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June 14, 2025 - Lex Fridman Podcast

03:14:23

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

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	The following is a conversation with Terence Tau, 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 It means the world.
	This is the Lex Friedman Podcast.
	To support it, please check out our sponsors in the description or at lexfriedman.com slash sponsors.
	And now, dear friends, here's Terrence Tao.
	What was the first really difficult research-level math problem that you encountered?
	One that gave you pause, maybe.
	I mean, in your undergraduate education, you learn about the really hard and possible problems like the Riemann hypothesis, the Trin-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 on the boundary between what we can do easily and what are hopeless.
	But what are problems where existing techniques can do 90% of the job and then you just need that remaining 10%?
	I think as a PhD student, the Kikeya problem certainly caught my eye, and it just got solved, actually.
	It's a problem I've worked on a lot in my early research.
	Historically, it came from a little puzzle by the Japanese mathematician Soji Kikeya in 1918 or so.
	The puzzle is that you have a needle on the plane.
	Think of it like driving on a road.
	You want to execute a U-turn.
	You want to turn the needle around.
	But you want to do it in as little space as possible.
	So you want to use this little area in order to turn it around.
	But the needle is infinitely maneuverable.
	So you can imagine just spinning it around as the unit needle.
	You can spin it around its center.
	And I think that gives you a disk of area, I think, pi over 4. Or you can do a three-point U-turn, which is what we teach people in their driving schools to do.
	And that actually takes area pi over 8. So it's a little bit more efficient.
	Then a rotation.
	And so for a while, people thought that was the most efficient way to turn things around.
	But Vesikovic showed that, in fact, you could actually turn the needle around using as little areas as you wanted.
	So 0.001, there was some really fancy multi back-and-forth U-turn thing that you could do.
	That you could turn the needle around.
	And in so doing, it would pass through every intermediate direction.
	Is this in the two-dimensional plane?
	This is in the two-dimensional plane.
	So we understand everything in two dimensions.
	So the next question is what happens in three dimensions?
	So suppose the Hubble Space Telescope is tube in space, and you want to observe every single star in the universe.
	So you want to rotate the telescope to reach every single direction.
	And here's the unrealistic part.
	Suppose that space is at a premium, which it totally is not.
	You want to occupy as little volume as possible.
	In order to rotate your needle around in order to see every single star in the sky, how small a volume do you need to do that?
	And so you can modify Besokovic'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 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 how fast does it go down?
	And the conjecture was that it goes down very, very slowly, like logarithically, roughly speaking.
	And that was proved after a lot of work.
	So this seems like a puzzle-wise and 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 wave propagation, you splash some water around, you create water waves and they travel in various directions.
	But waves exhibit both particle and wave type behavior.
	So you can have what's called a wave packet, which is like a very localized wave that is localized in space and moving a certain direction in time.
	And so if you plot it into 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 focuses at a single point later in time.
	You can imagine dropping a pebble into a pond and ripples spread out.
	But then if you time-reverse that scenario, and the equations of wave motion are time-reversible, you can imagine ripples that are converging to a single point and then a big splash occurs, maybe even a singularity.
	And so it's possible to do that.
	And geometrically, what's going on is that there's always light rays.
	So if this wave represents light, for example, you can imagine this wave as a superposition of photons all traveling at the speed of light.
	They all travel on these light rays, and they're all focusing at this one point.
	So you can have a very dispersed wave focus into a very concentrated wave at one point in space and time, but then it defocuses again and it separates.
	But potentially, if...
	So what that meant is that there's a very efficient way to pack tubes pointing in different directions into a very, very narrow region of a very narrow volume.
	Then you would also be able to create waves that start out, there'll be some arrangement of waves that start out very, very dispersed, but they would concentrate not just at a single point, but there'll be a lot of concentrations in space and time.
	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.
	And so in mathematical physics, we care a lot about whether certain equations and wave equations are stable or not, whether they can create these singularities.
	There's a famous unsolved problem called the Navier-Stokes regularity problem.
	So the Navier-Stokes equations are equations that govern the fluid flow of incompressible fluids like water.
	The question asks, if you start with a smooth velocity field of water, can it ever concentrate so much that the velocity becomes infinite at some point?
	That's called a singularity.
	We don't see that in real life.
	If you splash around water in the bathtub, it won't explode on you.
	Or have water leaving at a speed of light.
	But potentially, it is possible.
	In fact, in recent years, the consensus has drifted towards the The belief that, in fact, for certain very special initial configurations of, say, water, that singularities can form.
	But people have not yet been able to actually establish this.
	The Clay Foundation has these seven-millennium prize problems, has a million-dollar prize for solving one of these problems.
	This is one of them.
	Of these seven, only one of them has been solved.
	At the point, great conjecture.
	So, the Kakaia conjecture is not directly, directly related to the Navistokes 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, millennial price problem.
	Right.
	You've made a lot of progress on this one.
	In 2016, you published a paper, Finite Time Blowup, for an averaged three-dimensional Navier-Stokes equation.
	Right.
	So we're trying to figure out if this thing usually doesn't blow up.
	Right.
	But can we say for sure it never blows up?
	Right, yeah.
	So yeah, that is literally the million-dollar question.
	So this is what distinguishes mathematicians from pretty much everybody else.
	If something holds 99.99% of the time, that's good enough for most things.
	But mathematicians are one of the few people who really care about whether 100%, really 100% of all situations are covered by...
	So most fluid, most of the time, 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 set of equations that govern in the field of fluid dynamics, trying to understand how fluid behaves, and it actually turns out to be a really complicated...
	Extremely complicated thing to try to model.
	Yeah, so it has practical importance.
	So this clay price 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 has a lot of computational fluid dynamics.
	A lot of it is actually just trying to solve the Navier-Stokes equations as best they can.
	Also gathering a lot of data so that they can initialize the equation.
	There's a lot of moving parts.
	So it's a very important problem practically.
	Difficult to prove general things about the set of equations like it not blowing up.
	The short answer is Maxwell's Daemon.
	Maxwell's Daemon is a concept in thermodynamics.
	If you have a box of two gases, oxygen and nitrogen, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them, then they will mix.
	And they should stay mixed.
	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.
	Maybe there's a microscopic demon called Maxwell's demon.
	Every time an oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen drifts onto one side and the nitrogen goes to the other.
	You could have an extremely improbable configuration emerge, which we never see.
	Statistically, it's extremely unlikely.
	But mathematically, it's possible that this can happen, and we can't rule it out.
