomenon. I mean, it's a very subtle monkey. It's still an infinite monkey phenomenon. Yeah. If the primes were really genuinely random, if the primes were generated by monkeys, um, then yes, in fact, the infinite monkey theorem would, Oh, but you're saying that twin prime is it doesn't, you can't use the same tools, like the it doesn't appear random almost.

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

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

Uh, what are sexy primes? Primes that differ by six. The name is much less, the concept is much less exciting than the name suggests. Got it. Um, so you can make a conspiracy rule out one of these, but like once you have like 50 of them, it turns out that you can't rule out all of them at once. It just it requires too much energy somehow in this conspiracy space.

How do you do the bound part? How do you how do you develop a bound for the difference between the primes that, okay, so um that there's an infinite number of, so it's ultimately based on what's called the pigeonhole principle. Um, so the pigeonhole principle, uh, it's a statement that if you have a number of pigeons and they all have to go into into pigeonholes, and you have more pigeons than pigeonholes, then one of the pigeonholes has to have at least two pigeons in, so there has to be two pigeons that that are close together. So for instance, if you have 100 numbers and they all range from one to a thousand, um, two of them have to be at most 10 apart. Mhm. Because you can divide up the numbers one to 100 into 100 pigeonholes. Let's let's say you have if you have 101 numbers, 100 one numbers, then two of them have to be distance less than 10 apart because two of them have to belong to the same pigeonhole. So it's a basic um basic principle in mathematics. Um, so it doesn't quite work with the primes directly because the primes get sparer and sparser as you go out, that fewer and fewer numbers are prime. But it turns out that there's a way to assign weights to the to to numbers, like um, so there are numbers that are kind of almost prime, but they're not, they they don't have no factors at all other than themselves and one, but they have very few factors. Um, and it turns out that we understand almost primes a lot better than primes. Um, and so, for example, it was known for a long time that there were twin almost primes. This has been worked out. So almost primes are something we can understand. So you can actually restrict attention to a a suitable set of almost primes and uh whereas the primes are very sparse overall, relative to the almost primes, actually are much less sparse. They make um you can set up a set of almost primes where the primes have density like say 1%. Um, and that gives you a shot at proving by applying some sort of pigeonhole principle that that those pairs of primes are just only 100 100 apart. But in order to with the twin prime conjecture, you need to get the density of primes inside the also primes up to up to a fraction of 50%. Um, once you get up to 50%, you will get twin primes. But uh, unfortunately, there are barriers. Um, we know that that no matter what kind of good set of almost primes you pick, the density of primes can never get above 50%. It's called the parity barrier. Um, and I would love to find, yes. So one of my long-term dreams is to find a way to breach that barrier because it would open up not only the twin prime conjecture, the Goldbach conjecture, and many other problems in number theory are currently blocked because our current techniques would require improvements going beyond this theoretical um parity barrier. It's like it's like pulling past the speed of light. Yeah. So we just say a twin prime conjecture, one of the biggest problems in the history of mathematics, Goldbach conjecture also um they feel like next-door neighbors.

Uh, has there been days when you felt you saw the path? Oh, yeah. Um um, yeah, uh, sometimes you try something and it it works super well. Um, you you again again the sense of mathematical smell uh we talked about earlier, uh you learn from experience when things are going too well, because there are certain difficulties that you sort of have to encounter. Um um I think the way a colleague might put it is that um you know, like if if you are on the streets in New York and you put in a blindfold and you put in a car and and um after some hours um you the blindfold's off and you're in Beijing. Um, you know, I mean, that was too easy somehow, like like there was no ocean being crossed. Even if you don't know exactly what how what what was done, you're suspecting that that something wasn't right.

But is that still in the back of your head, to do you return to these to the prime, do you return to the prime numbers every once in a while to see? Yeah, when I have nothing better to do, which is less and less often, which is I get busy with so many things these days, but yeah, when I have free time and I'm not and I'm too frustrated to to work on my sort of real research projects and I also don't want to do my administrative stuff, I don't want to do some errands for my family, um, I can play with these these things um for fun, uh, and usually you get nowhere. Yeah, you have you have to learn to just say, okay, fine, once again, nothing happened, I I will move on. Um, yeah, very occasionally one of these problems I actually solved, or sometimes, as you say, you think you solved it, and then you're euphoric for maybe 15 minutes and then you think, I should check this because this is too easy, too good to be true, and it usually is.

