#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI

The following is a conversation with Terence 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.

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You know, what's fascinating is the mechanisms of human memory before we had widely adopted technologies and tools for writing and recording stuff, certainly before the computer. So you can look at medieval monks, for example, that would use the now well studied uh memory techniques like the memory palace, the spatial memory techniques to memorize entire books. That is certainly the effect of technology started by Google search and moving to all the other things like Notion that we're offloading more and more and more of the task of memorization to the computers.

Which I think is probably a positive thing because it frees more of our brain to do deep reasoning, whether that's deep dive, focused specialization, or the journalist type of thinking versus memorizing facts. Although I do think that there's a kind of background model that's formed when you memorize a lot of things, and from there, from inspiration, arises discovery. So I don't know. It could be a great cost to offloading most of our, uh, memorization to the machines.

But it is the way of the world. Try Notion AI for free when you go to Notion dot com slash Lex. That's all lowercase Notion dot com slash Lex to try the power of Notion AI today. This episode is also brought to you by Shopify, a platform designed for anyone to sell anywhere with a great looking online store. Our future friends has a lot of robots in it. Looking into that distant future, you have Amazon warehouses with millions of robots that move packages around.

You have Tesla bots everywhere in the factories and in the home and on the streets and, uh, the baristas, all of that. That's our future. Right now you have something like Shopify that connects a lot of humans in the digital space. But more and more, there will be a automated, digitized, AI fueled connection between humans in the physical space. Like a lot of futures, there's going to be negative things and there's going to be positive things.

And like a lot of possible futures, there's little we could do about stopping it. All we can do is steer it in the direction that, uh, enables human flourishing instead of hiding in fear or fear mongering. Be part of the group of people that are building the best possible trajectory of, uh, human civilization. Anyway, sign up for a one per month trial period at Shopify dot com slash Lex. That's all lowercase. Go to Shopify dot com slash Lex to take your business to the next level today.

This episode is also brought to you by NetSuite, an all in one cloud business management system. There's a lot of messy components to running a business, and I must ask, and I must wonder, at which point there's going to be an AI AGI like CFO of a company, an AI agent that handles most, if not all, of the financial responsibilities or all of the things that NetSuite is doing. At which point will NetSuite increasingly leverage AI for those tasks?

I think probably it will integrate AI into its tooling, but I think there's a lot of edge cases that we need the human wisdom, the human intuition grounded in years of experience in order to make the tricky decision around the edge cases. I suspect that running a company is a lot more difficult than people realize. But there's a lot of sort of paperwork type stuff that could be automated, could be digitized, could be summarized, integrated, and used as a foundation for the said humans to make decisions.

Anyway, that's our future. Download the CFO's guide to AI and machine learning at netsuite dot com slash lex. That's netsuite dot com slash lex. This episode is also brought to you by Element, my daily zero sugar and delicious electrolyte mix. You know, I run along the river, uh, often and, uh, get to meet some really interesting people. Uh, one of the the people I met was preparing for his first ultramarathon. I believe he said it was a hundred miles.

And that, of course, sparked in me the thought that, uh, I need for sure to, uh, to do one myself. Some time ago now, I was planning to do something with David Goggins, and I think that's still on the sort of to-do list between the two of us to do some crazy, uh, physical feat. Of course, the thing that is, uh, crazy for me is a daily activity for Goggins. But nevertheless, I think it's important in the physical domain, the mental domain, and and all domains of life to challenge yourself.

And athletic endeavors is one of the most sort of crisp, clear, well-structured way of challenging yourself. But there's all kinds of things. Writing a book, to be honest, having kids and marriage and relationships and friendships, all of those, if you take it seriously, if you go all in and do it right, I think, uh, that's a serious challenge. Because most of us are not prepared for it. And you learn along the way. And if you have the rigorous feedback loop of improving constantly and growing as a person and really doing a great job of the thing, I think, uh, that might as well be an ultramarathon.

Anyway, get a sample pack for free with any purchase. Try it to drink element dot com slash lex. And finally, this episode is also brought to you by AG One, an all in one daily drink to support better health and peak performance. I drink it every day. I'm preparing for a conversation on, uh, drugs in the Third Reich. And funny enough, it's a kind of way to analyze Hitler's biography, is to look at what he consumed throughout.

And, uh, Norman Ohler does a great job of analyzing all of that and tells the story of Hitler and the Third Reich in a way that hasn't really been touched by historians before. It's always nice to look at, uh, key moments in history through a perspective that's not often taken. Anyway, I mention that because I think Hitler had a lot of stomach problems, and so that was the motivation for getting a doctor, the doctor that eventually, uh, would, uh, fill him up with all kinds of drugs.

But, uh, the doctor earned Hitler's trust by giving him probiotics, which is a kind of revolutionary thing at the time. And so that really helped deal with whatever stomach issues that, uh, Hitler was having. All of that is a reminder that war is waged by humans, and humans are biological systems, and biological systems require fuel and supplements and all of that kind of stuff. And depending on what you put in your body will affect your performance in the short term and the long term.

With meth, that's true with, uh, Hitler, to his last days in the bunker in Berlin, all the cocktail of drugs that he was taking. So I think I got myself somewhere deep and I'm not sure how to get out of this. Uh, it deserves a multi-hour conversation versus a few seconds of mention. But yeah, all of that was sparked by my thinking of, uh, AG1 and how much I love it. I appreciate that you're listening to this and, uh, coming along for the wild journey that these, uh, ad reads are.

Anyway, AG1 will give you a one month supply of fish oil when you sign up at drink AG1 dot com slash Lex. This is the Lex Fridman podcast. To support it, please check out our sponsors in the description or at LexFreeman dot com slash sponsors. And now, dear friends, here's Terence 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 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 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 worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese mathematician Soichi 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, or or think of like like a like driving like on on a road, something. And you you want to 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. Uh, so you want to use as little area uh uh in order to turn it around. So um, but the needle's infinitely maneuverable. So you could imagine just spinning it around its um as 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 their driving schools to do.

Uh, and that actually takes area pi over eight. Um, 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 Mazakovich uh showed that in fact uh you could actually uh turn the needle around using as little area as you wanted. So point zero, zero one, there was some really fancy multi um uh um back and forth u u-turn thing that you could you could do that that you could turn the needle around, and in so doing it, it would pass through every intermediate direction.

Is this in the two dimensional plane? This is in the 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 a 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 the 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 Bazarovitch's 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 your needle 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, combinatorics.

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 w 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 all these sort 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's it separates.

But potentially, if the conjecture had a negative solution, so what that means is that there's there's a very efficient way to pack um tubes pointing in different directions to a very, very narrow region of of of uh n 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 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 blow up, 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 in wave equations are stable or not, uh, whether they can create um these singularities. There's a famous unsolved problem called the Navier Stokes regularity problem.

So the Navier Stokes equations, the 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 in the bathtub, it won't explode on you. Um, or or have p have water leaving at the speed of light or anything, but potentially it is possible.

Um, and in fact, in recent years, the the consensus has has drifted towards the, uh, the the belief that, uh, that in fact, for certain very special initial con uh 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. And this is one of them. Of these seven, only one of them has been solved.

Uh, the Poincare 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? So the existence and smoothness, like you said, Millennium Prize problem. 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.

So we're trying to figure out if this thing usually doesn't blow up. But can we say for sure it never blows up? Right, yeah, so yeah, that is literally the the the million dollar question. Yeah, so th this is what distinguishes mathematicians from pretty much everybody else. Like if if if something holds ninety nine nine of the time, um, that's good enough for most, you know, uh, for 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 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 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 computation of fluid dynamics. A lot of it is actually just trying to solve the Navier Stokes equations as as best they can. Um, also gathering a lot of data so that they can get they can initialize the equation.

There's a lot of moving parts, so it's a very important problem practically.

Why is it difficult to prove general things about the 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 with two gases in it, uh, oxygen and nitrogen, uh, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them, right? Then they will mix. Um, and they should stay mixed, right? There's 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 the nitrogen goes to the other. And uh, you could have an extremely improbable configuration emerge, uh, which we never see, um, and and which we statistically is 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. Uh, there should be no preference in the digits of pi to favor, let's say, seven over eight. Um, but maybe there's some demon in the digits of pi that that, like, every time you compute more and more in the digits of pi that that, like, every time you compute more and 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 if fluid is in motion, the energy gets transported around. And, uh, 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 it 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's there's some turbulence and and waves and so forth, but eventually it it settles down, and and and the the lower the amplitude, the the smaller the velocity, the the more calm it gets. Um, but potentially there is some sort of 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 effect of viscosity is relatively less.

And so it could happen that that it it creates us some sort of, um, um, what's called a self-similar blowup scenario where, you know, um, the energy of the fluid starts off at some, um, large scale, and then it all sort of, um, transfers energy into a smaller, um, region of of of the fluid, which then at a much faster rate, um, moves it 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, yeah, you you could, you could actually, uh, converge to to all the energy, um, concentrating in one point in a a finite amount of time. Um, and that that that's a that's a scenario is called finite time blowup. 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 this energy from one big eddy into one smaller eddy.

It will transfer into maybe three or four, and then those ones split up into maybe three or four smaller eddies of their own. And so the energy gets dispersed to the point where the viscosity can can then keep everything under control. Um, but if it can somehow, um, concentrate, um, all the energy, keep it all together, uh, and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blowup can occur.

So there are papers who have claimed that, oh, you just need to take into account conservation of energy and just carefully use the viscosity, and you can keep everything under control for not just the 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 finite time blowup, that velocity stays smooth, and they'd 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 blowup. 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, so 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 blowup 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 ob 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 that 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 interactions, I could force all the energy to to blow up in a 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 it's it's not just about finding, you know, taking a technique that is gonna 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 m you 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 counterexamples for nearby problems, um, kind of rules out, um, uh, it it saves you a lot of time because you 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, was, 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, it 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 unbalanced at small scales, but not in balance at large scales, or or vice versa.

Um, so Navier-Stokes is what's called super critical, so at at smaller and smaller scales, the transport terms are much stronger than the viscosity terms. So the viscosity terms 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, uh, Ladyzhenskaya, she in the sixties showed in two dimensions there was no blowup, and in two dimensions, the the Navier-Stokes equation is what's called critical.

The effect of transport and the effect of viscosity are 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 prove, 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 can create all kinds of blowup 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 that's a key qualitative feature that distinguishes some equations for being sort of nice and predictable, and and, you know, like like planetary motion, and I mean, there's there's certain equations that that you can predict for millions of years and, uh, or or thousands at least.

Again, it's not really a problem, but but f- there's a reason why we can't predict the weather past two weeks into the future, because it's a supercritical equation. Lots of really strange things are going on at at very fine scales.

So whenever there's some huge source of nonlinearity that Can create a huge problem for predicting what's gonna happen.

Yeah, and it's 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 many in in many equations, you can approximate things by bulk. Um, so, for example, planetary motion, you know, if you wanted to, uh, understand the orbit of the moon or Mars or something, you don't really need the microstructure of, uh, the 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, uh, and y- you j- just the aggregate behavior is important. Um, but if you wanna 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-scale 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 And then show that the halting problem from computation theory has, uh, consequences for fluid dynamics. So, uh, Show it in that way. Can you describe this this Idea?

so this came out of of this work of constructing th- this this av- this average equation that that blew up. Um, so o- one, um, as as part of how I had to to do this, so there's sort of this naive way to do it. You j- you you just keep t pushing, um, e- 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 naive way to to force blowup. Um, it turns out in five and higher 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 to just always keep trying to push, um, the energy into smaller and smaller scales, um, what happens is that the energy st- 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 directive bullsh- doesn't uh 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 airlocks. Uh, so, um, I I needed an equation which would start with a a fluid doing something at one scale, it would push this energy into the next scale, but it would s 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, uh, and and then you you push that in as well. So, um, b- 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 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, it was constructed like an el electronic circuit. So I I actually thanked my wife for this because she was trained as a electrical engineer. Um, and, um, you know, she she talked about, um, uh, you know, she had to design circuits and so forth. And you know, if you if you want a circuit that does a certain thing, like maybe and have a light that that flashes on and then turns off and then on and then off, you can build it from s from more primitive components, you know, capacitors and resistors and so forth.

And you have to build a diagram, and you you, um, and these diagrams, you can, you can sort of follow it with your eyeballs and say, oh, yeah, the the the current will 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, so 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.

Uh, kind of a Rube 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 it's really waterpump, uh, type of thing where, um, you know, so modern computers are electronic, you know, they're they're they're powered by, uh, by electrons, p- passing through very tiny wires and interacting with other other electrons and and so forth.

But instead of electrons, you 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, uh, they would come out with some new configuration, which is which would be something like an and gate or or gate, you know, that that it would it 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 Turing machine, and and then you could you have computers, uh, which are made completely out of water. Um, and if you have computers, then maybe you can do robotics, uh, uh, so, uh, you know, hydraulics and so forth. Um, and so you could create some h machine, which is basically a fluid analog of what's called a von Neumann machine. Uh, so, von Neumann proposed, if you wanna colonize Mars, the sheer cost of transporting people and 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 raw materials, smelt them, and build more copies of the same machine, um, then you could colonize the 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 rob- it's a fluid robot, okay, and what it would do it's it's it's purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state.