	And this is a situation that shows up a lot in mathematics.
	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 1s and 2s and 3s as 4s and 5s and 6s.
	There should be no preference in the digits of pi to favor, let's say, 7 over 8. But maybe there is some demon in the digits of pi that every time you compute more and more digits, it's of biases.
	One digit to another.
	This is a conspiracy that should not happen.
	There's no reason it should happen, but there's no way to prove it with our current technology.
	Getting back to Navier-Stokes, a fluid has a certain amount of energy.
	Because the fluid is in motion, the energy gets transported around.
	And water is also viscous.
	So if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and it will go to zero.
	And this is what happens when we actually experiment with water.
	You splash around, there's some turbulence and waves and so forth, but eventually it settles down and the lower the amplitude, the smaller the velocity.
	The more calm it gets.
	But potentially, there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale.
	it will move faster and faster, and at faster speeds the effect of viscosity is relatively less.
	It could happen that it creates some sort of self-similar blob scenario where the energy of the fluid starts off at some large scale and then it all transfers the energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth.
	You could actually converge to all the energy concentrating in one point in a finite amount of time.
	In practice, this doesn't happen.
	Water is what's called turbulent.
	It is true that if you have a big eddy of water, it will tend to break up into smaller eddies, but it won't transfer all the 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 small eddies of their own.
	And so the energy gets dispersed to the point where the viscosity can then keep everything under control.
	But if it can somehow concentrate all the energy Keep it all together and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blow-up can occur.
	So there were papers who had 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 Navier-Stokes, but for many, many types of equations like this.
	And so in the past, there have been many attempts to try to obtain what's called global regularity for Navier-Stokes, which is the opposite of final time blow-up, that velocity stays smooth.
	And it all failed.
	There was always some sign error or some subtle mistake, and it couldn't be salvaged.
	So what I was interested in doing was trying to explain why we were not able to disprove Valentine's Blower.
	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 the Navier Sox, basically, if I could turn off certain types of ways in which water interacts and only keep the ones that I want.
	So in particular, if there's a fluid and it could transfer its 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 one and direct it only into this smaller eddy while still preserving the law of conservation of energy.
	So you're trying to make a blow-up?
	Yeah.
	So I basically engineer a blow-up by changing the laws of physics, which is one thing that mathematicians are allowed to do.
	We can change the equation.
	How does that help you get closer to the proof of something?
	Right.
	So it provides what's called an obstruction in mathematics.
	So what I did was that basically if I turned off certain parts of the equation, 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 interactions, I could force all the energy to blow up in finite time.
	What that means is that if you wanted to prove global regularity for Navier-Stokes for the actual equation, you must use some feature of the true equation which my artificial equation does not satisfy.
	It rules out certain approaches.
	The thing about math is it's not just about taking a technique that is going to work and applying it, but you need to not take the techniques that don't work.
	And for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem, but 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 kind of rules out.
	It saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work.
	Is it that specific problem of fluid dynamics, or is it some more general intuition you build up about mathematics?
	Right, yeah.
	So the key phenomenon that 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 force coming from viscosity, and it's very well understood.
	It's linear.
	It calms things down.
	If viscosity was all there was, then But there's also transport, that energy in one location of space can get transported because the fluid is in motion to other locations.
	And that's a nonlinear effect, and that causes all the problems.
	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 large, then basically you get regularity, and if the transport term dominates, then Then we don't know what's going on.
	It's a very nonlinear situation.
	It's unpredictable.
	It's turbulent.
	Sometimes these forces are in balance at small scales, but not in balance at large scales, or vice versa.
	Navier-Stokes is what's called supercritical.
	At smaller and smaller scales, the transport terms are much stronger than the viscosity terms.
	The viscosity terms are things that calm things down.
	This is why the problem is hard.
	In two dimensions, so the Soviet mathematician Ladishinskaya, she, in the 60s, showed in two dimensions there was no blow-up.
	And in two dimensions, the Navi-Socos equations 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 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 blow-up examples.
	Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen.
	So this is sort of one of the main insights of this line of work, is that supercriticality versus criticality and subcriticality, this makes a big difference.
	I mean, that's a key qualitative feature that distinguishes.
	Some equations were being nice and predictable, like planetary motion.
	I mean, there are certain equations that you can predict for millions of years, or thousands at least.
	Again, it's not really a problem, but 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 very fine scales.
	So whenever there is some huge source of non-linearity, that can create a huge problem for predicting what's going to happen.
	Yeah.
	And if non-linearity is somehow more and more featured and interesting at small scales.
	I mean, there's many equations that are non-linear, but in many equations, you can approximate things by the bulk.
	So, for example, planetary motion, if you wanted to understand the orbit of the Moon or Mars or something, you don't really need the microstructure of the seismology of the Moon or exactly how the mass is distributed.
	You can almost approximate these planets by point masses.
	And just the aggregate behavior is important.
	But if you want to model a fluid, 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 really important.
	if we can just linger on the Navier-Stokes equations a little bit.
	So you've suggested, maybe you can describe it, that one of the ways to solve it, or to negatively resolve it, would be to...
	And then show that the halting problem from computation theory has consequences for fluid dynamics.
	So show it in that way.
	Can you describe this idea?
	Right, yeah.
	So this came out of this work of constructing this average equation that blew up.
	So as part of how I had to do this, so there's still this naive way to do it.
	You just keep pushing...
	This is the naive way to force blow up.
	In five and higher dimensions, this works.
	But in three dimensions, there was this funny phenomenon that I discovered that if you change the laws of physics, you just always keep trying to push the energy into smaller and smaller scales.
	What happens is that the energy starts getting spread out into many scales at once.
	So you have energy at one scale, you're pushing it into the next scale, and then 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.
	You're trying to do everything at once.
	And this spreads out the energy too much.
	And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything.
	So it turns out this directive motion doesn't actually work.
	There was a separate paper by some other authors that actually showed this.
	In three dimensions.
	So what I needed was to program a delay, so kind of like airlocks.
	I needed an equation which would start with a fluid doing something at one scale.
	It would push its energy into the next scale, but it would stay there until all the energy from the larger scale got transferred.
	And only after You pushed all the energy in, then you open the next gate and then you push that in as well.
	By doing that, the energy inches forward scale by scale in such a way that it's always localized at one scale at a time.
	Then it can resist the effects of viscosity because it's not dispersed.
	So in order to make that happen, I had to construct a rather complicated nonlinearity.