What's your gut say about when these problems would be uh solved, when prime and Goldbach? Prime, I think we'll keep getting keep getting more partial results. Um, it does need at least one, this parity barrier is is the biggest remaining obstacle. Um, there are simpler versions of the conjecture where we are getting really close. Um, so I think we will, in 10 years we will have many more, much closer results. May not have the whole thing. Um, yeah, so twin prime is somewhat close. Riemann hypothesis, I have no, I mean, it has to happen by accident, I think, so the Riemann hypothesis is a kind of more general conjecture about the distribution of prime numbers, right? Yeah, it's it's states are sort of viewed multiplicatively.

Like, for questions only involving multiplication, no addition, the primes really do behave as randomly as as you could hope. So there's a phenomenon in probability called square root cancellation that, um, you know, like if you want to poll, say, America upon on on some issue. Um, and you you ask one or two voters and you may have sampled a bad sample, and then you get you get a really imprecise um measurement of of the full average, but if you sample more and more people, the accuracy gets better and better, and it actually improves like the square root of the number of people you you sample. So yeah, if you sample a thousand people, you can get like a 2-3% margin of error.

So, in the same sense, if you measure the primes in a certain multiplicative sense, there's a certain type of statistic you can measure, and it's called the Riemann's zeta function, and it fluctuates up and down, but in some sense, um, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random. And there's a very precise way to quantify that, and the Riemann hypothesis is a very elegant way that captures this. But um, as with many others in mathematics, we have very few tools to show that something really genuinely behaves like really random. And this is actually not just a little bit random, but it's it's asking that it behaves as random as an actually random set; this square root cancellation. And we know actually, because of things related to the parity problem, actually that most of our usual techniques cannot hope to settle this question. Um, the proof has to come out of left field. Um, yeah, but uh what that is, yeah, no one has any serious proposal.

Um, yeah, and and there's there's various ways to sort of, as I said, you can modify the primes a little bit, and you can destroy the Riemann hypothesis. Um, so like it has to be very delicate. You can't apply something that has huge margins of error. It has to just barely work. Um, and like, um, there's like all these pitfalls that you have like dodge very adeptly. The prime numbers are just fascinating. Yeah. Yeah. What what to you is um most mysterious about the prime numbers?

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

Another incredibly surprisingly difficult problem is the Collatz conjecture. Oh, yes. Simple to state, beautiful to visualize in its simplicity, and yet extremely uh difficult to solve, and yet you have been able to make progress. Uh, Paul Erdős said about the Collatz conjecture that mathematics may not be ready for such problems. Others have stated that it is an extraordinarily difficult problem, completely out of reach—this is in 2010—out of reach of present-day mathematics, and yet you have made some progress. Why is it so difficult to make? Can you actually even explain what it is?

Oh, yeah. So, it's it's it's a problem that you can explain. Um, yeah, it um it helps with some um visual aids, but yeah, so you take any natural number like say 13. And you apply the the following procedure to it. So, if it's even, you divide it by two, and if it's odd, you multiply by three and add one. So, even numbers get smaller, odd numbers get bigger. So, 13 will become 40 because 13 * 3 is 39. Add one, you get 40. So, it's a simple process for odd numbers and even numbers. They're both very easy operations. And then you put it together. It's still reasonably simple. Um, but then you ask what happens when you iterate it. You take the output that you just got and feed it back in. So, 13 becomes 40. 40 is now even; divide by 2 is 20. 20 is still even; divide by 10, 2, 10, 5, and then 5 * 3 + 1 is 16. And then 8, 4, 2, 1. So, uh, and then from 1 it goes 1, 4, 2, 1, 4, 2, 1. It cycles forever. So this sequence I just described, um, yeah, 13, 40, 20, 10, so these are also called hailstone sequences because there's an oversimplified model of of hailstone formation, yeah, which is not actually quite correct, but it's somehow taught to high school students as a first approximation is that um like a a little nugget of ice gets gets an ice crystal forms in a cloud, and it it goes up and down because of the wind, and sometimes when it's cold, it acquires a bit more mass, and maybe it melts a little bit, and this process of going up down creates this slush of partially melted ice which eventually hailstone, and eventually it falls out of the earth.