It it wouldn't start just yet. Once it's ready, the big robot, kind of filled with water, would transfer all its energy into this the smaller configuration and then power down, okay, and then like like 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 once you do that, you can just keep iterating.

So this, in principle, would create a blowup, uh, for the actual Navier-Stokes, and this is what I managed to accomplish for this average Navier-Stokes. So it provided this sort of roadmap 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 b- 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 that this include 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, cause 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 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 p- now pursuing ways to make Navier-Stokes blow up, which are nowhere near as ridiculously complicated as this. Um, um, they they they actually are pursuing much closer to the the direct self-similar model, which can, uh, it 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 Navier-Stokes to this Turing 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 Turing 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 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's called cellular automata, 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 they die. Um, and this, i um, you know, um, when I was a uh, a student, uh, it was a very popular screensaver to actually just have these these animations, like, going. And and they they look very chaotic.

And, in fact, they look a little bit like turbulent flow 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, five cells, which evolves and it just moves at a certain direction. And that's like this this vortex rings this analog. Um, yeah, so this is an analogy, the Game of Life is kind of like a discrete equation, and and, um, the fluid Navier-Stokes is 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 of Life is a very simple system, it only has, like, three or four rules, um, to to do order, but but you can design all kinds of interesting configurations inside it. Um, there's some called called a glider gun that does nothing but 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, um, 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 gliders coming out if so, may maybe if both of of the, um, streams, um, have gliders, then there'll be an out output stream.

But if only one of them does, then nothing comes out. 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, that you can build a Turing machine. I mean, the it's 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 von Neumann machine, which over a lo- huge period of time and it always looked like glider guns inside doing these very steampunk calculations.

It would create another version of itself, which could replicate.

That's so incredible.

A lot of this was like community crowdsourced by, uh, 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 for Navier-Stokes. Um, now it was just a much as I said, analog is much worse than digital, like, I, uh, it's gonna be, um, you can't just directly take the constructions in the Game of Life, M, plonk them in, but again, it just it shows it's possible.

You know, there's a kinda emergence that happens with these cellular automata, local rules Maybe 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 amenable to mathematical analysis? Do we have the tools to say something profound about that?

The thing is, you can get this emergent, very complicated structures, but only with very carefully prepared initial conditions. Yeah, so so these these these glider guns and and gates and and cellular machines, if you just plonk down randomly some cells and you and look at that, you will not see any of these. Um, and th- that's the analogous situation with Navier-Stokes, again, you know, that that with with typical initial conditions, you're not you will not have any of this weird computation going on.

Um, but basically, through engineering, you know, by by by s s specially designing things in a very special way, you can pick 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 yeah, uh this Interesting.

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. Uh, they look like random, 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 if you just pick digits randomly, there's something called the law of large numbers that tells you you're gonna get as many ones as as twos, uh, 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 spent 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 n a natural numbers to the natural numbers, so so maybe, um, you know, two maps to four, three maps to six, and so forth. Um, some functions are 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, uh, multiply by a constant. If you multiply a number by 10, um, if you, if, 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 n um functions are additive. Some functions 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 2, and I take the integer part of that.

So 10 by square root of 2 is like 14 point something, so 10, map to 14, um, 20 map to 28. Um, so in that case, a additively is true then, so 10 plus 10 is 20, and 14 plus 14 is 28. But because of this rounding, uh, sometimes there's round off 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 1.

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, comp- completely structured, which is explaining this sort of partial pattern that you have.

Um, and so if you have these sort of 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, uh, in either case, you can make progress. Um, a good example of this is that there's this old theorem in mathematics called Szemerédi's theorem, uh, proven in the 1970s.

It concerns trying to find a certain type of pattern in a set of numbers, and the the pattern is arithmetic progression. Things like 3, 5 and 7, or or or 10, 15 and 20. And Szemerédi, Andrey Szemerédi proved that, um, any set of of numbers that is sufficiently big, um, what's called s 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 certain density 1/2, um, and they contain arithmetic progressions of any length.

Um, so in that case, it's obvious because the the, uh, the odd numbers are really, really structured. I can just take, uh, uh, 11, 13, 15, 17, I can just I can I can easily find arithmetic progressions in in in that set. Um, but, um, Szemerédi's theorem also applies to random sets. If I take the set of all numbers and I flip a coin, um, and I, uh, for each number, and I only keep the 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. M 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 theorems, but occasionally they they give colorful names. 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 script of Hamlet or any other finite string of text. Uh, it will just take some time, like quite a lot of time, actually, but if you have an infinite number, then it happens.

Um, so, um, basically the theorem says that if you take an infinite string of of digits, or whatever, um, eventually any finite pattern you wish will emerge. Uh, 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 uh, you 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 just an abstraction of, um, a finite number for which you d 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 a lot cleaner when you do that. I mean, in 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 s send some things to infinity, send some some things to zero.

Um, and um, and the mathematics becomes a lot simpler to work with there.

I wonder h- how often using infinity, uh, forces us to deviate from, um, the physics of reality.

Yeah, so there's a s a lot of pitfalls. Um, so, you know, we we spend a lot of time in our undergraduate math classes teaching analysis. Um, and analysis is often about how do you take limits and and whether you you know, so, for example, a plus b is always b plus a. Um, so when when you have a finite number of terms and you add them, you can swap them and this n there's no problem.

But when you have an infinite number of terms, there are 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 and th- there's there's a certain type of way of reasoning that helps you avoid mistakes.

Um, in r- more recent years, um, people have started taking results that are true in in infinite limits and tr- what's called and what's called finiteizing them. Um, so you know something's true eventually, but, um, you don't know when. Now give me your rate. Okay, so it's just that if I have don't have an infinite number of monkeys, but but a large finite number of monkeys, how long do you have to wait for Hamlet to come out?

Um, and, um, that's a more quali- 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 i 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 if, um, and so this is why you never see the monkeys create Hamlet, you can maybe see them create a four letter word, but not nothing that big.

And so I personally find once you finitezate an infinite statement, it's it does become much more intuitive, uh, and it's no longer so so weird. Um,

So even if you're working with infinity, it's good to finiteize so that you can have some intuition.

yeah, the downside is that the finite proofs are just much, much messier, and and, uh, yeah, so so the infinite ones are found first, usually, like decades earlier, uh, and then later on people finitez them.

So since we mentioned a lot of math and a lot of physics, uh, what do you see as 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 new perspective on circuits. So this different way of looking at the world, given that you've done mathematical physics, so you you've you've worn all the hats.

Right. So I think science in general is 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 h- how we think the world will works. Um, so, um, we can't directly access reality, okay? Uh, all we have are the o- observations, which are incomplete, and they they have errors, um, and, um, 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, ha- have the observation we'd like to make 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 is it, uh, you we we stay within the model and we ask, what are the consequences of that model? What observations, what what predictions would the model make of the of future observations, um, or past observations?

Does it fit o- observed data? Um, so there's definitely a symbiosis. Um, it's math I guess mathematics is is unusual among other disciplines is that we start from hypotheses like the axioms of a model and ask what conclusions come out from that 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 I want to to make money, I want to do this, okay?

And then you you you find the path to get there. Um, a lot there's and there's a lot less of speculation of, uh, suppose I did this, what would happen? Um, you know, planning and and and, uh, modeling, um, uh, speculative fiction, maybe, is 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 know, I mean, they want to know, you know, where is this asteroid going to go?

You know, or what is what what is the weather going to be tomorrow? Um, but, um, mathematics also has this other direction of of of going from the, uh, the axioms.

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

Well, you need both top down and bottom up. Um, yeah, as I said, it's it's a really an interaction between all these things, you know? So over time, the observations and the theory and the modeling should go both get get closer to reality. But initially, and it is in it, I mean, uh, this is, um, this is always the case, you know, they're they're always far apart to begin with. Um, but you need one to figure out, uh, where 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 to find more data to refine the models. Um, you know, so it it it goes it goes back and forth. Um, within mathematics itself, there's 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 wanna study prime numbers or whatever, they can just generate large data sets and with a comp- so once we had computers, um, we began to do it a little bit. Um, although even before well, like Gauss, for example, he discovered or he conjectured the most basic theorem in in number theory, it's called the prime number theorem, which predicts how many primes there are up to a million, up to 1 trillion.

It's not an obvious question. And basically what he did was that he computed, uh, I mean, m mostly, uh, um, by himself, but also hired human computers, um, people who whose professional job it was to do arithmetic, um, to compute the first 100,000 primes or something and made tables and made a prediction. Um, that's an early example of experimental mathematics. Um, but until very recently, it was not, um, yeah, I mean, theoretical mathematics was just much more successful.

I mean, because doing complicated mathematical computations is, uh, was just not not feasible, uh, until very recently. Uh, and even nowadays, you know, even though we have powerful computers, only some mathematical things can be, um, explored numerically. There's something called the combinatorial explosion. If you want us to study, for example, Szemerédi's theorem, you wanna study all possible subsets of the numbers 1 to a thousand, there's only one thousand numbers. How bad could it be? It turns out the number of different subsets of of 1 to a thousand is 2 to the power 1,000, which is way bigger than than than any computer can currently can can, any computer ever or ever can, um, enumerate.

Um, so if you have to, um, there are certain math problems that very quickly become just i in intractable to attack by direct brute force computation. Uh, chess is another, um, uh, famous example. Uh, the the number of chess positions, uh, we can't get a computer to fully explore. But now we have A I. Um, um, we have tools to explore this space not with 100% guarantees of s success, but with experiment, you know, so, like, um, we can empirically solve chess now, uh, for example, uh, we have we have a uh, very, very good AIs that that can, you know, they don't explore every single position in in the game tree, but they have found some very good approximation.

Um, and people are using actually these chess engines to to make, uh, to do experimental chess. Um, that, uh, they're they're revisiting old chess theories about, oh, you know, when you this type of opening, you know, uh, this is a g- this is a good type of move, this is not. And they can use these chess engines to actually, uh, refine, uh, in some cases overturn, um, um, conventional wisdom about chess. And, um, I do hope that, uh, that mathematics will will have a larger experimental component in the future, perhaps powered by A I.

We'll, of course, talk talk about that, but in the case of chess, and there's a similar thing in mathematics, the 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, and that you can intuit as a human being. And then from that, we humans can construct a theory of the matter. You've mentioned the Plato's cave allegory, 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, um, looking at shadows of reality. D 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, um, models. Um, a and technically they are distinct, and I think they will always be distinct. Um, but they can get closer, um, over time. Um, you know, so, um, and the process of getting closer often means that you you have to discard your initial intuitions. Um, so, um, uh, like astronomy, for for provides great examples, you know, like, you know, like, your an initial model of the world is that it's flat because it's it looks flat, you know?

Uh, and, um, and that it's and it's big, you know, and the rest of the universe, the sky is is not, you know, like the sun, for example, looks really tiny. Um, and so you 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 for, you know, but but over time, as you make more and more observations bringing it closer to to to reality, um, the model gets dragged along with it, you know?

And so over time, we had to realize that the earth was round, that it spins, it goes around the solar system, solar system goes around the galaxy, and so on and so forth, and the big bang part of the universe, you know, it's expanding. Um, the expansion is just self expanding, accelerating. And in fact, very recently, in this year or so, there's, uh, even the acceleration of the universe itself is, uh, this evidence that it is non constant.

And, uh, the explanation behind why that is It's it's catching is up. Um, it's catching up. I mean, it's still, you know, the dark matter, dark energy, this this kind of thing.

Yes. We have a we have a model that sort of explains that fits the data really well. It just has a few parameters that, um, uh, you have to specify. Um, but, so, you know, people say, oh, that's fudge factors, you know, with with enough fudge factors, you can explain anything. Um, yeah, but, uh, the mathematical point, uh, 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 ten parameters that explains ten, ten observations, that is a completely useless model. It's what's called overfitted. But, like, if you have a model with with, you know, two parameters and it explains a 1 trillion observations, which is basically, uh, so, uh, yeah, the the the dark matter model, I think it has like fourteen parameters, and it explains petabytes of data. Um, that that the astronomers have. Um, you can think of a theory, uh, like, uh, one way to think about, um, uh, a physical mathematical theory, uh, a theory is it's it's a compression of of the universe, um, and a a 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, uh, almost all of of your observations. I 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 the most incomprehensible thing about the universe is that it is comprehensible.

right? And not just comprehensible, you can do an equation like E equals M C squared.