	And it was basically like...
	I actually thank my wife for this because she was trained as an electrical engineer.
	She talked about how she had to design circuits and so forth.
	If you want a circuit that does a certain thing, like maybe have a light that flashes on and then turns off and then on and then off, you can build it from more primitive components, capacitors and resistors and so forth.
	You have to build a diagram.
	And these diagrams, you can sort of follow with your eyeballs and say, oh yeah, the current will build up here and then it will stop and then it will do that.
	So I knew how to build the analog of basic electronic components, like resistors and capacitors and so forth.
	And I would stack them together in such a way that I would create something that would open one gate and then there would be a clock.
	And then once the clock hits this in threshold, it would close it.
	It would become 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 you can imagine kind of a steampunk, but it's really waterpunk type of thing.
	So modern computers are electronic.
	They're powered by electrons passing through very tiny wires and interacting with other electrons and so forth.
	But instead of electrons, you can imagine these pulses of water moving at a certain velocity.
	Maybe there are two different configurations corresponding to a bit being up or down.
	Probably if you had two of these moving bodies of water collide, they would come out with some new configuration which would be something like an AND gate or OR gate.
	The output would depend in a very predictable way on the inputs.
	You could chain these together and maybe create a Turing machine and then you have computers which are made completely out of water.
	And if you have computers, then maybe you can do robotics, hydraulics and so forth.
	And so you could create some machine, which is basically a fluid analog, what's called a von Neumann machine.
	So von Neumann proposed, if you want to 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 more materials, smelt them, and build More copies of the same machine.
	Then you could colonize the whole planet over time.
	If you could build a fluid machine, it's a fluid robot, and what it would do, its purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state.
	It wouldn't start just yet.
	Once it's ready, the big robot, the water, would transfer all its energy into the smaller configuration and then power down.
	And then I clean myself up.
	And then what's left is this newer state, which would then turn on and do the same thing, but smaller and faster.
	And then the equation has a certain scaling symmetry.
	Once you do that, it can just keep iterating.
	So this, in principle, would create a blur 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 a pipe dream because there are so many things that are missing.
	For this to actually be a reality.
	So I can't create these basic logic gates.
	I don't have these special configurations of water.
	I mean, there's candidates that include vortex rings that might possibly work.
	But also, you know, analog computing is really nasty compared to digital computing because there's always errors.
	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 running of the smaller machine.
	But everything in principle can happen.
	It doesn't contradict any of the laws of physics.
	So it's sort of evidence that this thing is possible.
	There are other groups who are now pursuing ways to make Navisrox blow up, which are nowhere near as ridiculously complicated as this.
	They actually are pursuing much closer to the direct self-similar model, which can, it doesn't quite work as is, but there could be some simpler scheme than what I just described to make this work.
	There is a real leap of genius here to go from 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.
	Torian machine gets smaller, smaller, smaller, and somehow seeing how that could be used to say something about a blow-up.
	I mean, that's a big leap.
	So there's precedent.
	I mean, so the thing about mathematics is that it's really good at spotting connections between what you might think of as completely different problems.
	But if the mathematical form is the same, you can draw a connection.
	There's a lot of work previously on what's called cellular automator, 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.
	There's a very simple rule that tells you how these cells evolve.
	Sometimes cells live, and sometimes they die.
	A student who was a very popular screensaver to actually just have these animations going on.
	And they look very chaotic.
	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.
	So, for example, they discovered this thing called a glider.
	So a glider is a very tiny configuration of like four or five cells, which evolves and it just moves in a certain direction.
	And that's like this vortex rings.
	So this is an analogy.
	The game of life is kind of like a discrete equation.
	The fluid Navi-Sokes is a continuous equation, but mathematically, they have some similar features.
	Over time, people discovered more and more interesting things that you could build within the Game of Life.
	The Game of Life is a very simple system.
	It only has three or four rules to do it, but you can design all kinds of interesting configurations inside it.
	There's something called a glider gun that does nothing but spit out gliders one at a time.
	After a lot of effort, people Managed to create AND gates and OR gates for gliders.
	There's this massive ridiculous structure, which if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce a stream of gliders coming out.
	Maybe if both of the streams have gliders, then there'll be an 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 These basic gates, then just from software engineering, you can build almost anything.
	You can build a Turing machine.
	I mean, it's like an enormous steampunk type thing.
	They look ridiculous.
	But then people also generated self-replicating objects in the game of life.
	A massive machine, a phenomenal machine, which over a huge period of time, and it always looked like glider guns inside doing these very steampunk calculations, it would create another version of itself.
	That's so incredible.
	A lot of this was community crowdsourced by amateur mathematicians, actually.
	I knew about that work, and that is part of what inspired me to propose the same thing with Navier Stokes.
	As I said, analog is much worse than digital.
	You can't just directly take the constructions from the game of life and plunk them in.
	Again, it shows it's possible.
	Emergence that happens with these cellular automata, local rules, maybe it's similar to fluids, I don't know, but local rules operating at scale can create these incredibly complex dynamic structures.
	Do you think any of that is amenable to mathematical analysis?
	Do we have the tools to say something profound about that?
	The thing is, you can get these emergent, very complicated structures, but only with very carefully prepared initial conditions.
	These glider guns and gates and software machines, if you just plunk down randomly some cells, you will not see any of these.
	That's the analogous situation with Navier-Stokes again.
	With typical initial conditions, you will not have any of this weird computation going on.
	Basically, through engineering, Especially designing things in a very special way, you can pick clever constructions.
	I wonder if it's possible to prove the negative of basically prove that only through engineering can you ever create something interesting.
	This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness.
	That most objects that you can generate in mathematics are random.
	They look like the digits of pi.
	Well, we believe is a good example.
	But there's a very small number of things that have patterns.
	Now, you can prove something as a pattern by just constructing.
	If something has a simple pattern and you have a proof that it does something like repeat itself every so often, you can do that.
	And you can prove that most sequences of digits have no pattern.
	So if you just pick digits randomly, there's something called a little large numbers that tells you you're going to get as many ones as twos in the long run.
	We have a lot fewer tools.
	If I give you a specific pattern, like the digits of pi, how can I show that this doesn't have some weird pattern to it?
	Some other work that I spend a lot of time on is to prove what's called structure theorems or inverse theorems that give tests for when something is very structured.
	Some functions are what's called additive.
	If you have a function, the natural numbers are the natural numbers.
	So maybe two maps to four, three maps to six, and so forth.