So the conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, you go this process that that goes up if you're odd and down if you're even, eventually um goes down to to earth all the time. No matter where you start with this very simple algorithm, you end up at one, and you might climb for a while, right? Yeah, so yeah, if you plot it, um these sequences, they look like Brownian motion; um they look like the stock market, you know, they just go up and down in a in a seemingly random pattern, and in Usually that's what happens that that if you plug in a random number, you can actually prove at least initially that it would look like um random walk. Um, and that's actually a random walk with a downward drift. Um, it's like if you're always gambling on on roulette at at the casino with odds slightly weighted against you. So sometimes you you win, sometimes you lose, but over in the long run, you lose a bit more than you win. Um, and so normally your wallet will hit will go to zero um if you just keep playing over and over again. So statistically it makes sense. Yes. So, so the result that I I proved, roughly speaking, such that that statistically like 99% of all inputs would would drift down to maybe not all the way to one, but to be much much smaller than what you started. So, it's it's like if I told you that if you go to a casino, most of the time you end up, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started. That's kind of like the what the result that I proved.

So why is that result like can you continue down that thread to prove the full conjecture? Well, the problem is that um my I I used arguments from probability theory, um and there's always this exceptional event. So you know, so in probability we have this this law of large numbers um which tells you things like if you play a casino with a um a game at a casino with a losing um expectation, over time you are guaranteed or almost surely with probability as close to 100% as you wish, you're guaranteed to lose money. But there's always this exceptional outlier. Like it is mathematically possible that even in when the game is is the odds are not in your favor, you could just keep winning slightly more often than you lose. Very much like how in Navier-Stokes there could be, you know, um most of the time um your waves can disperse. There could be just one outlier choice of initial conditions that would lead you to blow up. And there could be one outlier choice of um um special number that they stick in that shoots off to infinity while all other numbers crash to earth uh crash to one. Um, in fact, um there's some mathematicians um who Alex Kontorovich for instance who've proposed that um that actually um these Collatz uh iterations are like the similar automata um actually if you look at what they happen on in binary, they do actually look a little bit like like these Game of Life type patterns. Um, and in an analogy to how the Game of Life can create these these massive like self-replicating objects and so forth, possibly you could create some sort of heavier-than-air flying machine, a number which is actually encoding this machine which is just whose job it is is to encode is to create a version of itself which which is larger, heavier-than-air machine encoded in a number that flies forever.

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

What would you say is out of these within reach famous problems is the hardest problem we have today? Is there a Riemann hypothesis? We want is up there. Um, P versus NP is a good one because like uh that's that's a meta-problem, like if you solve that in the um in the positive sense that you can find a P=NP algorithm, that potentially this solves a lot of other problems as well, and we should mention some of the conjectures we've been talking about. You know, a lot of stuff is built on top of them. Now there's ripple effects. P equals NP has more ripple effects than basically any other, right? If the Riemann hypothesis is disproven, um that would be a big mental shock to the number theorists, uh but it would have follow-on effects for um cryptography um because a lot of cryptography uses number theory, um it uses number theory constructions involving primes and so forth, and um it relies very much on the intuition that number theorists are built over many many years of what operations involving primes behave randomly and what ones don't. Um, and in particular, our encryption um methods are designed to turn text with information on it into text which is indistinguishable from um from random noise. So um and hence we believe to be almost impossible to crack um at least mathematically. Um, but uh if something as core to our belief as the Riemann hypothesis is is wrong, it means that there are there are actual patterns in the primes that we're not aware of, and if there's one, there's probably going to be more. Um, and suddenly a lot of our crypto systems are in doubt. Yeah. But then how do you then say stuff about the the primes? Yeah. That you're going towards the Collatz conjecture again. Um, because if I I you you want it to be random, right? You want it to be randomly. Yeah. So more broadly, I'm just looking for more tools, more ways to show that that that things are random. How do you prove a conspiracy doesn't happen, right?

Is there any chance to you that P equals NP? Is there some Can you imagine a possible universe? It is possible. I mean, there's there's various uh scenarios. I mean, there there's one where it is technically possible but in practice is never actually implementable. The evidence is sort of slightly pushing in favor of no, that P probably is not equal to NP. I mean, it seems like it's one of those cases similar similar to the Riemann hypothesis that I think the evidence is leaning pretty heavily on the no. Certainly more on the no than on on the yes. The funny thing about P versus NP is that we have also a lot more obstructions than we do for almost any other problem. Um, so while there's evidence, we also have a lot of results ruling out many many types of approaches to the problem. Uh, this is the one thing that the computer scientists have actually been very good at. It's actually saying that that certain approaches cannot work. No-go theorems. It could be undecidable. We don't Yeah, we don't know.