There is actually a some mathematical possible explanation for that. Um, so there's this phenomenon in mathematics called universality. So many complex systems at the macroscale are coming out of lots of tiny interactions at the other macroscale. And normally, because of the combinatorial explosion, you would think that, uh, the macroscale equations must be, like, infinitely exponentially more complicated than than the, uh, the microscale ones. And they are, if you want to solve them completely exactly. Like, if you want to model, um, all the atoms in a box of of air, uh, 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. But certain laws emerge at the microscopic scale that almost don't depend on what's going on at the macro scale, or or only depend on a very small number of parameters. So if you want to model a gas, um, of, you know, quintillion particles in a 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 you need to know about these ten to the twenty three or whatever particles. Um, so we we have, um, we we don't understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do, um, have a good understanding of why universa- universality occurs. Um, uh, most basic one is 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, uh, famous bell curve.

Uh, there's now even a meme with this curve.

And even the meme applies broadly, the universality to the meme.

Yeah, y- yes, you can call it that, uh, if you if you like. But there are many, many processes, for example, you you can take lots of independent, um, random variables and average them together, um, uh, in in various ways. Uh, you you can take a simple average or more complicated average, and we can prove in various cases that that these these bell curves, these Gaussians, emerge. And it is a satisfy satisfying explanation. Um, sometimes they don't, um, so so if you have many different inputs and they're all correlated in some s systemic way, then y- you can get something very far from a bell curve show up.

Uh, and this is also important to know when this this agreement fails. So universality is not 100% reliable thing to rely on. At that, um, um, that the global financial crisis was a a famous example of this. Uh, people thought that, uh, um, mortgage de- defaults, um, uh, had this sort of, um, Gaussian type behavior that that if you if you ask, if you have a population of of of, uh, you know, 100,000 Americans with mortgages, ask what what proportion of them will default on their mortgages.

Um, if everything was decorrelated, it would be a nice bell curve and and like you you can, you can manage risk with options and derivatives and so forth. And, um, and it is 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, uh, that's very non-Gaussian behavior. Um, and, uh, this wasn't fully accounted for in the 2008. Um, now I think there's some more awareness that this is a systemic risk is actually a much bigger issue.

And, uh, just because the model is pretty, uh, and nice, uh, it may not, you match reality, uh, so so the mathematics of working out what models do is really important. Um, but, um, also the the science of validating when the models, uh, fit reality and when they don't. Um, I mean, that you need both, um, and but mathematics can help because it it can, it, uh, for example, the central limit theorem, it it tells you that that if you have certain axioms like like like, uh, non correlation, that if all the inputs were not correlated to each other, um, then you have these Gaussian behaviors, so things are fine.

It it tells you where to look for weaknesses in the model. So, be if you have a mathematical understanding of s central limit theorem, and someone proposes to use these Gaussian copulas or whatever to mo- to model, um, default risk, um, if you're mathematically, um, trained, you would say, okay, but what is the systemic correlation between all your inputs? And so then that then you can ask the economists, you know, well, how how 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 this synergy between science and and mathematics.

A little bit on the topic of universality. 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.

um, but anyway, so you 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 ar- areas of mathematics? Is there kind of deep underlying structure, uh, 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, um, 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, Euclid w- uh, worked both on 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 you you could say that 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, uh, who developed, you know, what we now call analytic geometry, that you can you can parameterize the plane, a geometric object, by, um, by two real numbers, uh, every point can be.

Uh, and so geometric problems can be turned into into problems about numbers. Um, and the the today, this feels almost trivial, like, there's like there's there's there's no content to this. Like, of course, uh, you you know, um, a plane is x x and y, and because that's what we teach and it's internalized, um, but it was an important development that these these two fields are are were unified. Um, and this process has just just gone on throughout mathematics over and over again, and algebra and geometry were separated, and now we have this weird algebraic geometry that connects them.

And 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 fox. A fox knows many things a little bit, but a 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, uh, who can play both roles. Um, and I think, like, ideal collaboration between mathematicians in involves a very you need some diversity, like, um, a fox working with many hedgehogs or or vice versa.

So, yeah, but but I identify mostly as a fox, uh, certainly. I I I like, uh, arbitrage somehow, you know, like, like, um, learning how one field works, learning the tricks of that wheel, and then going to another field, which people don't think of, is related, but I can 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, they are really, they're really hedgehogs, you know, they they they know everything about one field, and they're much faster and and and more effective in that field. But I can I can give them these extra tools.

I mean, you said that you can be both a hedgehog and a and the fox, depending on the context and 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 with the, uh, the, uh, the fox paradigm, yeah. So, um, yeah, I I like looking for analogies, narratives. Um, I I spend a lot of time. If if there's a result, I see it 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, um, 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 of the tools that are used in in in that field. So it's very exploratory, very doing crazy things in crazy fields, and and like, yeah, reinventing the wheel a lot. Yeah, whereas sort of the hedgehog style is, uh, I think, much more scholarly, you know, you you very knowledge based, you you stay up to speed on, like, all the developments in this field.

You you know all the history. Um, you have a very good understanding of of exactly the strengths and weaknesses of of each particular, uh, technique. Um, yeah, uh, I think you you'd rely a lot more on sort of c- calculation than sort of trying to find narratives. Um, so yeah, I mean, I could do that too, but, uh, there are other people who are extremely good at that.

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

That's a good question. Um, when I came to graduate school, uh, in Princeton, um, so John Conway was there at the time, he he passed away a few years ago, but, uh, I remember one of the very first research talks I I went to was a talk by Conway on what he called extreme proof. So Conway had just had this this amazing way of of thinking about all kinds of things in a in a way that you wouldn't normally think of.

So, um, he thought of proofs themselves as occupying some sort of space, you know, so so, um, if you want to prove something, let let's say that there's infinitely many primes, okay, there's all these different proofs, but you could you could rank them in different axes, like some proofs are elegant, some proofs are long, some proofs are are, um, are elementary and so forth. Um, and so there's this cloud, uh, so the space of all proofs itself has some sort of shape.

Um, and so he he was interested in in extreme points of this shape, like, out of all all these proofs, what is the one that is the shortest at the ex at the expense of everything 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 it's not just getting a proof for a result. W 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's certainly informed my writing style, um, that, you know, like, when you do your your math assignments and as a undergraduate, your homework and so forth, you you're sort of encouraged to just write down any proof that works okay and then 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 generalized other, um, things. Um, it's it's the same in in many other disciplines, like like coding. It's an 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 and dirty, and it works. But, uh, there's lots of good principles for for, um, writing, uh, 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, uh, similar things with mathemat- mathematics, so

yeah, the i for 1st of all, there's so many beautiful things there, and and Conway is one of the great minds, uh, in mathematics ever and computer science. Uh, just even considering the space of proofs and saying, Okay, what does this space look like? And what are the extremes? Uh, like you mentioned, coding is an analogy. It's interesting because there's also this activity called, uh, Code Golf, which I also find beautiful and fun. Uh, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task.

Then I believe there's even competitions on this. 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 a different notation or whatever to use to to accomplish a different task.

yeah, you learn a lot. I mean, it it may seem like a frivolous exercise, but it it it can generate all these insights, which if you didn't have this artificial, um, objective to to p- to pursue, you you might not see.

What do 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 equals M C squared, so when when a few concepts come together, that's why the Euler identity is often considered, uh, the most beautiful equation in mathematics. Do you do you find beauty in that one, in the Euler identity?

Yeah, well, as I said, I mean, what I find most appealing is is connections between different things that you that, um, so the if you, uh, e to the pi i equals minus one, um, so yeah, people, e i all these uses all the fundamental constants. Okay, that that that's, I mean, that's cute, um, but but to me, so the exponential function was is interesting, or, eh, to measure exponential growth, you know, something compound interest or decay, you know, a 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 degrees, uh, you need to rotate by pi radians. And i, complex numbers, represents this rotation if you wanna imagine axes of a ninety degree rotation, so a change in direction. So the exponential function represents growth and decay in the direction that you already are. Um, when you stick an i in the exponential, it it it now it's it's instead of motion in the same direction as your current position, it's the motion as at right angles to your current position, so rotation, um, and then so e to e to the pi i equals minus one tells you that if you

rotate for a 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 by i. So it it connects together all these strands of mathematics. Yeah, yeah, that connects geometry and complex and complex and, um, the complex numbers. They're all considered almost yeah, they're all next door neighbors in mathematics, uh, because of this iden- identity.

do do you think the thing you mentioned is cute? The the the collision of notations from these disparate fields, um, is 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, uh, in the 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, y- sometimes you, because your p- 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, uh, but it turns out okay.

So actually, uh, with physics, okay, so E equals M C squared. Okay, so, uh, 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 and 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, you know, F equals M A was the a famous, uh, Newton's second law of motion.

So those were the the primary objects, so they gave them the central billing 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, and particular momentum and energy. Um, uh, and th- it's not obvious that things have an energy. Like, it's not something you can directly measure the same way you can measure mass and and and velocity and so forth. But over time, people realized that this was actually a really fundamental concept.

Hamilton, eventually, in the 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 the dynamics, like, what what happens to to all 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 i um, the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their n- Newtonian thinking because, you know, everything was a particle and so forth to to to quantum mechanics, you know, because I I think because it was a wave, it it just w- looked really, really weird. Um, like, you, you ask, what is the quantum version of E equals M A? 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 this there's also an object called the Hamiltonian. It's a dif- 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 Schrodinger's equation that tells you exactly how quantum systems evolve once you have the 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. So, for example, in in classical mechanics, there's this thing called Noether'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 ten steps to the left, I experience the same laws of physics as if I was here.

And that corresponds to conservation of mo- momentum. Um, if I turn around by n- by some angle, again, I experience the same laws of physics. Uh, this corresponds to conservation of angular momentum. If I wait for 10 minutes, um, I s still have the same laws of physics. Um, so there's time translation invariance. This corresponds to the law of 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, uh, 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 Euclidean type spaces, and it has to be s, um, you know, and you know, w- we kind of know that at very tiny scales, uh, um, there's gonna be quantum fluctuations, there's a

space space time foam, um, and trying to to use Cartesian coordinates x y z is gonna 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, re- um, um, concepts. Um, the analog of a Hamiltonian that sort of organized everything,

Does your gut say that there is a theory of everything? So this is p- 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 that of unification, much like mathematics, um, over the years, you know, electricity and magnetism were were separate theories, and then Maxwell unified them. You know, Newton unified the the motion of the heavens for the motions of of objects on the earth and so forth. So it should happen, uh, it's just that the um, uh, again, to go back to this model of of the observations and and and theory.

Part of our problem is that physics is a victim of its own success. That our two big theories of of of physics, general relativity and quantum mechanics, are so are so good now. Is 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 c- combine them together. Um, but I have faith that we, you know, we've, 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 you will be a mathematician that develops a theory of everything?

What often happens is that when the physicists need, uh, um, some theory of math 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, You know, 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, uh, I think, uh, Riemann came up with something.

Um, and so, yeah, Riemann had developed, you know, Riemannian geometry, um, which is precisely, um, you know, a a a theory of spaces that are curved in in various general ways, which turned out to be almost exactly what was needed, um, for Einstein's theory. This is going back to to Wittgenstein'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 useful ways.

It just feels like you might need to go to some weird land that's very hard to to intuit. 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, cause it's it's not matching experiment. Uh,

So one of the big challenges, of course, like you said, is experiment is very tough 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 spacetime, 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, you mentioned, and now we're just it's.

It's very hard to use our limited ape descendants of a a cognition to intuit what that reality really is like.

This is why analogies are so important. You know, I mean, so, yeah, um, the round earth is not intuitive, cause we're we're stuck on it. Um, but, you know, but y- y- 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 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 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 flatland 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 r- r- round rock, flying through space. 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 it 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 and more suspicious because of that. So, you know, we, 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, there's, there's, there's, there's lots of science things that you can do at home.

I I there's lots of YouTube videos. I did a YouTube video recently with Grant Sanderson, who talked about this earlier. That, uh, you know, uh, 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 really in 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, shrouded in mystery. 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 and try to make some kind of general statements about distance, places.

Change of perspective is really important. You say travel broadens the mind, uh, 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 hypotheses, spherical cows, whatever, you know, speculate, um, and you know, this is this is what mathematicians do, and some, uh, you know, some 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 when you don't have much to figure out Well, stuff with.

if you propose axioms, then the mathematics lets you, you know, follow those axioms to to their conclusions. And sometimes you can get quite a lot, quite a long way from, you know, initial hypotheses.

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, uh, from a sort of mathematical standpoint, uh, w- 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 spacetime gravity itself, but of certain fields that might exist on top of spacetime. Um, so Einstein's equations of relativity just describe spacetime itself, um, but then there's other fields that live on top of that. Uh, there's the elect- electromagnetic field, um, there's, uh, things called Yang-Mills fields, and there's this whole hierarchy of different equations, of which Einstein's 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 yeah, so it's pointing in in different directions, um, but they propagate like waves. If if if you wiggle an arrow, it will, it will propagate and cre- and make all the arrows move kinda like, uh, sheaves of wheat in a 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 c- to c- 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, uh, the energy had to disperse a little bit.