	Some functions are also called additive, which means that if you add For example, I multiply by a constant.
	If you multiply a number by 10, if you multiply a plus b by 10, that's the same as multiplying a by 10 and b by 10 and adding them together.
	Some functions are additive.
	Some functions are kind of additive, but not completely additive.
	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 up to 14. 20 up to 28. So in that case, additivity is true then, so 10 plus 10 is 20, and 14 plus 14 is 28. But because of this rounding, sometimes there's round-off errors, and sometimes when you add A plus B, this function doesn't quite give you the sum of the two individual outputs, but the sum plus or minus 1. So it's almost additive, but not quite additive.
	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 a function exhibits some structure like this, then there's a reason for why it's true, and the reason is because there's some other nearby function which is actually completely structured, which is explaining this partial pattern that you have.
	If you have these inverse theorems, it creates this dichotomy that either the objects that you study have no structure at all, Or they are somehow related to something that is structured.
	And in either way, in either case, you can make progress.
	A good example of this is that there's this old theorem in mathematics called Zamoretti's theorem, proven in the 1970s.
	It concerns trying to find a certain type of pattern in a set of numbers.
	the patterns have made progression.
	Things like 3, 5, and 7, or 10, 15, and 20. And André, André, has already proved that any set of numbers that are sufficiently big For example, the odd numbers have a set of density one-half, and they contain arithmetic progressions of any length.
	In that case, it's obvious because the odd numbers are really, really structured.
	I can just take 11, 13, 15, 17. I can easily find arithmetic progressions in that set.
	But Xamarinism also applies to random sets.
	If I take The set of all numbers, and I flip a coin for each number, and I only keep the numbers for which I got a heads.
	So I just flip coins, I just randomly take out half the numbers, I keep one half.
	So that's a set that has no patterns at all.
	But just from random fluctuations, you will still get a lot of ethnic progressions in that set.
	can you prove that there's arithmetic progressions of arbitrary length Have you heard of the infinite monkey theorem?
	Usually, mathematicians give boring names to theorists, but occasionally they give colourful names.
	The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each typewriter, they type out text randomly.
	Almost surely, one of them is going to generate the entire script of Hamlet or any other finite string of text.
	It will just take some time.
	Quite a lot of time, actually.
	But if you have an infinite number, then it happens.
	Basically, the theorem says that if you take an infinite string of digits or whatever, eventually any finite pattern you wish will emerge.
	It may take a long time, but it will eventually happen.
	In particular, arithmetic progressions of any length will eventually happen, 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 an abstraction of a finite number for which you do not have a bound for.
	So nothing in real life is truly infinite.
	You can ask yourself questions like, what if I had as much money as I wanted?
	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.
	And often the mathematics becomes a lot cleaner when you do that.
	I mean, in physics we joke about assuming spherical cows.
	You know, like, real-world problems have got all kinds of real-world effects, but you can idealize, send something to infinity, send something to zero.
	And the mathematics becomes a lot simpler to work with there.
	I wonder how often using infinity forces us to deviate from the physics of reality.
	Yeah, so there's a lot of pitfalls.
	We spend a lot of time in undergraduate math classes teaching analysis, and analysis is often about how to take limits.
	So for example, A plus B is always B plus A. So when you have a finite number of terms and you add them, you can swap them and there's no problem.
	But when you have an infinite number of terms, they're these sort of show 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.
	You have to introduce these epsilons and deltas, and there's a certain type of way of reasoning that helps you avoid mistakes.
	In more recent years, people have started taking results that are true in infinite limits and what's called finalizing them.
	You know that something's true eventually, but you don't know when, now give me a rate.
	If I don't have an infinite number of monkeys, but a large finite number of monkeys, how long do I have to wait for Hamlet to come out?
	That's a more quantitative question.
	This is something that you can attack by purely finite methods, and you can use your finite intuition.
	In this case, it turns out to be exponential in the length of the text that you're trying to generate.
	And so this is why you never see the monkeys create Hamlet.
	You can maybe see them create a four-letter word, but nothing that big.
	And so I personally find once you finalize an infinite statement, it does become much more intuitive.
	And it's no longer so weird.
	So even if you're working with infinity, it's good to finalize so that you can have some intuition.
	Yeah.
	The downside is that the finalize groups are just much, much messier.
	So the infinite ones are found first, usually.
	Decades earlier.
	And then later on, people finalized them.
	So since we mentioned a lot of math and a lot of physics, what is the difference between mathematics and physics as disciplines, as ways of understanding, of seeing the world?
	Maybe we can throw in engineering in there.
	You mentioned your wife is an engineer.
	Give it a new perspective on circuits.
	So this is a different way of looking at the world, given that you've done mathematical physics.
	So you've worn all the hats.
	Right.
	So I think science in general is interaction between three things.
	There's the real world, there's what we observe in the real world, our observations, and then our mental models as to how we think the world works.
	We can't directly access reality.
	All we have are the observations, which are incomplete, and they have errors.
	There are many, many cases where we want to know, for example, what is the weather like tomorrow, and we don't yet have the observation and we'd like to predict.
	And then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things.
	Those are the mathematical models.
	Mathematics is concerned with the models.
	Science collects the observations and it proposes the models that might explain these observations.
	What mathematics does, we stay within the model and ask what are the consequences of that model?
	What predictions would the model make of future observations?
	Or past observations, does it fit observed data?
	So there's definitely a symbiosis.
	I guess mathematics is unusual among other disciplines.
	We start from hypotheses, like the axioms of a model, and ask what conclusions come up from that model.
	In almost any other discipline, you start with the conclusions.
	I want to do this.
	I want to build a bridge.
	I want to make money.
	I want to do this.
	You find the path to get there.
	There's a lot less speculation about it.
	Suppose I did this, what would happen?
	Planning and modeling.
	Speculative fiction, maybe, is one other place.
	But that's about it, actually.
	Most of the things we do in life is conclusions-driven, including physics and science.
	They want to know, where is this asteroid going to go?
	What is the weather going to be tomorrow?
	But SpaceX also has this other direction.
	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.
	It's a really interaction between all these things.
	So over time, the observations and the theory and the modeling should both get closer to reality.
	But initially, and I mean, this is...
	They're always far apart to begin with.
	You need one to figure out where to push the other.
	If your model is predicting anomalies that are not picked up by experiment, that tells experimenters where to look to find more data, to refine the models.
	It goes back and forth.
	Within mathematics itself, there's also a theory and experimental component.