There's a funny story I read that when you won the Fields Medal, somebody from the internet wrote you and asked uh you know, what are you going to do now that you've won this prestigious award? And and then you just quickly, very humbly said that, you know, this shiny metal is not going to solve any of the problems I'm currently working on. So, I'm just I'm going to keep I'm going to keep working on them. It's just first of all, it's funny to me that you would answer an email in that context, and second of all, it um it just shows your humility. But anyway, uh maybe you could speak to the Fields Medal, but it's another way for me to ask uh about Grigori Perelman. What do you think about him famously declining the Fields Medal and the Millennium Prize, which came with a $1 million prize money? He stated that I'm not interested in money or fame. The prize is completely irrelevant for me. If the proof is correct, then no other recognition is needed.

Yeah. No, he's he's somewhat of an outlier. Um, even among mathematicians who tend to uh to have uh somewhat idealistic views. I've never met him. I think I'd be interested to meet him one day, but I I never had the chance. I know people who met him, but he's always had strong views about certain things. Um, you know, I mean, it's it's not like he was completely isolated from the math community. I mean, he would he would give talks and write papers and so forth. Um, but at some point, he just decided not to engage with the rest of the community. He was he was disillusioned or something. Um, I don't know. Um, and he decided to to uh uh to peace out uh and you know, collect mushrooms in St. Petersburg or something. And then that's that's fine, you know, and you can you can do that. Um, I mean that's another sort of flip side. I mean, we are not—a lot of our problems that we solve, you know, they some of them do have practical application, and that's that's great, but uh like if you stop thinking about a problem, you know, so he's he hasn't published since in in this field, but that's fine; there's many many other people who've done so as well.

Um, yeah, so I guess one thing I didn't realize initially with the Fields Medal is that it it sort of makes you part of the establishment. Um, you know, so you know, most mathematicians, you there's uh just career mathematicians, you know, you just focus on publishing the next paper, maybe getting one to promote one one rank, you know, and and starting a few projects, maybe taking some students or something. Yeah. But then suddenly people want your opinion on things, and uh you have to think a little bit about you know things that you might just so foolishly say because you know, no one's going to listen to you. Uh, it's more important now. Is it constraining to you? Are you able to still have fun and be a rebel and try crazy stuff and well play with ideas?

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

At the same time, I also want to give sort of respect to somebody like Grigori Perelman who is critical of awards in his mind. Those are his principles, and any human that's able for their principles to like do the thing that most humans would not be able to do. It's beautiful to see. Some recognition is is necessarily important. Uh, but yeah, it's it's also important to not let these things take over your life, um and like only be concerned about uh getting the next big award or whatever. Um, I mean, yeah, so again, you see these people try to only solve like really big math problems and not work on on on things that are less uh sexy, if you wish, but but but actually still interesting and instructive, as you say, like the way the human mind works, it's um we understand things better when they're attached to humans um and also uh if they're attached to a small number of humans, like this this way our human mind is is wired; we can comprehend and the relationships between you know 10 or 20 people, you know, but once you get beyond like 100 people, like there there's a there's a limit I think there's a name for it um beyond which uh it just becomes the other um and so we have you have to simplify the poll master, you know, 99.9% of humanity becomes the other um and uh often these models are are incorrect, and this causes all kinds of problems, but um so yeah, so to humanize a subject, you know, if you identify a small number of people and say, you know, these representative people of the subject, role models for example, um that has some role um but it can also be um uh yeah, too much of it can be harmful because it's I'll be the first to say that my own career path is not that of a typical mathematician um I the very accelerated education I skipped a lot of classes um I think I was had very fortunate mentoring opportunities um and I think I was at the right place at the right time, just because someone doesn't have my

Um, trajectory, you it doesn't mean that they can't be good mathematicians. I mean, they be ma good mathematician in a very different style. Uh, and we need people of a different style. Um, and you know, even if and sometimes too much focus is given on the on the person who does the last step to complete um a project in mathematics or elsewhere; that's that's really taken, you know, centuries or decades with lots and lots of building lots of previous work.