And the moment it dispersed a little bit, it would, it would, it would stay regular. Yeah, this was back in 2000. That was part of why I got interested in Navier-Stokes afterwards, actually, yeah, so I developed some techniques to, um, to solve that problem. So part of it is it, it was, um, this problem is really nonlinear, uh, because of the curvature of the sphere. Um, there's there, there was a certain nonlinear effect, which was a non-perturbative effect.

It was when you sort of looked at it normally, it 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 kinda 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 sort of 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 s 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 theories. And I, I couldn't do it with pen and paper, um, and I had no 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, uh, yeah, my aunt walked in on on me while I was doing that, and she was asking, What are, what am I doing, doing this? It's complicated, And yeah,

is the answer.

yeah, and, you know, she's like, 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, uh, the the. The beauty of mathematics is that is that you get to change the rule, change the problem, change the rules as you wish. Uh, like, this is, 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, this river, you can't say I wanna build this a bridge over here instead, or I wanna build it out of paper in 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, where there's unlimited cheat codes available. Uh, and so, you know, you, you can, you can set this, uh, so, uh, there's a dimension that's too large. I'll set it to one. I'll solve the one dimensional problem first. Or there's a main term and an error term. I'm gonna s make a spherical cow assumption, as I've seen, 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, uh, the way you should, you should, uh, approach, um, uh, any reasonable math problem is that you, if, if, there. If there are ten 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 solve that, um, and then that just fig- so you you install nine cheats.

Okay, if you install ten cheats, then then the game is trivial, but you install nine cheats, you solve one problem. That, that's, 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 on, and then you solve that one. And after you, you know how to solve the ten problems, ten difficulties separately, then you have to start merging them. Uh, if you do it at a time, um, I, I.

As a kid, I watched a lot of these Hong Kong action movies, um, as as from my culture, um, and, uh, one thing is that every time there's the fight scene, you know, so maybe the the hero gets swarmed by 100 bad guy goons or whatever. But it'll always be choreographed so that he'd always be only fighting one person at a time, and then it would defeat that person and move on. And and, uh, because of that, he could, he could defeat all of them, right?

But whereas if they had fought a bit more intelligently, you know, and t you know, just swarmed the guy at once, uh, it would make for much, uh, much worse choreo- uh, um, cinema, but, uh, um, but they would win.

are you usually, uh, pen and paper? are you working, uh, with computer and LaTeX?

Um, mostly pen and paper, actually. So, in 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 on 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 that's the beauty of blackboards, you you erase, and it's, it's, it's a very organic thing. Um, I'm beginning to use more and more computers, um, partly because, uh, 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 had to to remember how to set up a Python program.

And And how does a for loop work? And and and debug it, and it would take an two hours and so forth, and and now I can do it in 10, 15 minutes, it's it's much. Um, yeah, I'm, I'm using more and more, uh, computers to do simple ex- 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 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, uh, traditional language, but it can also produce certificates.

So a software like like Python might do a computation and give you the answer is seven. Okay, that it does a sum of three plus four is equal to seven. But, uh, Lean can produce not just the answer, but but a proof that, uh, how it got the the answer of seven as three plus four, uh, and all the steps involved in in um, so it's, uh, 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 called are called proof assistants, and so they provide languages for which you you can create quite complicated, um, intricate, um, mathematical proofs. And, um, they produce these certificates that just give a 100um guarantee that your arguments are correct if you trust the compiler of Lean. But they made the compiler really small, and you can, there are several different compilers available for the same for, um,

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, um, individual lines of code resemble individual lines of a mathematical argument. Like, you might want to introduce a variable, you might want to prove a contradiction, you, you're, um, um, there are various standard things that you can do. And and it's it's written so that, ideally, it should be like a one to 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 a a natural number, or or or a function or something? Um, it, if you write things informally, it's, um, it's up in the type context.

Uh, you say, you know, um, clearly x is equal to, uh, 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, uh, can you tell me more about what this object is? Uh, what, what type of object it is? You see, you have to think more, um, like, at a philosophical level, well, not just sort of the computations that you're doing, but sort of what each object actually, um, is in some sense.

Is it using something like LLMs to do, uh, the type inference, or, like you mentioned, with the real numbers?

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 algorithms to ma- match one tree to another tree.

So it's actually doable to figure out if something is a a real number or a natural because, yeah, every number.

object sort of comes with a history of where it came from, and you can, you can kinda t trace Oh, I see, it. Um, yeah, so it's, that does make it's. It's designed sense, 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 in Lean, um, often there's a step. Okay, now I want to use, um, the fundamental theorem of calculus, say, okay, to do the next step.

So the Lean developers have built this, this massive project called Mathlib, a a collection of tens of thousands of useful facts, ah, about mathematical objects. And somewhere in there is the fundamental theorem of calculus, but you need to find it. So a lot of the burden of tech 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, okay, um, I need the fundamental theorem of calculus at this point, and it'll say, Okay, uh, um, uh, for example, um, when I code, I have GitHub Copilot installed as a plugin to my ID, and it scans my text and it sees what I need. Says, You know, I might even type, Okay, now I need to use the fundamental theorem of calculus. Okay, and then it it might just 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, it's close enough that I can say, Oh, yeah, if I just change it here and here, it will work. And then, like, half the time, it gives me complete rubbish. Um, so, but people are beginning to use AIs a little bit on top. Um, uh, mostly on the level of basically fancy autocomplete.

Um, that, uh, you can type half of one line of a proof, and it will find, it will tell you,

yeah, but but a fancy, especially fancy, with a 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 ten 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 just I mean, it having a pedantic coworker?

right, yeah, if if that was the only aspect of it, okay, but um, okay, there, there's some, there's there's some cases where it's actually more pleasant to do things formally. So there's a there's a theorem I formalized, and there was a certain constant twelve, um, that that came out of that, um, in in the final statement. And so this twelve had to be carried all through the proof, um, and like everything had to be checked that it goes through.

All the, uh, all these other numbers, uh, had to be consistent with this final number, twelve. And then so we wrote a paper proving this theorem with this number twelve. And then a few weeks later, someone said, Oh, we can actually improve this twelve to an eleven by reworking s 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.

I mean, there can be subtle things that you didn't quite realize, some properties on number 12 that you didn't even realize that you were taking advantage of. Um, so a proof can break down at a subtle place. Um, so we had formalized the proof with this constant twelve, and then when this this new paper came out, uh, we said, Okay, let's, uh. So that took like three weeks to formalize, uh, and and like twenty people to formalize this, this, this original proof.

So, open now, now, let's let's, um, uh, uh, let's update the B to twelve to eleven. And what you can do with Lean is that you just in your high-level theorem, you've you changed your twelve to an eleven. 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 these, 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 red. 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, uh, you, 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, uh, because this is a very quick process, you know, um, you, you make a change, there are ten things now that don't work, for each one, you, 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 twenty seven and you look at paragraph six and you, you have a line of of of text of math, I often can't read it immediately because it it assumes various definitions which I have to to go back and and maybe on ten 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 know, open up a novel halfway through and start reading. There's a lot of context. But with a proof in Lean, if you put your cursor on a line of code, every single object there, you can hover over it and it will, it will say what it is, where it came from, what, where stuff is justified. You can trace things back much easier than sort of flipping through a 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 wanna collaborate with another mathematician, um, either you do it at a blackboard where you, um, you can really interact, but if you're doing it sort of by email or something, um, basically, yeah, yeah, you have to segment it. Say, I'm gonna, I'm gonna finish section three, you do section four, but, uh, you can't really sort of work on the same thing, uh, collaboratively at the same time.

But with Lean, you can be trying to formalize some portion of the proof and say, Oh, I got stuck at at line sixty seven here. I need to prove this thing, but it it doesn't quite work. Here's the 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 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 proof they give me, but Lean gives me a certificate of of of trust. Um, so, yeah, I can do, I can do trustless mathematics.

So there's so many interesting questions there's, uh, so, uh, 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 just its implementation is the main difficulty. You have to figure out even what is the right path, right? 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, no engineer can actually do this, like I say, they have fixed requirements. Yeah, so there's this sort of jam sessions always at the beginning where you try all kinds of crazy things and you, 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 subproblems which you don't know how to do, but then you f- uh, you focus on on on the sub ones, uh, and sometimes different collaborators are better at at working on on certain things. Um, so one of my theorems I'm known for is a theorem of Ben Green, which is now 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 of Szemerédi, and the way we collaborated was that Ben had already proven a similar result for progressions of length three. Um, he showed that just like the primes contain, lossless progressions of length three, um, even and even, um, subsets of the primes, 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 cert- some set relating to the primes, like there's there's a certain technical condition which if I could have it, if if Ben could supply me with this fact, I could give, I could conclude the theorem. But I what I asked was a really difficult question in number theory, which, um, he said, No, there's no way we can prove this.

Can you? So he said, Can you prove your part of the theorem using a weaker hypothesis that I have a chance of proving? And he proposed something which he could prove, but it was too weak for me. Uh, I said, I can't use this. Um, so there's this, there was this conversation going back And forth. Um, Yeah, so so the Different cheats Hacking, yeah, Too. Different Yeah, yeah. I I wanted to cheat Techniques. More, he wanted to cheat less, I think, Uh, but eventually we found a a property which A he could prove, uh, and B I could use, um, and then we we could prove our theorem.

Uh, and um, yeah, so there's there's a there all kinds of dynamics, you know, I mean, it's, it's, it's every, every, um, collaboration has a, has a, has some story. There's no two of the same.

And then on on the flip side of that, like you mentioned with lean programming, now that's, uh, almost like a different story because you can do, you can create, I think you've mentioned a kind of a blueprint for a problem, and then you can really do a divide and conquer with Lean, where you're working on separate parts 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 and trustable. Um, yeah. So currently, no 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 boop, a blueprint, in a sense. It is taking a a difficult statement, like big theorem, and breaking up into maybe a hundred 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 possible, and to try to make each step kind of self-contained. Um, and or depending on only a very specific number of of previous statements that have been proven, so that each node of this blueprint graph that gets generated can be tackled independently of 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 wanna create an iPhone or some other complicated object, um, no one person can can build up, um, it's a single object, but you can have specialists who who just 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, cause you can have, if you can find problems that could be, Uh, broken down this way, then you can have, you know, thousands of contributors, right? It can be completely Yes, yes, yeah. So Distributed.

I told you before about this split between theoretical and experimental mathematics, and right now most mathematics is theoretical and only a tiny bit is experimental. I think the platform that Lean and and other software tools, uh, so, um, and GitHub and things like that, um, allow, uh, they will allow ex experimental mathematics to be to scale up, um, to a much greater degree than we can do now. So right now, if you want to, um, um, do any mathematical exploration, uh, of some mathematical p pattern or something, you need some code to write out the pattern.

And I mean, sometimes there are some computer algebra packages that can 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 is un unreliable. Um, so it's these are, uh, so you get these bespoke, uh, spaghetti code that written by non not professional programmers, but mathematicians, you know, and they're clunky and and and slow, and um, and so because of that, it's it's it's hard to to really mass produce experimental results.

Um, but, um, 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 Equational Theories Project. Basically, we generated about 22 million little problems in abstract algebra. Maybe I should back up and and tell you what what the project is. Okay, so abstract algebra studies operations like multiplication and addition and their abstract properties. Okay, so multiplication, for example, is commutative.

X times y is always y times x, at least for numbers, um, and uh, it's also associative. X times y times z is the same as x times y times z. Um, so, um, these operations obey some laws and they, they, they d d they don't obey others. For example, x times x is not always equal to x, so that law is not always true. So, given any 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 could satisfy. And our question was, 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 you can show that commutativity does not imply associativity. But some other laws do imply other laws by substitution and so forth.

Uh, and you can write down some some algebraic proof. So we looked at all the pairs between these 4,000 laws, and there's about twenty to 22 million of these pairs. And for each pair, we ask, does this law imply this? So, um, law, if so, give a give, uh, give a proof, if not, give a counterexample. Um, so 22 million problems, each one of which you could give to, like an an undergraduate algebra student, and they had a decent chance of solving the problem.

Although there are a few, out of these 22 million, there are like a hundred or so that are really quite hard. Okay, but a lot are easy. And the project was just to to work out, to determine the entire graph, like, 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 it would not have been feasible, you know, I mean, the state of the art in the literature was like, you know, fifteen equations and sort of highly implied. Uh, that's sort of at the limit of what a human with pen and paper can do. So, so you need to scale that up, so you need to crowdsource, but you also need to trust all the, um, you know, I i I mean, uh, uh, no one person can check 22 million of These proofs, right?

You you need 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 the project is almost complete. Um, so of these 22 million, all but two have been settled. Um, And, uh, well, actually, and of those two, uh, we have a pen and paper proof of the two, uh, and we we're formalizing it. In fact, I was, uh, this morning, I was working on on finishing It.

Yeah, um, so we're almost done on this. Um, it's

that's incredible.