	It's just that until very recently, Theory has dominated almost completely.
	99% of mathematics is theoretical mathematics, and there's a very tiny amount of experimental mathematics.
	I mean, people do do it.
	If they want to study prime numbers or whatever, they can just generate large data sets.
	So once we had computers, we began to do it a little bit.
	Although even before, like Gauss, for example, he discovered, he conjectured the most basic theorem in number theory, it's called the prime number theorem.
	Which predicts how many primes are up to a million, up to a trillion.
	It's not an obvious question.
	Basically, what he did was he computed mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic, to compute the first 100,000 primes or something, and made tables and made a prediction.
	That was an early example of experimental mathematics.
	But until very recently, it was not Yeah, I mean, theoretical mathematics was just much more successful.
	I mean, because doing complicated mathematical computations was just not feasible until very recently.
	And even nowadays, even though we have powerful computers, only some mathematical things can be explored numerically.
	There's something called the combinatorial explosion.
	If you want us to study, for example, you want to study all possible subsets of numbers 1 to 1,000.
	There's only 1,000 numbers.
	How bad could it be?
	It turns out the number of different subsets of 1 to 1,000 is 2 to the power of 1,000, which is way bigger than any computer can enumerate.
	There are certain math problems that very quickly become intractable to attack by direct brute force computation.
	Chess is another famous example.
	The number of chess positions we can't get a computer to fully explore.
	But now we have AI.
	We have tools to explore this space, not with 100% guarantees of success, but with experiment.
	We can empirically solve chess now.
	For example, we have very good AIs that don't explore every single position in the game tree, but they have found some very good approximation.
	People are using these chess engines to do experimental chess.
	They're revisiting old chess theories about, oh, this type of opening, this is a good type of move, this is not.
	And they can use these chess engines to actually refine, in some cases overturn, conventional wisdom about chess.
	And I do hope that mathematics will have a larger experimental component in the future, perhaps powered by AI.
	We'll, of course, talk about that.
	But in the case of chess, and there's a similar thing in mathematics, I don't believe it's providing a kind of Formal explanation of the different positions.
	It's just saying which position is better or not, that you can intuit as a human being.
	And 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 looking at shadows of reality?
	Is it possible for us to truly access reality?
	Well, there are these three ontological things.
	There's actual reality, there's our observations, and our models.
	And technically, they are distinct, and I think they will always be distinct.
	But they can get closer over time.
	The process of getting closer often means that you have to discard your initial intuitions.
	Astronomy provides great examples.
	An initial model of the world is flat because it looks flat and it's big.
	The rest of the universe, the sun, for example, looks really tiny.
	And so you start off with a model which is actually really far from reality, but it fits the observations that you have.
	But over time, as you make more and more observations, bringing it closer to reality, the model gets dragged along with it.
	And so over time, we had to realize that the Earth was round, that it spins, it goes around the solar system, the solar system goes around the galaxy, and so on and so forth.
	And the universe is expanding.
	Expansions are self-expanding, accelerating.
	And in fact, very recently in this year, I saw this, uh, even the explosion of the universe itself is, uh, this evidence that is non-constant.
	And, uh, the explanation behind why that is.
	It's catching up.
	I mean, it's still the dark matter, dark energy, this kind of thing.
	We have a model that sort of explains, that fits the data really well.
	It just has a few parameters that you have to specify.
	So people say, oh, that's fudge factors.
	With enough fudge factors, you can explain anything.
	But the mathematical point of the model is that you want to have fewer parameters in your model than data points in your observational set.
	So if you have a model with 10 parameters that explains 10 observations, that is a completely useless model.
	It's what's called overfitted.
	But like, if you have a model with, The dark matter model has 14 parameters, and it explains petabytes of data that the astronomers have.
	One way to think about physical mathematical theory is that it's a compression of the universe, a data compression.
	You have these petabytes of observations.
	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 almost all of your observations, I mean, the more compression that you make, the better your theory.
	In fact, one of the great surprises of our universe and of everything in it is that it's compressible at all.
	It's the unreasonable effectiveness of mathematics.
	Yeah.
	Einstein had a quote like that.
	The most incomprehensible thing about the universe is that it is comprehensible.
	Right.
	And not just comprehensibly.
	You can do an equation like E equals MC squared.
	There is actually some mathematical possible explanation for that.
	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 macroscale.
	And normally, because of the common form of explosion, you would think that the macroscale equations must be infinitely exponentially more complicated than the macroscale.
	And they are, if you want to solve them completely exactly.
	If you want to model all the atoms in a box of air, Avogadro's number is humongous.
	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 macroscopic scale that almost don't depend on what's going on at the macroscale, only depend on a very small number of parameters.
	So if you want to model a gas of, you know, You just need to know its temperature and pressure and volume and a few parameters, like five or six, and it models almost everything you need to know about these 10 to 23 or whatever particles.
	We don't understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs.
	The most basic one is the central limit theorem.
	That explains why the bell curve shows up everywhere in nature, that so many things are distributed by what's called a Gaussian distribution, a famous bell curve.
	There's not even a meme with this curve.
	And even the meme applies broadly, the universality to the meme.
	Yes, you can go meta if you like, but there are many, many processes.
	For example, you can take lots and lots of independent random variables and average them together in various ways.
	You can take a simple average or more complicated average, and we can prove in various cases that These bell curves, these calcians, emerge.
	And it is a satisfying explanation.
	Sometimes they don't.
	So if you have many different inputs and they're all correlated in some systemic way, then you can get something very far from a bell curve to show up.
	And this is also important to know when it fails.
	So universality is not a 100% reliable thing to rely on.
	The global financial crisis was a famous example of this.
	People thought that mortgage defaults.
	We had this sort of Gaussian-type behavior that if you ask a population of 100,000 Americans with mortgages, ask what proportion would default in the mortgages.
	If everything was decorrelated, it would be a nice bell curve, and you can manage risk with options and derivatives and so forth.
	It is a very beautiful theory, but if there are systemic shocks in the economy that can That's very non-Gaussing behavior.
	This wasn't fully accounted for in 2008.
	Now I think there's some more awareness that systemic risk is actually a much bigger issue.
	Just because the model is pretty and nice, it may not match reality.
	The mathematics of working out what models do.
	It's really important.
	But also, the science of validating when the models fit reality and when they don't.
	I mean, you need both.
	But mathematics can help because it can, for example, these central limer theorems, it tells you that if you have certain axioms like non-correlation, that if all the inputs are not correlated to each other, then you have these classical behaviors and things are fine.