Um, but that's a a story that's difficult to tell um if you're not an expert because, you know, it's easier to just say one person did this one thing. You know, it makes for a much simpler history. I think on the whole it um is a hugely positive thing to to talk about Steve Jobs as a representative of Apple; when I personally know and of course everybody knows the incredible design, the incredible engineering teams, just the individual humans on those teams. They're not a team. They're individual humans on a team. And there's a lot of brilliance there. But it's just a nice shorthand, like a very like pi. Yeah. Steve Jobs. Yeah. Yeah. As as a starting point, you know, as a first approximation; that's how you and then read some biographies and then look into much deeper. First approximation. Yeah. That's right.

Uh, so you mentioned you were a Princeton to um Andrew Wiles at that time. He's a professor there. It's a funny moment how history is just all interconnected. And at that time he announced that he proved the Fermat Last Theorem. What did you think, maybe looking back now with more context about that moment in math history?

Yes. So I was a graduate student at the time. I mean, I I vaguely remember, you know, there was press attention and uh um we all had the same um we had pigeon holes in the same mail room, you know. So we all picked our mail and like suddenly Andrew Wiles' mailbox exploded to be overflowing. That's a good that's a good metric. Yeah. um, you know, so yeah, we we all talked about it at at tea and so forth. I mean, we we didn't understand; most of us didn't understand the proof um we understand sort of high-level details um fact there's an ongoing project to formalize it in Lean, right? Kevin Buzzard. Yeah, can can we take that small tangent? Is it is it how difficult does that cuz as as I understand the Fermat Last the proof for uh for Fermat Last Theorem has like super complicated objects, yeah, really difficult to formalize now.

Yeah, I guess yeah, you're right; the objects that they use um you can define them. Uh, so they've been defined in Lean. Okay. So so just defining what they are can be done. Uh, that's really not trivial, but it's been done. But there's a lot of really basic facts about um these objects that have taken decades to prove and that they're in all these different math papers and so lots of these have to be formalized as well. Um, Kevin's uh Kevin Buzzard's goal, actually, he has a five-year grant to formalize Fermat's Last Theorem, and his aim is that he doesn't think he will be able to get all the way down to the basic axioms, but he wants to formalize it to the point where the only things that he needs to rely on as black boxes are things that were known by 1980 to um to number theorists at the time. Um, and then some other person, some other work would have to be done to to to get from there.

Um, so it's it's a different area of mathematics than um the type of mathematics I'm used to. Um um in analysis, which is kind of my area, um the objects we study are kind of much closer to the ground. We study I study things like prime numbers and and functions and things that are within scope of a high school um math education to at least uh define. Um, yeah, but then there's this very advanced algebraic side of number theory where people have been building structures upon structures for quite a while. Um, and it's it's a very sturdy structure. It's it's been it's been very um at the base at least is extremely well developed in the textbooks and so forth. But um um it does get to the point where um if you if you haven't taken these years of study and you want to ask about what what is going on at um like level six of of this tower, you have to spend quite a bit of time before they can even get to the point where you can see you see something you recognize.

What uh inspires you about his journey that we similar as we talked about seven years mostly working in secret?

Yeah. Uh, that is a romantic uh Yeah. So it kind of fits with sort of the the romantic image I think people have of mathematicians, to the extent they think of them at all, as these kind of eccentric uh, you know, wizards or something. Um, so that certainly kind of uh uh accentuated that perspective, you know. I mean, it is a great achievement; his style of solving problems is so different from my own um but which but which is great. I mean, we we need people; speak to it like what uh in in terms of like the you like the collaborative I like moving on from a problem if it's giving too much everybody.

Um, got it. But you need the people who have the tenacity and the fearlessness. Um, you I've collaborated with with people like that where where I want to give up uh cuz the first approach that we tried didn't work and the second one didn't approach, but they're convinced and they have the third, fourth, and the fifth approach works. Um, and I have to eat my words. Okay. I didn't think this was going to work, but yes, you were right all along. And we should say for people who don't know, not only are you known for the brilliance of your work, but the incredible productivity, just the number of papers, which are all of very high quality. So there's something to be said about being able to jump from topic to topic.

Yeah, it works for me. Yeah, I mean, there also people who are very productive and they focus very deeply on Yeah. I think everyone has to find their own workflow. Um, like one thing which is a shame in mathematics is that we have mathematics; there's sort of a one-size-fits-all approach to teaching mathematics um and you know, so we have a certain curriculum and so forth. I mean, you know, maybe like if you do math competitions or something you get a slightly different experience, but um I think many people um they don't find their their native math language uh until very late or usually too late, so they they stop doing mathematics and they have a bad experience with a teacher who's trying to teach them one way to do mathematics. They don't like it.