It's, it's, yeah, a a fantastic

A How many people were able to Part fifty, Get, uh

Um, which in mathematics is is considered a huge number.

It's a huge number, that's Crazy.

So we're gonna have a paper with fifty authors, uh, and a big appendix of who contributed. What?

Here's an interesting question, not to maybe speak even more generally about it. When you have this pool of people, is there a way to, uh, organize the contributions by level of expertise of the people, of the contributors? Now, okay, uh, I'm asking a lot of pothead questions here, but I 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 of these lean projects is is that automatically you get all this data, you know? So, like, like, everything's been uploaded to this GitHub, and GitHub tracks who contributed what. Um, so you could generate statistics from, yeah, at any at any later point in time. You could say, Oh, this person contributed this, made this many lines of code or whatever. I mean, these are very crude metrics. Um, I would, I would definitely not want this to become, like, you know, part of your tenure review or something.

Uh, um, uh, but, um, I mean, uh, I think already in in in enterprise computing, right? People do use some of these metrics as part of of the assessment of of, uh, performance of a of an employee. Um, again, this is a direction which is a bit scary for academics to go down. We we we don't like metrics so much.

And yet academics use metrics, they just use old ones, number of papers.

Yeah, yeah, uh, it's true, it's true that, yeah, I mean, um,

It feels like this is a metric while flawed is is going in the more in the right direction, right? It's an interesting, uh, 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 of whether these are better predictors. Um, there's this problem called Goodhardt'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, no, I mean, G always It's, it's, it's rational.

Gaming the system.

So what we've done for this par project is is self-report. So, um, there are actually standard categories, um, from the sciences of what types of contributions people give. So there's there's, uh, concept and validation, and resources, and and coding, and and so forth. Um, so we, we, we, there's a standard list of twelve or so categories. Um, and we just ask each contributor to there's a big matrix of all the of all the authors and 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, uh, 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, uh, 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. Um, and this was a huge bottleneck, actually. Um, but still, we had projects that were, you know, ten authors 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.

Um, so we created this fictional character called D H J Polymath, in the spirit of Bourbaki. Bourbaki is is, uh, the pseudonym for a a famous, uh, group of mathematicians of the twentieth century. But, um, and so the paper was author 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. Uh, 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, uh, as your submitted as your 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 referred to the most famous person who was involved in the project. Oh, yeah, so this was Tim Gowers' Polymath project, This was Terence Tao's Polymath project, and not mentioned the the other nineteen or whatever people that were involved. So we're trying something different this time around, where w- we have everyone's an author, um, but we will have an ap- an appendix with this matrix, and we'll see how that well, I Works.

mean, uh, so both pro 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, uh, the Lean programming language just a few years ago, and he was saying that, uh, this might be the future of mathematics. And so it's also exciting that you're embracing, uh, one of the greatest mathematicians in 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, Uh, versus graduate level, uh, problems?

yeah, the the difficulty in increases exponentially with the the number of steps involved in the proof. It's a combinatorial explosion, right? So the thing with large language models is is that they make mistakes, and so if a proof has got twenty steps and your large language model has 10% failure rate, um, at each step, um, of c- of going in the wrong direction, like, uh, it's it's just extremely unlikely to actually, um, reach the end.

Actually, uh, 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 know, if if you get one little thing wrong, um, like the whole thing is is is nonsense. Um, even formal to formal is is is very hard. There are, there are different incompatible, um, f- uh, proof assistant languages.

Uh, there's Lean, but also Coq and Isabelle and and so forth. And actually, even converting from a formal language to formal language, um, is, uh, is an unsolved, basically unsolved problem.

that is fascinating. Okay, so, uh, but once you have an informal language, they're using, um, their RL trained model, so some 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, uh, over time become kind of normalized. Um, so, yeah, n- now, somehow it's, oh, of course, geometry is a solvable problem.

right? That's true, that's true. I mean, it's still beautiful, it's Still, it's worth it, you know?

is. It's a great work. It shows what's possible. I mean, um, it, it, um, the push doesn't scale currently. Yeah, three days of Google's server's server time to solve one, uh, high school math problem. Yeah, this this is not a scalable, uh, prospect, um, especially with the exponential increase in, um, a as as the complexity, um, increases.

We should mention that they got a silver medal performance,

The equivalent of, I mean, yeah, it's the equivalent it, uh, of a silver I mean, they, uh. medal. So, first of all, they took way more time than was, uh, allotted, um, and they had this assistance where where the humans started helped by by formalizing, um, but, uh, yeah, also they, they're given with those full marks for the solution, which I guess is formally verified. So I guess that that's, that's fair. Um, yeah, um, there there are efforts there was.

There will be a proposal at some point to actually have an an AI math Olympiad, where, at the same time as the human contestants, get the the actual olympiad, um, 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 it will have to be written in natural language rather than formal language.

Oh, I hope that happens. I hope that this IMO happens. I hope, I hope next it's, it won't one.

happen this IMO. The performance is not good enough in in in the time period. And and uh, um, but there are smaller competitions, uh, yeah, there are competitions where the the answer is a is a number rather than a a long form proof. Um, and that's that's um. AIs are actually a lot better at, um, problems where there's a specific numerical answer, um, because it's it's. It's easy to to to, uh, to reinforce, do reinforcement learning on it.

Yeah, you got the right answer, you got the wrong answer. Uh, it's, it's. It's a very clear signal. But a long form proof either has to be formal and then you c- uh, the the lean can give it thumbs up, thumbs down, or it's informal. Um, but then you need a human to grade it to tell, uh. And if you're trying to do billions of of reinforcement learning, um, you know, um, um, runs, you're, you're not.

You can't hire enough humans to, uh, to grade those. Um, I mean, it's already hard enough for for the large language models to do reinforcement learning on on just the regular text that that people get. But now you actually hire people not just to give thumbs up, thumbs down, but actually check the the output mathematically. Yeah, uh, that's too expensive,

So if we, uh, just explore this possible future, what, what? What is the thing that humans do that's most special in, uh, in mathematics? So that you could see AI, uh, not cracking for a while? So, I inventing new theories, so coming up with new conjectures versus, uh, proving the conjectures, building new abstractions, new representations, maybe, uh, an AI, turnstile with, uh, seeing new connections between disparate fields.

That's a good question. Um, I 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's really complicated, uh, calculations, you know, but it's all automated, been automated for centuries. Uh, we, we don't need that anymore, you know? But they used to navigate to do spherical navigation, spherical trigonometry to navigate how to get from from, um, the old world to the new.

So, like a very complicated calculations that again need to be automated. Um, you know, even a lot of undergraduate mathematics, even before AI, um, like Wolfram Alpha, for example, uh, is is 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 a problem and, um, and say, here's a problem in partial differential equations. Could you solve it using any of the 20 standard techniques?

Um, and they, they, they'll say, Yes, I've tried all 20 and here are the 100 different permutations, and and here's my results. Um, and th- that type of thing, I think it will work very well. Um, type of scaling to once you've solved one problem to to, uh, ma- make the AI attack a hundred adjacent problems. Um, the things that humans do still, you know, 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 gonna solve this problem, I'm gonna split up this problem into, um, into these two cases, I'm gonna try this technique. And um, sometimes, if you're lucky and it's it's a simple problem, it's the right technique and you solve the problem. And sometimes it will, it will get, it will have a problem. It w- it would propose an approach which is just complete nonsense, um, and but like, it looks like a proof, um, so this, this is one annoying thing about LLM generated mathematics.

Is so, um, we, we've, we've had human generated mathematics at a very low quality. Um, uh, like, you know, s submissions where 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 it's partly because that's what the reinforcement learning has actually trained them to do, uh, to to make things to, to produce text that looks like cu- um, uh, what is correct, which for many applications is good enough.

Um, uh, so the the errors are often really subtle, and then when you spot them, they're, they're really stupid. Um, like, you know, like, like, no human would have actually made that mistake.

yeah. It's actually really frustrating in the programming context, cause I I program a lot. And yeah, when a human makes, when a low quality code, there's something called code smell, right? You, you can tell, you can tell immediately, yeah, by like, okay, there's signs. But with with AI generated code, it neoterous. 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, so have to work.

yeah. So the sense of smell, okay, uh, this, this is. This is one thing that humans have, um, and th th. There's a metaphorical mathematical smell that, uh, this, we. It's not clear how to get the AIs to do, but, you know, eventually, um, I mean, so the the the way, um, AlphaZero and so forth. They 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 enunciate 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, so so you can say, I, I'm gonna try to to break up this problem into two smaller subtasks. And then it can say, Oh, this looks good. The the the 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, m. You made the problem worse because each of the two sub problems is actually harder than your original problem, which is actually what normally happens. If you try a random uh thing to try, you normally you actually. It's very easy to transform a problem into an even harder problem. Hm, very rarely do you prob- transform to s a simpler problem. Um, yeah. So if they can pick up a sense of smell, then they could maybe start competing with, uh, uh, human level mathematicians.

So so this is a hard question, but not competing, but collaborating. 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, yeah, go on.

what would that oracle? what would you like that oracle to be able to do? Would you like it to, uh, maybe be a verifier, like check do the code? Sm- like, you're, yes, uh, a professor child, this is the correct, this is a good, this is a p- promising, fruitful direction. Or or would you like it to, uh, generate possible proofs, and then you see which one is the right one? Um, or would you like it to maybe generate different representation, different, totally different ways of seeing this problem?

Yeah, I think all of the above. Um, a a lot of it is, we don't know how to use these tools. Because because it's a paradigm that is is not. Um, yeah, we have not had in the past. Assistants that are competent enough to understand complex instructions, um, that can work at massive scale, but are also unreliable. Uh, like, it's, it's a interesting, uh, but but unreliable in subtle ways, whilst we whilst providing s sufficiently good output.

Um, it's a interesting combination. Um, you know, I mean, you have re, you have, like, graduate students that you work with, who are, like, kind of like this, but n not at scale. Um, you know, and and and we had 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 Gowers, actually, you, I mean, imagine he, he actually s foresaw, like in in 2,000, he was envisioning what mathematics would look like in in in actually two and a half decades.

Um, and yeah,

that's funny.

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, as he's trying to solve a problem. And they would have to have a conversation, they. He. They. Sometimes the human would would propose an idea, and the AI would would, uh, evaluate it, uh, and sometimes the AI would propose an idea. Um, and uh, and sometimes a conversation was required, and the AI would just go and say, Okay, I've.

I've checked the the 100 cases needed here, or, um, uh, the first, uh, uh, you, you said this is true for all n. I've checked it for n up to 100, um, and it looks good so far. Or, hang on, there's a problem. It n equals 46. You know, so just a free form conversation where you don't know in advance where things are gonna go, but just based on on. I think ideas are good, proposed on both sides, calculations are proposed on both sides.

I've had conversations with AI where I say, Okay, let's we're gonna 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 I suggest using this tool and it. It'll find this. This lovely argument using a completely different tool, which eventually goes, you know, into the weeds and say, no, no, no, try using this, okay? And it might start using this and then it'll go back to the to the tool that it wanted to to to do before.

Um, and like, you have to keep railroading it, um, onto the path you want. And I, I could eventually force it to give the proof I wanted, um, but it was like herding cats, um, like. And the amount of personal effort I had to take to not just sort of prompt it, but, uh, also check its output. Cause it a lot of what it looks like is gonna work. And I know there's a problem on line 17 and I basically arguing with it, um, like, it was more exhausting than doing it, uh, 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 herding cats. And maybe you'll surprise us how quickly that comes, I, I you know,

I believe so. Um, so, in formalization, I I mentioned before that it takes ten times longer to formalize a proof than to write it by hand. With these modern AI tools, it's it, uh, and also just better tooling, the the um, the lean, um um, developers are doing a great job adding more and more features and making it u- user friendly. You know, it's going from nine to eight to seven. Okay, no big deal, but one day it'll drop below one, um, and that's a phase shift.

Because suddenly, um, it makes sense when you write a paper to f- to write it in lean first, um, or through a conversational AI which is generating lean, um, on the fly with you. And it becomes natural for journals to accept, uh, you know, maybe they'll offer, uh, expedited refereeing, you know, like, uh, if if a paper's already been formalized in 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 it it has been certified. Um, papers are getting longer and longer in mathematics, and it keeps getting harder and harder to get good ref refereeing for, um, the the really long ones, unless they're really important. Uh, it it is actually an an issue which the formalization is coming in just the right time. K- for this to be

and they easier and easier to guess because of the tooling and all the other factors. Then you're gonna see much more like math lib will grow potentially exponentially. It's a it's a it's a virtuous c- uh, cycle. Okay,

I mean, one phase shift of this type that happened in the past was, uh, the adoption of LaTeX, so so LaTeX is this typesetting language that all mathematicians use now. So in the past, people used all kinds of word processors and typewriters and and whatever. But at some point, LaTeX became easier to use than all other competitors. And that, if people would switch, you know, within a few years, like, like, it was just a dramatic um phase 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 c- I can imagine if it was a metal, writing paper, having some AI assistance in writing it, you know, uh, just, you know. Like, the autocomplete alone is already I I. I use it like it speeds up my my own writing. Um, um, like, you know, you, you. You can have a theorem and you have a proof, and the proof has three cases. And I, I. I write down the proof of the first case, and the autocompletes suggests, Oh, yeah, that now here's how the proof of the second case could work.