	It tells you where to look for weaknesses in the model.
	If you have a mathematical understanding of central limit theorem, and someone proposes to use these Gaussian copulas or whatever to model default risk, if you're mathematically trained, you would say, okay, but what is the systemic correlation between all your inputs?
	Then you can ask the economists how much of a risk is that, and then you can go look for that.
	There's always this synergy between science 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.
	But anyway, so you are known for this ability to go both deep and broad in mathematics, so you're the perfect person to ask, do you think there are threads that connect all the disparate areas of mathematics?
	is their kind of deep underlying structure to all of mathematics?
	There's certainly a lot of connecting threads and a lot of the progress of mathematics can be represented by taking An ancient example is geometry and number theory.
	In the times of ancient Greeks, these were considered different subjects.
	I mean, mathematicians worked on both.
	Euclid worked both on geometry, most famously, but also on numbers.
	But they were not really considered related.
	I mean, a little bit.
	You could say that this length was five times this length because you could take five copies of this length.
	But it wasn't until Descartes who really realized that you can parametrize the plane, a geometric object, by two real numbers.
	So geometric problems can be turned into problems about numbers.
	And today, this feels almost...
	There's no content to this.
	Of course, a plane is x, x, and y, because that's what we teach, and it's internalized.
	But it was an important development that these two fields were unified.
	This process has just gone on throughout mathematics over and over again.
	Algebra and geometry were separated, and now we have a fluid algebraic geometry that connects them over and over again.
	That's certainly the type of mathematics that I enjoy the most.
	So I think there's sort of different styles to being a mathematician.
	I think hedgehogs and foxes.
	A fox knows many things a little bit, but a hedgehog knows one thing very, very well.
	And in mathematics, there's definitely both hedgehogs and foxes.
	And then there's people who can play both roles.
	I think ideal collaboration between Methodicians involves very a fox working with many hedgehogs or vice versa.
	But I identify...
	I like arbitrage somehow.
	Learning how one field works, learning the tricks of that field, and then going to another field, which people don't think is related, but I can adapt the tricks.
	So see the connections between the fields.
	So there are other mathematicians who are far deeper than I am.
	They're really hedgehogs.
	They know everything about one field, and they're much faster and more effective in that field.
	But I can give them these extra tools.
	I mean, you said that you can be both a hedgehog and the fox, depending on the context, depending on the collaboration.
	So can you, if it's at all possible, speak to the difference between those two ways of thinking about a problem?
	Say you're encountering a new problem.
	You know, searching for the connections versus, like, very singular focus.
	I'm much more comfortable with the Fox paradigm.
	So, yeah, I like looking for analogies, narratives.
	I spend a lot of time, 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.
	It uses types of mathematics that I'm not super familiar with.
	I often try to re-prove it myself using the tools that I favor.
	Often my proof is worse, but by the exercise of doing so, I can say, oh, now I can see what the other proof was trying to do.
	From that, I can get some understanding of the tools that are used in that field.
	it's very exploratory very doing crazy things and crazy builds and like reinventing the wheel a lot whereas the hedgehog style is I think much more scholarly You stay up to speed on all the developments in this field.
	You know all the history.
	You have a very good understanding of exactly the strengths and weaknesses of each particular technique.
	I think you'd rely a lot more on calculation than trying to find narratives.
	I could do that too, but there are other people who are extremely good at that.
	Let's step back and maybe look at a bit of a romanticized version of mathematics.
	So, I think you've said that early on in your life, math was more like a puzzle-solving activity when you were young.
	When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it?
	That's a good question.
	When I came to graduate school in Princeton, John Conway was there at the time.
	He passed away a few years ago.
	I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof.
	Conway just had this amazing way of thinking about all kinds of things in a way that you wouldn't normally think of.
	he sort of proves themselves as occupying some sort of space.
	So if you want to prove something, There are all different proofs, but you could rank them in different axes.
	Some proofs are elegant, some proofs are long, some proofs are elementary, and so forth.
	The space of all proofs itself has some sort of shape.
	He was interested in extreme points of this shape.
	Out of all these proofs, what is one that is the shortest at the expense of everything else, or the most elementary, or whatever?
	He gave some examples of well-known theorems, and then he would give what he thought was the extreme proof in these different aspects.
	I just found that really eye-opening.
	It's not just getting a proof was interesting, but once you have that proof, trying to optimize it in various ways.
	That proofing itself had some craftsmanship to it.
	It's something for my writing style.
	When you do your math assignments and undergraduate, your homework and so forth, you're encouraged to just write down any proof that works and hand it in.
	As long as it gets a tick mark, you move on.
	But if you want your results to actually be influential and be read by people, it can't just be correct.
	It should also be a pleasure to read, motivated.
	It's the same in many other disciplines, like coding.
	There's a lot of analogies between math and coding.
	I like analogies, if you haven't noticed.
	You can code something spaghettical that works for a certain task, and it's quick and dirty and it works, but there's lots of good principles for writing code well so that other people can use it, build upon it, and so on, it has fewer bugs and whatever.
	Similar things with mathematics.
	Yeah.
	First of all, there's so many beautiful things there.
	And Karl is one of the great minds in mathematics ever and computer science.
	Just even considering the space of proofs.
	Yeah.
	And saying, okay, what does this space look like?
	And what are the extremes?
	Like you mentioned, coding is an analogy.
	It's interesting because there's also this activity called Code Golf, which I also find beautiful and fun where people use different programming languages to try to write the shortest possible program that accomplishes a particular task.
	And I believe there's even competitions on this.
	Yeah, yeah, yeah.
	And it's also a nice way to stress test not just the...
	Maybe that's a different notation or whatever to use to accomplish a different task.
	Yeah, you learn a lot.
	I mean, it may seem like a frivolous exercise, but it can generate all these insights, which if you didn't have this artificial objective to pursue, you might not see.
	What do you use the most beautiful or elegant equation in mathematics?
	I mean, one of the things that people often look to in beauty, It's the simplicity.
	So if you look at E equals MC squared, so when a few concepts come together, that's why the Euler identity is often considered the most beautiful equation in mathematics.
	Do you find beauty in that one, in the Euler identity?
	Yeah, well, as I said, what I find most appealing is connections between different things.
	So if the pi i equals minus one.
	So yeah, people use all the fundamental constants.
	Okay, that's cute.
	But to me, the exponential function was to measure exponential growth.