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

Um, you know, mathematics provides a common language. So, we can still talk to each other even if we are thinking in in different ways. But you can tell there's a different set of subsystems being used in the thinking process; like they take different paths. They're very quick at things that I struggle with and vice versa. Um, and yet they still get to the same goal. Um, that's beautiful. And yeah, but I mean the way we educate; unless you have like a personalized tutor or something. I mean, education sort of just by natural scale has to be mass-produced, you know; you have to teach to 30 kids and you know, if they have 30 different styles you can't you can't teach 30 different ways.

On that topic, what advice would you give to students uh young students who are struggling with math and but are interested in it and would like to get better? Is there something in this Yeah. um in this complicated educational context, what what would you Yeah, it's a tricky problem. One nice thing is that there are now lots of sources for mathematical enrichment outside the classroom. Um, so in in in my day there already there are math competitions. Um, and you know, there also like popular math books in the library. Um, yeah, but but now you have, you know, YouTube uh there there are forums just devoted to solving, you know, math puzzles and um and math shows up in other places, you know, like um for example, there there are hobbyists who play poker for fun uh and um they they you know, they for very specific reasons are interested in very specific probability questions um and and they actually know there's a community of amateur proists in in in poker um in chess, in baseball. I mean, there's there's there's uh yeah um there's math all over the place.

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

Um, you know, you don't need to be a PhD holder to just work on one atomic thing. There's something about the formalization here that also at as a very first step opens it up to the programming community too. The people who are already comfortable with programming. It seems like programming is somehow maybe just the feeling but it feels more accessible to folks than math. Math is seen as this like extreme, especially modern mathematics, seen as this extremely difficult to enter area and programming is not. So that could be just an entry point; you can execute code and you can get results. You know, you can print a hello world pretty quickly.

Um, you know, like if uh if programming was taught as almost entirely theoretical subject where you just taught the the computer science, the theory of functions and and and routines and so forth and and outside of some some very specialized homework assignments, you're not actually program like on the weekend for fun. Yeah. Or Yeah. They would be as considered as hard as math. Mhm. Um Yeah. Yeah. So, as I said, you know, there are communities of non-mathematicians where they're deploying math for some very specific purpose, you know, like like optimizing their poker game and and for them then math becomes fun for them.

Uh, what advice would you give in general to young people how to pick a career, how to find themselves? Like that's a tough tough tough question. Yeah. So um there's a lot less certainty now in the world, you know. I mean, I there was this period after the war where uh at least in the west, you know, if you came from a good demographic you uh you know, like you there was a very stable path to to a good career; you go to college, you get an education, you pick one profession and you stick to it; becoming much more a thing of the past. So I think you just have to be adaptable and flexible. I think people have to get skills that are transferable, you know, like like learning one specific programming language or one specific subject of mathematics or something. It's it's it's that itself is not a super transferable skill, but sort of knowing how to um reason with with abstract concepts or how to problem solve when things go wrong. So these are things which I think we will still need even as our tools get get better and you know, you you would be working with AI support and so forth. But actually you're an interesting case study. I mean, you're like a one of the great living mathematicians, right? And then you had a way of doing things and then all of a sudden you start learning; I mean, first of all, you kept learning new fields, but you learn Lean; that's not that's a non-trivial thing to learn; like that's a that's a for a lot of people that's an extremely uncomfortable leap to take, right?