And actually, it was exactly correct. That was great. Saved me like five, ten minutes of, uh, of of typing.

But in that case, the AI system doesn't get the Fields medal, right? Are we talking 20 years, 50 years, 100 years? What do you think,

okay, uh, so I I gave a prediction in print also by 2026, which is now next year. Um, there will be math collaborations, you know, with AI, so not fields medal winning, but but like actual research level mathematics.

like, we sh- like, published ideas that are 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, there, there Has that already are, there's

happened, right?

already happened. Yeah, there there are. There are problems that were solved, uh, by a complicated process, convers- conversing with with AI to propose things, and then the human goes and tries it, and it and the can, of course, like, doesn't work. Um, but they, yeah, 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 mathematician, a human mathematician and an AI involved.

Um, but, uh, 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, thirty, forty, so 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 to code in Python. I'm not, I'm not a native, um, you know, professional, um, programmer, um, but um, the with AI, the 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, it's still a problem with, um, hallucinating, you know, the references that don't exist. Um, but this, I think, is a solvable problem. Uh, 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, you know, you, um, you should. In a few years, get to the point where you, you have a a lemma that you need, and, uh, say, has anyone proven this lemma before?

And it will. It will do basically a fancy web search, AI assisted, and say, Yeah, yeah, there are these six papers where something similar has happened. Uh, and I mean, it'll, 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, uh, 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 really 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, it's it's buried in in a list of options, of which the that other,

are bad? I mean, it being able to automatically generate a r- related work section that is correct, that's actually a beautiful thing. That might be another phase shift because it assigns credit correctly, it does. It breaks you out of the silos of

yeah, yeah, yeah, no, I mean, thought, you know, yeah, no, but it. There's a big hump to overcome right now. I mean, it's, 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 the last mile problem, um, with a lot, a lot of AI applications. Um, that, uh, you know, they can develop tools that, uh, 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 medal 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, a real breakthrough, something it doesn't, you know, like, feels metal or even hypothesis. It could be like, really, just this alpha zero moment would go, that kind of thing.

right. Um, yeah, this d uh, this decade, I can, I can see it. Like making a conjecture between two unrelated, two, two things that people thought were unrelated,

Oh, interesting, generating a conjecture, that's a beautiful conjecture,

yeah, and and actually has a re real chance of being correct and and and meaningful. And um,

because that's actually kind of doable, I suppose. But the where the data is is, yeah, no, that would be truly well, it's, amazing. Yeah,

um, the current models struggle a lot. I mean, so, um, a version of this is, um, I mean, the the physicists have a dream of getting the AIs to discover new l- new laws of physics. Um, uh, you know, a the the dream is you just feed it all this data, okay, uh, uh, and and it says, here, here is a new pattern 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, I mean, uh, or if it does, uh, there's a big concern of c- contamination, that that it did it only because, like, somewhere in its training data, it had already

seen some new, um, you know, Boyle's law, whatever law you're trying to to to reconstruct. Um, part of it is that we, 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 different 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 end up up being verified, um, or maybe counterexamples 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 realize that it was the wrong conjecture, and then they, they, they said, Oh, but we, uh, we should actually change, um, our claim to modify it in 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, cause it, it's embarrassing.

Uh, we make mistakes, and and we only like to publish our, 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, 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, Grigori Perelman. You mentioned that you try to be careful in your work and not let a problem completely consume you. Just you've really fallen in love with the problem and really cannot rest until you solve it. But you also hasten to add that sometimes this approach actually can be very successful. And the example you gave is Grigori Perelman, who proved the Poincare Conjecture and did so by working alone for seven years, with basically little contact with the outside world.

Can you explain this one Millennium Prize problem that's been solved, the Poincare Conjecture, and maybe speak to the journey that Grigori Perelman's been on?

all right, so it's it's a question about curved spaces. Earth is a good example. So I think I think it was a two d surface. Interesting thing around you, it could maybe be a torus with a hole in it, or it could have many holes. And there there are many different topologies a priori that that a surface could have. Um, even if you assume that it's it's bounded and and, uh, and and smooth and so forth.

So we we have figured out how to classify surfaces. As a first approximation, uh, everything's determined by something called the genus, how many holes it has. So a sphere has genus zero, 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 piece of a rope, you can contract it to a point and while staying on the surface.

And the sphere has this property, but a torus doesn't. Uh, if you're on a torus and you take a rope that goes around, say, the the outer diameter of The torus, there's no way it can't get through the hole, there's no way to contract it to a point. So it turns out that the the s the sphere is the only surface with this property of contractibility. I mean, up to, like, continuous deformations of the sphere. So, um, so things that are what are called topologically, um, equivalent to the sphere.

So, Poincare asked the same question in higher dimensions, um, so this it becomes hard to visualize, uh, because, um, surfaces, you can think of as embedded in three dimensions, but a curved three space, we don't have good intuition of four dimensional space to to to live in. And then there there are also three dimensional 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 three dimensional space now, which also has this simply connected property that every loop can be contracted, can you turn it into a three dimensional version of a sphere?

And so this is the Poincare conjecture. Weirdly, in higher dimensions, four and five, it actually was actually easier. So, uh, it was solved first in higher dimensions. There's somehow more room to do the deformation. It is easier to to to move things around to to create a sphere, but three was really hard. So people tried many approaches. There's sort of commentary approaches where you chop up the the the surface into little triangles or or tetrahedron, and you, you, you just try to argue based on how the faces interact with each other.

Um, there were, um, algebraic approaches, uh, there's there's various algebraic objects, uh, like things called the fundamental group that you can attach to these homology and cohomology, and and all these very fancy tools. Um, they also didn't quite work, um, but Richard Hamilton proposed a, um, partial differential equations approach. So you take, um, you take, so the problem is that you so you. You have this object, which is sort of secretly, is a sphere, but it's given to you in a in a really, um, in a in a weird way.

So it's like the. I 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, uh, you could, for example, think of it as the surface of a balloon. You could try 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, uh, unless, of course, it was a torus or something, in which case it would get stuck at some point. Like, if you inflate a torus, it would, there would, there'd be a point in the middle. When the inner ring shrinks to zero, you get a, you get a singularity, and you, you can't blow up any further. Uh, you can't flow any further. So he created this flow, which is now called Ricci flow, which is a way of taking a 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 either, uh, this process would give you a sphere, or it would create a singularity. Um, like, you very much like how PDEs either have, they have global regularity or finite time blow up, like it. It basically is almost exactly exactly the same thing, it's all connected. Um, and so, and and he showed that for two dimensions, two dimensional surfaces, um, uh, if you start with something connected, th- th- no singularities ever form.

Um, you, you never run into trouble, and you could flow, and and it will give you a sphere. And it so he he got a new proof of the two dimensional as well.

but by the way, that's a beautiful explanation of Ricci 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 of, Uh, 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 two D that that that helped, but in three D, the problem was that, uh, th- this equation was actually supercritical. It's the same problem as Navier Stokes. As you blow up, um, maybe the curvature could get concentrated in, in finite, smaller and smaller regions, and it, it, um, it looked more and more nonlinear, and things just looked worse and worse.

And there could be all kinds of singularities that that showed up. Um, some singularities, um, like if, if, uh, there's these things called neck pinches, where, where the, uh, the surface sort of creates, behaves like a, like a, like a, uh, barbell, and it it pinches at a point. Some s 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 evolve them separately.

But there's, there's a the pro, the 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, uh, 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 Perelman did was, first of all, he he made the problem, he turned the problem, a supercritical problem, to a critical problem.

Um, I said before about how, um, the invention of the of of energy, the Hamiltonian, that really clarified, um, Newtonian mechanics, um, uh, so he introduced, uh, something which is now called Perelman's reduced volume and Perelman's entropy. Um, he introduced new quantities, kind of like energy, that looked 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 Poincare conjecture. So, um, quite, uh, 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 a model proposing this idea as one of hundreds of different things to try.

Um, but the other ninety nine would be complete dead ends, but you'd only find out after months of of work. He must have had some sense that this was the right track to pursue, because, like, you know, it takes years to get from 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, you've done s similarly difficult things. What? What can you infer from the process he was going through, cause 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. W What happens to you? You're sitting in your office when you realize the thing you did for the last few days, maybe weeks, is a failure.

well, for me, I switch to different problems. Uh, so, uh, as I said, I'm, I'm, I'm, I'm a fox, I'm not a hedgehog.

but you'll generally, that is a break that you can take is is to step away and look at it from a problem.

yeah, you can modify the problem too. Um, I mean, um, yeah, you can add some cheat. Uh, if if there's a specific thing that's blocking you, that that just, 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, you know, 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, but if this is the only problem that you know, then everything else checks out, you know, then it's still worth fighting. Um, so, yeah, 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, yeah, um, eventually. sometimes, actually, it's it's even productive to make mistakes. So, um, one of the, I mean, um, there was a project which actually, uh, we won some prizes for, actually, but, um, uh, for other people. Um, we worked on this PDE problem a again, actually, this blow up regularity type problem, um, and it was considered very hard. Um, Jean Bourgain, um, uh, was another Fields medalist who worked on this 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 ev everything fit, and we were excited. Uh, we were planning celebration, um, to 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 s um, we have to estimate these thirteen terms that that show up in this expansion, and we estimated twelve of them.

But in our notes, I can't find the the estimation of the thirteenth. Can you can someone supply that? And I said, Sure, I'll look at this, and I give, it's, yeah, we didn't cover that. We completely omitted this term, and this term turned out to be worse than the other twelve terms put together. Um, in fact, we could not estimate this term, um, and we tried for a few more months, uh, 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 in 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, it solved the problem, so we we solved the problem after two years.

But if we hadn't had that initial false dawn of nearly solving the 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, you know, sometimes actually having the incorrect, you know, it's. It's like Columbus traveling to new new world. He had the incorrect version of the measurement of the size of the earth.

Um, he thought he was going to find a new trade route to India. Um, or at least that was how he s 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 most like that, you know, cause it, the this stuff, it feels like math. It's 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 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 work, you go on 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 are called mathematical diseases, where, where, where 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 the the career suffers and so forth. But they say, Oh, but I could have this big win, this will, you know, once I once I finish this problem, I'll, I will, it'll make up for all the years of of lost opportunity. And that that's, that's, I mean, occasionally, occasionally, it works, but I, I, um, I really don't recommend it for people who aren't 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, uh, we sort of say we will, we will study this set of problems. But even then, we don't promise definitely by five years, I will supply a proof of all these things. You know, we, 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, you can say something new about.

Uh, 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's, But

um, well, again, so I mean, the problem is like the Riemann hypothesis, those are so far out of reach. I why do you I,

think so?

yeah, there there's no even viable stragegy, like, even if I activate all my all the cheats that I know of, um, in this problem, like it, there's just still no way to get me to B. Um, like, it's, 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 it would be a useful thing to transport into this problem.

so we, we should s maybe s step back for a little bit and just talk about prime numbers. So they're often referred to as the atoms of mathematics. Can you just speak to the structure that these, uh, atoms so, provide?

the natural numbers have two basic operations attached to them, addition and mu multiplication. Um, so if you wanna 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. One, two, three, four, five. Or you can take the prime number. If you wanna generate it multiplicatively, you can take all the prime numbers two, three, five, seven, and multiply them all together.

Um, and together they, they, they gives you all the the the natural numbers, except maybe for one. So they're these two separate ways of thinking about the natural numbers, from an additive point of view and a mul multiplicative point of view. Um, and separately, they're not so bad. Um, so, like any question about that, if natural numbers only involves addition is relatively easy to solve, and any question that only involves multiplication is relatively 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 as, uh, undecidable. There are certain polynomials in some number of variables, you know, if there's a solution in the natural numbers, and the answer depends on on an undecidable statement. Um, like, like, whether, um, the axioms of of math are consistent or not.

Um, but, um, yeah, but even the s. 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. 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, which is mind blowing that you could prove something like that.

yeah. So what we've realized because of th- this, this type of of research is that these 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, three, five, seven, eleven and so forth, there are some twins in there, eleven and thirteen 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, uh, the twins, they show up and they're infinitely many of them, but they're actually reasonably sparse.

Um, not there's, there's not. I mean, initially, there's quite a few, but once you go to the millions, the trillions, they become rarer and rarer. And you could actually just, you know, like, if if someone was given access to the database of primes, you just edit out a few, a few primes here and there. They could make the twin prime conjecture false by just removing like point zero one 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 things 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, um, edited primes.