	Compound interest or decay, anything which is continuously growing, continuously decreasing, growth and decay or dilation or contraction is modeled by the exponential function.
	Whereas pi comes around from circles and rotation.
	If you want to rotate a needle, for example, 100 degrees, you need to rotate by pi radians.
	And I, complex numbers, represent.
	When you stick an eye in the exponential, instead of motion in the same direction as your current position, it's motion as a right angle as your current position, so rotation.
	If the pi equals minus 1, it 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 complexification, rotation by I. So it connects together all these fields in mathematics, you know, dynamics, geometry, and complex, and the complex numbers, they're all considered almost, they're all next-door neighbors in mathematics because of this identity.
	Do you think the thing you mentioned is cute, the collision of notations from these disparate fields is just a frivolous, Right.
	And when the notation, all our old friends come together.
	Well, it's confirmation that you have the right concepts.
	So when you first study anything, you have to measure things and give them names.
	And initially, sometimes, because your model is, again, too far off from reality, you give the wrong things the best names.
	You only find out later what's really important.
	Physicists can do this sometimes.
	I mean, but it turns out, okay.
	So actually, with physics, so E equals mc squared, okay, so one of the big things was the E, right?
	So when Aristotle first came up with his laws of motion and then Galileo and Newton and so forth, you know, they saw the things they could measure.
	They could measure mass and acceleration and force and so forth.
	Newtonian mechanics, for example, F equals MA was the famous Newton's second law of motion.
	Those were the primary objects.
	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, in particular, momentum and energy.
	It's not obvious that things happen in energy.
	It's not something you can directly measure the same way you can measure mass and velocity.
	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 dynamics, like what happens to all the states.
	It really was a central actor, which was not obvious initially.
	This change of perspective really helped when quantum mechanics came along.
	Because the early physicists who studied quantum mechanics had a lot of trouble trying to adapt their Newtonian thinking because everything was a particle and so forth to quantum mechanics.
	I think because there was a wave, it just looked really, really weird.
	You asked, what is the quantum vision of F equals MA?
	It's really, really hard to give an answer to that.
	But it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics, also is the key object in quantum mechanics.
	There's also an object called a Hamiltonian.
	It's a different type of object.
	It's what's called an operator rather than a function.
	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 a Hamiltonian.
	Side by side, they look completely different objects.
	One involves particles, one involves waves, and so forth.
	But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.
	For example, in classical mechanics, there's this thing called Noether's theorem.
	Every time there's a symmetry in a physical system, there is a conservation law.
	The laws of physics are translation invariant.
	If I move 10 steps to the left, I experience the same laws of physics as if I was here.
	Corresponds to conservation of momentum.
	If I turn around by some angle, again, I experience the same laws of physics.
	This corresponds to conservation of angular momentum.
	If I wait for 10 minutes, I still have the same laws of physics.
	So there's time transition invariance.
	This corresponds to the law of conservation of energy.
	So there's this fundamental connection between symmetry and conservation.
	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.
	Every time the Hamiltonian has a symmetry, the equations will have a conservation law.
	Once you have the right language, it actually makes things 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.
	For example, we have to give up the notion of space and time being these almost Euclidean-type spaces.
	We kind of know that at very tiny scales, there's going to be quantum fluctuations, there's space-time foam, and trying to use Cartesian coordinates XYZ is a non-starter.
	But we don't know what to replace it with.
	We don't actually have the mathematical concepts.
	The analog or the Hamiltonian that sort of organized everything.
	Does your gut say that there is a theory of everything, so this is even possible to unify, to find this language?
	That unifies general relativity and quantum mechanics?
	I believe so.
	I mean, the history of physics has been that of unification, much like mathematics over the years.
	Electricity and magnetism were separate theories, and then Maxwell unified them.
	Newton unified the motions of the heavens for the motions of objects on the Earth and so forth.
	So it should happen.
	Again, to go back to this model of observations and theory, part of our problem is that physics is a victim of its own success.
	Two big theories of physics, general relativity, and quantum mechanics are so good now.
	Together, they cover 99.9% of all the observations we can make.
	You have to either go to extremely insane particle accelerations or the early universe or things that are really hard to measure in order to get any deviation from either of these two theories to the point where you can actually figure out how to combine them together.
	I have faith that we've We've been doing this for centuries.
	We've made progress before.
	There's no reason why we should stop.
	Do you think you'll be a mathematician that develops a theory of everything?
	What often happens is that when the physicists need some theory of mathematics, there's often some precursor that the mathematicians worked out earlier.
	When Einstein started realizing that space was curbed, He went to some mathematician and asked, is there some theory of curved space that the mathematicians already came up with that could be useful?
	And he said, oh yeah, I think Riemann came up with something.
	And so, yeah, Riemann had developed Riemannian geometry, which is precisely a theory of spaces that are curved in various general ways, which turned out to be almost exactly what was needed for Einstein's theory.
	This has gone back to Wigner's unreasonable effectiveness on mathematics.
	I think the theories that work well to explain the universe Ultimately, they're just both ways of organizing data in useful ways.
	It just feels like you might need to go to some weird land that's very hard to intuit.
	You have string theory.
	Yeah, that was a leading candidate for many decades.
	I think it's slowly falling out of fashion because it's not matching experiment.
	So one of the big challenges, of course, like you said, Experiment is very tough.
	Yes.
	Because of how effective both theories are.
	But the other is, you're talking about, you're not just deviating from space-time.
	You're going into some crazy number of dimensions.
	You're doing all kinds of weird stuff.
	To us, we've gone so far from this flat Earth that we started at, like you mentioned.
	Yeah, yeah, yeah.
	And now we're just, it's very hard to use our limited descendants of cognition to intuit what that reality really is like.
	This is why analogies are so important.
	I mean, so, yeah, the round Earth is not intuitive because we're stuck on it.
	But round objects in general, we have pretty good intuition over it.
	And we have intuition about light works and so forth.
	It's actually a good exercise to work out how eclipses and phases of the sun and the moon and so forth can be really easily explained by round Earth and round moon models.
	And you can just take a basketball and a golf ball and a light source and actually do these things yourself.
	So the intuition is there.
	But you have to transfer it.
	That is a big leap, intellectual, for us to go from flat to round earth.
	Because, you know, our life is mostly lived in flat land.
	Yeah.
	To load that information, and we're 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 in a rock.
	Yeah.
	Flying through space.
	That's a big leap.
	And you have to take a chain of those leaps the more and more and more we progress.