Yeah, mathematicians um first of all, I've always been interested in new ways to do mathematics; I I I feel like a lot of the ways we do things right now are inefficient. Um, I I I I spend me my colleagues, we spend a lot of time doing very routine computations or doing things that other mathematicians would instantly know how to do and we don't know how to do them. Uh, and why can't we search and get a quick response? And so that's why I've always been interested in exploring new workflows. About four or five years ago, I was on a committee where we had to ask for ideas for interesting workshops to run at a math institute. And at the time, Peter Scholze had just formalized one of his his um new theorems. And um there are some other developments in computer-assisted proof that look quite interesting. And I said, "Oh, we should we should uh um we should run a workshop on this. This be a good idea." Um, and then I was a bit too enthusiastic about this idea. So I I got volunteered. Um, so I did with a bunch of other people, Kevin Buzzard and Jordan Ellenberg and and a bunch of other people. Um, and it was it was a a nice success. We brought together a bunch of mathematicians and computer scientists and other people and and we got up to speed and state um and it was really interesting um developments that that most mathematicians didn't know was going on. Um, that lots of nice proofs of concept, you know, just sort of hints of of what was going to happen. This was just before ChatGPT, but there was even then there was one talk about language models and the potential um capability of those in the future. So that got me excited about the subject. So I started giving talks um about this is something we should more of us should start looking at um now that I arranged to run this conference and then ChatGPT came out and like suddenly AI was everywhere and so uh I got interviewed a lot um about about this topic um and in particular um the interaction between AI and formal proof assistance and I said, yeah, they should be combined; this this is this is um this perfect synergy to happen here. And at some point, I realized that I have to actually do not just talk the talk but walk the walk, you know, like you know, I don't work in machine learning I and I don't work in proof formalization and there's a limit to how much I can just rely on authority and saying, you know, I I'm a I'm a worn mathematician just trust me, you know, when I say that this is going to change mathematics and I'm not doing it any when I don't do any of it myself, so I felt like I had to actually uh uh justify it. Yeah, a lot of what I get into actually I don't quite see in advance as how much time I'm going to spend on it and it's only after I'm sort of waist-deep in in in in in a project that I I I realized by that point I'm committed.

Well, that's deeply admirable that you're willing to go into the fray, be in some small way a beginner, right? Or have some of the sort of challenges that a beginner would, right? New concepts, new ways of thinking, also, you know, sucking at a thing that others I think I think in that talk you could be a Fields Medal-winning mathematician and undergrad knows something better than you.

Yeah. Um, I think mathematics inherently I mean mathematics is so huge these days that nobody knows all of modern mathematics. Um, and inevitably we make mistakes and um you know uh you can't cover up your mistakes with just sort of bravado and and uh I mean because people will ask for your proofs and if you don't have the proofs you don't have the proofs. Um, I don't love math. Yeah. So it does keep us honest. I mean, not not I mean you can still it's not a perfect uh panacea, but I think uh we do have more of a culture of admitting error than because we're forced to all the time.

Big ridiculous question. I'm sorry for it once again. Who is the greatest mathematician of all time? Maybe one who's no longer with us. Uh, who are the candidates? Euler, Gauss, Newton, Ramanujan, Hilbert.

So, first of all, as as mentioned before, like there's there's some time dependent on the day. Yeah. Like like if if you if you if you plot cumulatively over time, for example, Euclid like like sort of like is is one of the leading contenders. Um, and then maybe some unnamed anonymous mathematicians before that um you know whoever came up with the concept of of numbers, you know, you know um do mathematicians today still feel the impact of Hilbert?

Just oh yeah, directly of everything that's happened in the 20th century. Yeah, Hilbert spaces; we have lots of things that are named after him; of course, just the arrangement of mathematics and just the introduction of certain concepts. I mean, 23 problems have been extremely influential; there's some strange power to the declaring ing which problems are hard to solve. The statement of the open problems. Yeah. I mean, this is bystander effect in everywhere. Like if no one says you should do X, everyone just moves around waiting for somebody else to to uh to do something and and like nothing gets done. Um, so and and like it like it's one one thing that actually uh you have to teach undergraduates in mathematics is that you should always try something. So um you see a lot of paralysis um in an undergraduate trying a math problem; if they recognize that there's a certain technique that that can be applied they will try it, but there are problems for which they see none of their standard techniques obviously applies, and the common reaction is then just paralysis; I don't know what to do or um or I think there's a quote from the Simpsons: I've tried nothing and I'm all out of ideas um so you know like the next step then is to try anything like no matter how stupid um and in fact almost as stupid the better um which you know and one a technique which is almost guaranteed to fail, but the way it fails is going to be instructive um like it fails because you you you're not at all taking into account this hypothesis; oh, this hypothesis must be useful; that's a clue.

I I think you also suggested somewhere this this fascinating approach which really stuck with me; I started using it and really works; I think you said it's called structured procrastination.

No, yes; it's when you really don't want to do a thing. Do you imagine a thing you don't want to do more? Yes. That's worse than that. And then in that way, you procrastinate by not doing the thing that's worse.