And so it, it, it must use some very, um, subtle, delicate feature of the primes that you can't just get from, like, like, aggregate s s statistical analysis. okay, so that's out Yeah, on the other hand, asymptotic progressions has 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 any 90% of the one, and, uh, it turns out, and s another thing we proved is that you still get arithmetic progressions.

Um, arithmetic progressions are much, you know, they're like cockroaches. of arbitrary length Yes, now, yes, that's crazy. I mean, yeah,

so so this, uh, 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, it's an infinite monkey type phenomenon. For any fixed length of your set, you don't get arbitrary length progressions, you only get quite short progressions.

But you're saying twin prime is not an infinite monkey a phenomena. I mean, it's a very subtle monkey. it's still an infinite monkey phenomenon.

right, yeah, yeah, if 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. Uh, yeah, we, we, we s. We believe the primes behave like a random set. And so the reason why we care about the prime hypothesis is that it's it's 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 with 100%, probably, uh, or 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 think it looks random or looks um structured, like periodic, in both cases, um, the arithmetic progressions appear, but for different reasons. Um, and this is basically all the ways in which the things uh, there are many proofs of of these sort of arithmetic progression type 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 you're happy, you win. But if the 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, uh, as I understand, progress on the k-tuple version.

Right, yeah, so, um, the the funny thing about conspiracies is that any one conspiracy theory is really hard to disprove. Uh, that, you know, if If you believe the world is run by by lizards, and say, here's some evidence that that it it not run by lizards, well, but that that evidence was planted by the lizards. Right, you know, you might even counter that, uh, this kind of phenomenon. Yeah, so So, like, like, um, a pure like there's there's almost no way to definitively rule out a conspiracy, and the same is true in mathematics, that a conspiracy is to solely devote devoted to learning twin primes.

You know, like, you would you would have to also infiltrate other areas of mathematics to sort of but but, like, i it could be made consistent, at least as far as we know. But there's a weird phenomenon that you can make one con- um, uh, one conspiracy rule out other conspiracies. So, you know, if The if the world is is run by lizards, it can't also be run by aliens, essentially. So, one unreasonable thing 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, uh, n no two which had, um, so the infinite pairs of primes which differ by at most, um, 246, actually is is is a is the current record.

Oh, so there's like a bound On the

Right, so, like, there's twin primes, there's these things called cousin primes that differ by by four, um, there's something called sexy primes that differ by six, Uh, what are There's

Sexy primes?

Primes that differ by six. Eh, the the name the name is much less ex- it the concept is much less exciting than the name suggests. 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. Um,

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, uh, 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 it. So there has to be two pigeons that are that are close together. So, for instance, if you have 100 numbers and they all range from 1 to a thousand, um, two of them have to be at most 10 apart Because you, you can divide up the numbers from 1 to 100 into 100 pigeonholes.

Let's let's say you have 101 numbers. If you have 101 numbers, then two of them have to be, uh, distance less than 10 apart, because two of them have to belong to the same pigeonhole. So it it's a basic, um, basic feature of, uh, a basic principle in mathematics. Um, so it doesn't quite work with the primes directly, because the primes get sparser and sparser as you go out, that that fewer and fewer numbers are prime.

But it turns out that, uh, there's a way to assign weights to the p- to to numbers, like, um, so there are numbers that are kind of, uh, almost prime, uh, but they're not. They they don't have n- 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 we understand primes. Um, and so, uh, for example, it was known for a long time that there were twin almost primes.

Uh, this has been worked out. So almost primes are something we we kind of understand. So you can actually restrict attention to a s a suitable set of almost primes. And, uh, whereas the primes are very sparse overall, uh, relative to the almost primes, they're actually, um, uh, much less sparse. They may, um, you can set up a s 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 there's pairs of primes that are just only a hundred, hundred apart.

But in order to prove the twin prime conjecture, you need to get the density of primes inside the almost primes up to up to a threshold of 50%. Um, once you get up to 50%, you would 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 yeah, 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, but the Goldbach conjecture, and many other problems in number theory are currently blocked because our current techniques would require improve going beyond this theoretical, um, parity barrier. It's it's like it's like going faster than the speed of light.

Yeah, so we should say the 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, in the sense of metaphorical 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 of of colleague might put it is that, um, you know, like, if if you are on the streets of New York and you put in a blindfold and you're put in a car, and and, uh, after some hours, um, you the blindfold's off and and you're in Beijing.

Um, you know, I mean, that was too easy somehow, like, uh, like, There was no ocean being crossed. Even if you don't know exactly what, how, what, what was done, uh, you're suspecting that there's something, what 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 now, t time, 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 or I don't wanna 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, I once again, uh, nothing happened. I will I will move on. Um, yeah, very occasionally, one of these problems I actually solved, uh, or sometimes, as you say, you think you solved it, and then you're you're for it for, uh, 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 will be, uh, solved? Twin prime and Goldbach

Prime, I think we'll keep

Ten years?

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. It may not have the whole thing. Um, yeah, so twin primes is somewhat close. Yeah. Riemann hypothesis, uh, I have no, I mean, uh, it has to happen by accident, I think, um,

So the the Riemann hypothesis is a kind of a more general conjecture about the distribution of prime numbers, right?

right, yeah, it's it's it's states that sort of viewed multiplicatively, like, for for questions only involving multiplication, no addition, the primes really do behave, uh, as randomly as as you could hope. So there's a phenomenon in probably called square root cancellation that, um, you know, like, if you want to poll, say, America, uh, up on s 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 a full average.

But if you sample more and more people, the accuracy gets better and better, and it the accuracy improves like the square root of the number of of people you, uh, you sample. So, yeah, if you sample, um, a thousand people, you can get like a two, 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 y you can measure, and it's it's called the Riemann zeta function, and it fluctuates up and down.

But, uh, in some sense, um, as you keep averaging more and more, if you sample more and more, the fluctuations 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 areas 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 a d actually random set, this, this, this square root cancellation.

And we know, uh, you know, because of things related to the parity problem, actually, that that most of us 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, as I said, you you can modify the primes a little bit, and you can destroy the Riemann Riemann hypothesis.

Um, so, like, it has to be very delicate. Um, you can't apply something that has huge margins of error. It has to, it just barely work. Um, and like, um, there's like all these pits, pitfalls that you have to like, dodge very adeptly.

The prime numbers is just fascinating To me. What what to you is, um, most mysterious about the prime numbers?

That's a that's a good question. So 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 these sort of 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 it 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 numerics. It tells us what to predict. Like, I can tell you with complete certainty, the twin prime conjecture is true. Uh, the random model gives overwhelming odds that it's 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 ran, kind of random, because there's so no reason for them to be, um, uh, to have any kind of secret pattern. But yeah, what is mysterious is what is the mechanism that really f- forces the randomness to happen. Uh, and this is just absent.

Another incredibly surprisingly difficult problem is the Collatz conjecture. 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, uh, Paul Erdos 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? It's easy to

So it's it's a it's a problem that you can explain, um, you know, it, 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 it by three and add one. So even numbers get smaller, odd numbers get bigger. So 13, uh, would become 40, because 13 times three 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 together, it's still reasonably simple. Um, but then you you ask what happens when you iterate it? You you take the output that you just got and feed it back in. So 13 becomes 40, 40 is now even, divide by two is 20, 20 is still even, divide by two, 10, 5, and then 5 times three plus one 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 the sequence I just described, um, you know, 13, 40, 20, 10 and so on, these are also called Hailstone sequences, because there's an oversimplified model of of hailstone formation, you know, which is not actually quite correct, but it's sort of somehow taught to high school students as a first approximation. Is that, um, like, a a little nugget of ice gets gets, uh, a nice crystal, forms in in cloud, and it goes up and down because of the wind, uh, and and sometimes when it's cold, it, it acquires a bit a bit more mass, and maybe it melts a little bit.

And this process of going up and down creates this sort of partially melted ice, which eventually becomes hailstone, and eventually falls down to 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, there's this process that that goes up if you're odd and and down if you're even, it eventually c um, goes down to 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 And come down.

It's 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 fact, usually that's what happens, uh, that that if you plug in a random number, you can actually prove, at least initially, that it would look like, um, a random walk. Um, and it's actually a random walk with, um, a downward drift.

Um, it's like if you're always gambling, um, on a roulette at the 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 That it would do Yeah, so It.

So the result that I I proved, roughly speaking, is that that statistically, like, 99% of all inputs would would drift down to, no, maybe not all the way to one, but to be much, much smaller than what you started. So, i it's it's like if I told you that if you go to a casino, most of the time you will end up if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started.

Uh, that's kind of like the was the result that I proved.

So why is that result? Like, can you continue down that thread to prove the full conjecture?

Well, the 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 these 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, well, almost surely, you know, with probability 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 in 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, a special number that you you 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, uh, Alex Kontorovich, for instance, who proposed that, um, that actually, um, these collatz, uh, iterations are like these cellular automata, um, that y- y- actually, if you look at what they happen on 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.

Now, a number which is actually encoding this machine, which is just whose job it is is to encode, is to create, um, a version of itself which is 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. So, Conway, um, so similar, in fact, uh, that was one of my inspirations for the Navi- Navier-Stokes project. Uh, Conway studied generalizations of the Collatz problem, where instead of multiplying by three and adding one or dividing by two, you you have a more complicated branch of rules. 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 a factrat as a play on on fractat, uh, and he showed that that you you can, um, you you can program, it was Turing complete. You could you could you could, uh, um, you could make a program that if if your number you inserted 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, uh, s having a fra- mathematical framework to say anything about cellular automata, maybe the same kind of framework is required For Collatz conjecture.

Yeah, if you want to do it not statistically, but you really want one hun- 100% all inputs to to for the earth. Yeah, so what might be feasible is is, yeah, sifting 99%, you know, going to Go to one, but the like everything, yeah, uh, that looks hard.

What would you say is out of these within reach, famous problems, is the hardest problem we have today? Is the Riemann hypothesis?

Well, it's up there. Um, P equals NP is is a good one because, like, uh, that's a 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 equals NP algorithm, then 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'd be a big mental shock to 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 have built over many, many years of what operations involving primes behave randomly and what ones don't. Um, and in particular, um, 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, but we believe to be almost impossible to crack, um, at least, uh, mathematically. Um, but, uh, if something as core to our beliefs as the Riemann hypothesis is is wrong, it means that there are there are actual patterns to the primes that we 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. Um,

But then how do you then say stuff about the the primes? Now you're going towards the uh, Collatz con- conjecture again. Um, because if I I I do, you do. You want it to be random, right? You want it to be random. Like,

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's there's one where it is technically possible, but in practice never actually implementable. The evidence is sort of slightly pushing in favor of no, that we probably P is not equal to NP.

I mean, it seems like it's one of those cases, similar similar to Riemann hypothesis, it. I think the evidence is le- leaning pretty heavily on the no.

Certainly more on the no than on the on the yes. The funny thing about P is that we have also a lot more obstructions than we do for almost any other problem. Um, so while there's evidence, uh, 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 science has actually been very good at. It's actually saying 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 gonna do now that you've won this prestigious award? And then you just quickly, very humbly said that, you know, this a shiny medal is not gonna solve any of the problems I'm currently working on. So I'm just, I'm gonna keep, I'm gonna 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, um, maybe you could speak to the Fields Medal, but it's another way for me to ask, uh, about, uh, Grigori Perelman. What do you think about him? Famously declining the Fields Medal and the Millennium Prize, which came with 1 million dollar of 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. Um, I've never met him. Uh, I think I'd be interested to meet him one day, but I, I never had the chance. I, I know people who've met him, but he, 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 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 s or something. Um, I, 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 and 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 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, 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 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 know, there's, uh, they just career mathematicians.

You know, you just focus on publishing the next paper, maybe getting one just to promote to 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, uh, you know, things that you might just have foolishly say, because you know, no one's gonna listen to you. Uh, it's, it's, so 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, uh, 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. C. Obviously, I have the option to sort of, uh, decline, uh, so I decline a lot of things. I I dec- I could decline even more, um, or I could acquire a reputation of being so unreliable that people don't even ask anymore. Uh, I this is this,

love, I love the different algorithms here. This is great.

uh, this is it's. It's always an option. Um, but, you know, um, there are things that are, you know, like, I, I mean, sort of, 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 uh, yeah, as you advance in your career, it's sort of the more soft skills.

So math somehow front loads all the technical skills to the early stages of your career. So, um, yeah, so it's, uh, as a postdoc, you publish or perish, you, you, you're incenti- you're incentivized to basically focus on on proving very technical theorems to sort of prove yourself, um, as well as prove the theorems. Um, but then as as you get more senior, you have to start, you know, mentoring and and giving interviews, uh, and uh, and trying to shape, um, direction of the field, both research wise.

And and you know, uh, sometimes you have to, uh, t, uh, you know, do various administrative things. And it's kind of the the right social contract, because you, you need to to work in the trenches to see what can help mathematicians.

The 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. And it's like, this is how the human mind works. This is where I would probably, uh, say that I like the Fields Medal. Th- that it does inspire a lot of p young people. Somehow, I don't. This is just how human brains work.

At the same time, I also wanna give sort of respect to somebody like Grigori Perelman, who is critical of awards in his mind. Those are his principles, and any human is 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 necessary and 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, but yeah, so, like, again, you see these people try to only solve like the really big math problems and and not work on on on things that are less, uh, sexy, if you wish, but but. But actually, uh, still interesting and and instructive, as you say, like, uh, the way the human mind works, i 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 so,

uh, there's, there's, the way our humans, um, mind is, is wired. We can comprehend the relationships between, you know, ten or twenty people, you know, but but once you get beyond, like, uh, a hundred people, like, there's, 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, uh, so we have. You have to simplify the whole mass of, you know, ninety, nine, nine of humanity becomes the other.

Um, and, uh, and often these models are, 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 are representative people of the subject, you know, I said, role models, uh, 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, I'll be the first to say that my own career tra- path is not that of a typical mathematician.

Um, I the very accelerated, uh, education, I skipped a lot of classes. Um, I think I was had very fortunate mentoring, uh, opportunities, um, and I think I was at the right place at the right time. Just because someone does doesn't have my um, trajectory, it doesn't mean that they can't be good mathematicians. I mean, they may be better mathematician in in a very different style, uh, and we need people with, with different styles. Um, and you know, even if a, 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 on lots of previous work. Um, but that's a s a story that's difficult to tell, um, if you're not an expert, cause, you know, it's. It's easy 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 is Steve Jobs yeah, yeah, as as pi.

a starting point, you say, you know, as a fir, as a first approximation, that's how, that's how you and then read would.

some biographies and then look into much deeper first approximation. That's right. Uh, so you mentioned you were a Princeton to, um, Andrew Wiles at that time was 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's last theorem. What did you think, maybe looking back now, with more context about that moment in, uh, math history?

yeah. 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, uh, we all had pigeonholes in the same mailroom, you know, so we would all pick up mail. And and like, suddenly, Andrew Wiles's mailbox exploded to be overflowing. That's a good, that's Um,

a good metric.

yeah, um, you know, so, yeah, we all, we all talked about it at at tea and so forth. I mean, uh, we, we didn't understand, most of us sort of understand the proof. Um, we understand sort of high level details. Um, in fact, there's an ongoing project to formalize it in Lean, right? Kevin Buzzard is actually.

Yeah, can can we take that small tangent? Is it? Is it? How difficult is that? Cause, as as I understand, the Fermat's last, the the proof for, uh, Fermat's last theorem has, like, super complicated objects, that's really difficult yeah, the to formalize, no,

yeah, I guess, yeah, you're right, the 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, if that's really not trivial, but it's been done. Uh. But there's a lot of really basic facts about, um, these objects that have taken decades to prove and that they're they're in all these different math papers. And so a lot of lots of these have to be formalized as well.

Um, Kevin's, uh, Kevin 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 or some other work will have to be done to f 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 and functions and and things that are within scope of a high school, um, uh, 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 on structures for for quite a while.

Um, and it's it's a very sturdy structure. There's, it's, it's, it's been, it's been very um, the the the at the base, at least, it's extremely well developed, and there's textbooks and so forth. But um, um, it does get to the point where, um, if you're, if you haven't taken these years of study and you, you wanna ask about what, what is going on at, um, like, level six of of this tower, uh, 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 was similar as we talked about seven years, mostly s working in secret.

Yeah, uh, yes, that is a a romantic, uh, yeah. So it kind of fits with the sort of the the romantic image that I think people have of mathematicians to the extent that they think of them at at all as these kind of eccentric, uh, you know, wizards or something. Um, so that that certainly kind of, uh uh, accentuated that perspective. You know, I mean, it's, it is a great achievement. Yeah, his style of solving problems is so different from my own.

Um, but but which is great. I mean, we we need people Can you speak like that.

to it? Like, what, uh, in in in terms of like, uh, you like the collaborative

I like moving on from a problem if it's giving too much difficulty. Um, but you need the people who have the tenacity and the the fearlessness. Um, you know, I, I've, I've collaborated with with people like that, where where I want to give up, uh, cause the the first approach that we tried didn't work, and the second one didn't approach. But they're convinced, and they have third, fourth and the fifth approach works. Um, and I have to eat my words.

Okay, I didn't think this was gonna work, but uh, yes, you were right all along.

And w- 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, but there are also people who are very productive and they 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, 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, uh, or usually too late. So they 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, that they, they don't like it. Um, my theory is that, um, humans don't come evolution has not given us a math center of our brain directly, like we have a vision center and a language center and some other centers, um, which have e- evolution has honed.

But we it doesn't. We don't have an innate sense of mathematics, um, but our other centers are sophisticated enough that different people, uh, uh, uh, 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 repurpose their their language center, and they think very symbolically. Um, you know, um, some people like, if if they are very competitive and they they like gaming, uh, there there's a type.

There's this part of your brain that's very good at at at, uh, uh, at solving puzzles and games, and and that can be repurposed. But, uh, like, when I talk to other mathematicians, you know, they don't quite think they. I can tell that they're using some of different styles of of thinking than I am. I mean, not not disjoint, but they they may prefer visual. I'm I, I. I don't actually prefer visual so much.

I 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 s set of subsystems being used in the thinking Yeah, like, like process.

they, 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 s, yeah, but I mean, the way we educate, and unless you have, like, a personalized tutor or something, I mean, education, sort of just by nature of 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 in in this complicated educational context? What? What would yeah, it's a you advise?

tricky problem. One nice thing is that it there are now lots of sources for mathematical enrichment outside the classroom. Um, so, in in in my day, there already there were math competitions, um, and you know, there were also, like, popular math books in the library. Um, you know, but but now you have, you know, YouTube, uh, you there. There are forums just devoted to solving, you know, math puzzles, and, um, and math shows up in in other places, you know, like, um, for example, there there are hobbyists who play poker, uh, for fun, uh, and um, they they, you know, they, for very specific reasons, are are interested in very specific probability questions.

Um, and and, uh, they they actually, you know, there's a community of amateur probabilists 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, uh, with these new sort of tools of for lean and so forth, that actually we can incorporate the a broader public into math research projects. Um, like this is o- almost is, uh, is doesn't happen at all currently.

So in the sciences, there's some scope for citizen science, like astronomers, uh, the amateurs who discover comets, and there's biologists, there're people who who identify butterflies and so forth. Um, and in math, in um, there are a small number of activities 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 probably it would just be be time consuming, cause 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've contributed to to Mathlib. Um, you know, you don't need to be a PhD holder to just work on s one atomic thing.

There's something about the formalization here that also, uh, at at at the as a very first step, opens it up to the programming community too. The people who are already comfortable with program, 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, you can print hello world pretty quickly. Um, you know, like, if, uh, if programming was taught as an almost entirely theoretical subject, where you just taught the the computer science, the the theory of functions and and and, uh, and and routines and so forth. And and outside of some some very specialized homework assignments, you're not actually programming, like on the weekend for fun, or, yeah, that would be as considered as hard as math.

Um, yeah, so, as I said, you know, uh, there are s, s 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, uh, their 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 Yeah, that's they can be good at?

a 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, 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 a to a good career. You go to college, you get an education, you pick one profession and and you, you stick to it.

It's 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, mathematics or something. It 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 is. So, anyway, these are things which I think we will still need, even as our tools get get better.

And you know, you'll, you'll be working with AI support and so forth.

But actually, you're an interesting case study. I mean, you're like, 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, 1st of all, you kept learning new fields, but you learned lean. That's not that's a non trivial thing to learn. Like, that's a that's a for for a lot of people, that's an extremely uncomfortable leap to take, right? yeah, so A lot of mathematicians,

um, 1st of all, I've. 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, like, I, I, I spend, like me and 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 on?

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, uh, formalized one of his his, um, new theorems, and um, there were some other developments in computer assisted proof that looked quite interesting. And I said, Oh, we should, we should, uh, um, we should run a workshop on this.

This would be a good idea. Um, and then I, I was a bit too enthusiastic about this idea, and so I, I got voluntold to actually run it. 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 an, uh, a a a nice success. We brought together a bunch of mathematicians and computer scientists and other people, and and we got up to speed on state of the art.

Um, and it was really interesting, um, to about it that, but most mathematicians didn't know what 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 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 I've managed to run this conference. And 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, the um, the interaction between AI and and formal proof assistants. And I said, Yeah, they, they should be combined. This, this is, this is, um, this perfect synergy to 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 say, you know, I, I, I'm a, I'm a well known mathematician. Just trust me, you know, when I say that this is going to change mathematics. And I'm not doing it any, and I don't do any of it myself. So I felt like I had to actually, uh uh, justify it. Yeah, there's a lot of what I get into actually. Uh, I don't quite see an advice as how much time I'm gonna spend on it.

And it's only after I'm sort of waist deep in in a project that I I realize, but 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? It's in new new concepts, new ways of thinking, also, you know, sucking at a thing that others. I think, I think in that talk, you know, you could be a fields, medal, medal winning mathematician, and an 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 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, it, it does keep us honest. I mean, not, not, I mean, you can still, uh, it's not a perfect, uh, panacea, but I think, uh, 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?

Uh, so, 1st of all, as I, as I mentioned before, like, there's, uh, there's some time dependence, but O on the day, yeah, uh, like, like, if, if you, if you, if you plot cumulatively over time, for example, Euclid, like, like, sort of like, is, is, is one of the contenders. Um, and then maybe some unnamed, un 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 directly, of everything that's happened in the 20th century?

yeah, yeah. Hilbert spaces, uh, have lots of things that are named after him, uh, 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 which problems are hard to solve, the statement of the open problems.

yeah, it, I mean, you know, this is bystander effect in in everywhere, right? Like, if, if no one says you should do x, everyone just sort of mills around, waiting for someone else to to, uh, to do something. And and like, nothing gets done. Um, so, and and like it, like, it's the 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. I, oh, um, was, 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 the stupider the better.

Um, which, you know, I want a technique which is almost guaranteed to fail, but the way it fails is gonna be instructive. Um, like it, it fails because you're, you're. 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 as started using, and it really works. I think you said it's called structured procrastination. It's when you really don't wanna do a thing, but you imagine a thing you don't wanna do more, that's worse than that, and then in that way, you procrastinate by not doing the thing that's worse. That's a nice, it's a nice hack, it actually works.

yeah, yeah, there's, um, I mean, w with anything like, you know, I mean, like, if, um, psychology is really important. Like you you talk to athletes like marathon runners and so forth, and you know, you know, they talk about what's the most important thing is it, their training regimen or their diet? And so forth. Actually, so much of it is actually psychology. Um, you know, J. Just tricking yourself to to think that the problem is feasible, um, so that you can get motivated to do it.

is there something our human mind will never be able to comprehend?

Hm, well, I sort of, I guess, a mathematician, I mean, you know, it's just by induction, right? A. A. And it's originally, there must be some. It's originally a large number that you can't understand. That was the 1st thing that came to mind.

So that, but even broadly, is there are we lim? Is there something about our mind that's we're going to be limited, even with the help of mathematics,

Well, okay, I mean, it's 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 the English language, is a technology. Uh, it's, uh, it's one that's been very internalized.

So you're right, they're 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? So we're already like a collective intelligence.

yes, yeah, I guess, uh. So, so, humanity, plural, has much more intelligence in principle, on its good days, than than than the individual humans put together. Uh, it can all have less. Okay, but, uh, um, yeah, so yeah, math, math, the mathematical community, plural, is is is incredibly super intelligent, uh, entity, um, that, uh, no single human mathematician can can come close to to to replicating. You see it a little bit on these like question analysis sites, um, uh, 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, it's, it's. It's a pleasure to watch, actually, as as an, as an expert.

I'm a fan spectator of that, uh, of that site, just seeing the brilliance of the different people there. Um, the depth of knowledge that some 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, the um, the younger generation is always like, like, really creative and enthusiastic and and inventive. Um, I, I. It's a pleasure working 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, it's, and I mean, like it was like navigation, you know, just just knowing where you were on on the planet was this horrendous problem.

People people died, um, you know, uh, and or or lost fortunes because they couldn't navigate, you know. And we have devices in our pockets that do this automatically for us, like, it's 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, 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, because in the next hundred 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 your teenage self or something, you know, I mean, you know, um, just the internet and and and our AI, I mean, that, uh, again, they've, they've been int, they're beginning to internalize. And say, yeah, of course, uh. An AI can understand our voice and and give reasonable, you know, slightly incorrect answers to to any question.

But, you know, I this was mind blowing even two years ago.

And in the moment, it's hilarious to watch on the internet and so on. The the drama, uh, people take everything for granted very quickly, and then they we humans seem to entertain ourselves with drama. Well, 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 R progress of robotics, 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 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 fan of. There's a lot of really smart people there. And thank you, uh, 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, uh, 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, today.

it was a pleasure.

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