Yeah. Yeah. It's a nice It's a nice hack. It actually works. Yeah. Yeah. This um I mean with anything like you know I mean like you um psychology is really important; like you you talk to athletes like marathon runners and so forth and and they talk about what's the most important thing is it their training regimen or the diet and so forth. Actually, so much of it is actually psychology. Um, you know, just tricking yourself to to think that the problem is feasible um so that you can you're motivated to do it.

Is there something our human mind will never be able

To comprehend? Well, I sort of, as a mathematician, I mean, you there must be some suffer that you can't understand. That was the first thing that came to mind. So that, but even broadly, is there are we li—is there something about our mind that's—we're going to be limited even with the help of mathematics?

Well, okay, I mean, like how much augmentation are you willing—like, like, for example, if if I didn't even have pen and paper, um, like if I had no technology whatsoever. Okay, so I'm not allowed blackboard, pen, and paper. Right? You're already much more limited than you would be—incredibly limited. Even language, the English language is a technology. It's a—it's one that's been very internalized.

So, you're right. There really the the the formulation of the problem is incorrect because there really is no longer a just a solo human. We're already augmented in extremely complicated, intricate ways, right?

Yeah.

Yeah. We're already like a collective intelligence.

Yes.

Yeah. Yes. So, humanity plural has much more intelligence, in principle, on its good days than than the individual humans put together. It can also have less. Okay. But uh, um, yeah, so yeah, mathematical community plural is is is incredibly super intelligent uh entity um that uh no single human mathematician can can come closer to to replicating.

You see it a little bit on these like question analysis sites. Um, so this math overflow, which is the math version of stack overflow, and like sometimes you get like this very quick responses to very difficult questions from the community. Um, and it's it's it's a pleasure to watch actually as a as an expert. I'm a fan, spectator of that uh of that site, just seeing the brilliance of the different people, the um the depth of knowledge that people have and the the willingness to engage in the in the rigor and the nuance of the particular question. It's pretty cool to watch. It's fun. It's almost like just fun to watch.

Uh, what gives you hope about this whole thing we have going on, human civilization? I think uh, yeah. Um, the younger generation is always like like really creative and enthusiastic and and inventive. Um, it's a pleasure working with with with uh with uh with young students. Um, you know, the uh the progress of science tells us that the problems that used to be really difficult can become extremely, you know, can become like trivial to solve. You know, I mean, like it was like navigation, you know, just just knowing where you were on the planet was this horrendous problem. People died, um, you know, or or lost fortunes because they couldn't navigate, you know, and we have devices in our pockets that do this automatically for us. I guess, a completely solved problem, you know. So things that are seem unfeasible for us now could be maybe just sort of homework exercises for—

Yeah.

But one of the things I find really sad about the finiteness of life is that I won't get to see all the cool things we create as a civilization. You know, that cuz in the next 100 years, 200 years, just imagine showing showing up in 200 years.

Yeah.

Well, already plenty has happened, you know, like if if you could go back in time and and talk to your teenage self or something, you know what I mean?

Yeah.

And just the internet and and our AI. I mean, again, they they've been in—they're beginning to be internalized and say, yeah, of course, an AI can understand our voice and and give reasonable, you know, slightly incorrect answers to to any question. But yeah, this was mind-blowing even 2 years ago, and in the moment it's hilarious to watch on the internet and so on. The the drama—people take everything for granted very quickly, and then they—we humans seem to entertain ourselves with drama out of anything that's created—somebody needs to take one opinion, another person needs to take an opposite opinion, argue with each other about it. But when you look at the arc of things, I mean, just even in progress of robotics.

Yeah.

Just to take a step back and be like, "Wow, this is beautiful that we humans are able to create this."

Yeah.

When the infrastructure and the culture is is healthy, you know, the community of humans can be so much more intelligent and mature and and and rational than the individuals within it.

Well, one place I can always count on rationality is the comment section of your blog, which I'm a big fan of. There's a lot of really smart people there. And thank you, of course, for uh for putting those ideas out on the blog, and it's—I can't tell you how uh honored I am that you would spend your time with me today. I was looking forward to this for a long time, Terry. I'm a huge fan. Um, you inspire me. You inspire millions of people. Thank you so much for talking.

Oh, thank you. It was a pleasure.

Thanks for listening to this conversation with Terrence Tao. To support this podcast, please check out our sponsors in the description or at lexfreedman.com/sponsors. And now, let me leave you with some words from Galileo Galilei: Mathematics is a language with which God has written the universe. Thank you for listening and hope to see you next time.