	Right.
	So modern science is maybe, again, a victim of its own success.
	In order to be more accurate, it has to move further and further away from your initial intuition.
	And so for someone who hasn't gone through the whole process of science education, it looks more and more suspicious because of that.
	So we need more grounding.
	I mean, you know, there are scientists who do excellent outreach, but there's lots of science things that you can do at home.
	There's lots of YouTube videos.
	I did a YouTube video recently with Grant Sanderson.
	We talked about this earlier, that, you know, how the ancient Greeks were able to measure things like the distance to the moon, distance to the Earth, and, you know, using techniques that you could also replicate yourself.
	It doesn't all have to be like fancy space telescopes and really intimidating mathematics.
	Yeah, 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 things, and you try to, like, localize yourself in this world.
	Yeah, yeah.
	And try to make some kind of general statements about distance to places.
	Change of perspective is really important.
	You say travel broadens the mind.
	This is intellectual travel.
	Put yourself in the mind of the ancient Greeks or some other person in some other time period.
	Make hypotheses, spherical cows, whatever.
	Speculate.
	This is what mathematicians do and some artists do, actually.
	It's just incredible that given the extreme constraints, you can 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?
	If you propose axioms, then the mathematics lets you follow those axioms to their conclusions, and sometimes you can get quite a long way from initial hypotheses.
	If we stay in the land of the weird, you mentioned general relativity.
	You've contributed to the mathematical understanding of Einstein's field equations.
	Can you explain this work from a mathematical standpoint?
	What aspects of general relativity are intriguing to you, challenging to you?
	I have worked on some equations.
	There's something called the wave maps equation, or the sigma field model, which is not quite the equation of space-time gravity itself, but of certain fields that might exist on top of space-time.
	So science equations of relativity just describe space and time itself.
	But then there's other fields that live on top of that.
	There's the electromagnetic field, there's things called Yang-Mills fields, and there's this whole hierarchy of different equations, of which Einstein is considered one of the most non-linear and difficult.
	But relatively low on the hierarchy was this thing called the wave maps equation.
	So it's a wave which, at any given point, is fixed to be like on a sphere.
	So I can think of a bunch of arrows in space and time, and yeah, so it's pointing in different directions.
	But they propagate like waves.
	If you wiggle an arrow, it will propagate and make all the arrows move like sheaves of wheat in the wheat field.
	I was interested in the global regularity problem again.
	Is it possible for all the energy here to collect at a point?
	The equation I considered was actually what's called a critical equation, where the behavior at all scales is roughly the same.
	I was able barely to show that That you couldn't actually force a scenario where all the energy concentrated at one point.
	The energy had to disperse a little bit, and the moment it dispersed a little bit, it would stay regular.
	This was back in 2000.
	That was part of why I got interested in Nari-Sogs afterwards.
	I developed some techniques to solve that problem.
	Part of it is that this problem is really non-linear because of the curvature of the sphere.
	There was a certain nonlinear effect which was a non-perturbative effect.
	When you sort of looked at it normally, it looked larger than the linear effects of the wave equation.
	And so it was hard to keep things under control, even when the energy was small.
	But I developed what's called a gauge transformation.
	So the equation is kind of like an evolution of heaves of wheat, and they're all bending back and forth.
	So there's a lot of motion.
	But if you imagine stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion, and under this stabilized flow, the flow becomes a lot more linear.
	I discovered a way to transform the equation to reduce the amount of nonlinear effects, and then I was able to solve the equation.
	I found this transformation while visiting my aunt in Australia, and I was trying to understand the dynamics of all these fields, and I couldn't do it with pen and paper.
	And I had no enough facility of computers to do any computer simulations.
	So I ended up closing my eyes, being on the floor, and just imagining myself to actually be the specter field, and rolling around to try to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion.
	And yeah, my aunt walked in on me while I was doing that.
	And she was asking, what am I doing doing this?
	It's complicated is the answer.
	Yeah, yeah.
	And she said, okay, fine.
	You're a young man.
	I don't ask questions.
	I have to ask about the, you know, how do you approach solving difficult problems?
	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?
	A lot of pen and paper.
	One thing you pick up as a mathematician is sort of, I call it cheating strategically.
	So the beauty of mathematics is that you get to change the problem, change the rules as you wish.
	You don't get to do this for any other field.
	If you're an engineer and someone says, build a bridge over this river, you can't say, I want to build this bridge over here instead, or I want to build it out of paper instead of steel.
	But in mathematician, you can do whatever you want.
	It's like trying to solve a computer game where there's unlimited cheat codes available.
	And so, you know, you can set...
	I'll set it to one.
	I'd solve the one-dimensional problem first.
	There's a main term and an error term.
	I'm going to make a spherical car assumption.
	I'll assume the error term is zero.
	And so the way you should solve these problems is not in this Iron Man mode where you make things maximally difficult.
	But actually, the way you should approach any reasonable math problem is that if there are ten things that are making life difficult, find a version of the problem that turns off nine of the difficulties but only keeps one of them.
	And solve that.
	So you install nine cheats.
	If you solve ten cheats, then the game is trivial.
	But you solve nine cheats, you solve one problem, that teaches you how to deal with that particular difficulty.
	And then you turn that one off, and you turn something else on, and then you solve that one.
	And after you know how to solve the ten problems, ten difficulties separately, then you have to start merging them a few at a time.
	As a kid, I watched a lot of these Hong Kong action movies from my culture.
	And one thing is that every time it's a fight scene, maybe the hero gets swarmed by a hundred bad guy goons or whatever.
	But it would always be choreographed so that you'd always be only fighting one person at a time, and then it would defeat that person and move on.
	And because of that, he could defeat all of them.
	But whereas if they had fought a bit more intelligently and just swarmed the guy at once, it would make for much fun.
	Are you usually pen and paper?
	Are you working with computer and LaTeX?
	Mostly pen and paper, actually.
	In my office, I have four giant blackboards, 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 only make sense to me.
	It's a beautiful blackboard that you raise.
	It's a very organic thing.
	I'm beginning to use more and more computers, partly because AI makes it much easier to do simple coding things.
	If I wanted to plot a function before which is moderately complicated, some iteration or something, I'd have to...
	And now I can do it in 10-15 minutes.
	I'm using more and more computers to do simple explorations.
	Let's talk about AI a little bit if we could.
	Maybe a good entry point is just talking about computer-assisted proofs in general.
	Can you describe the lean, formal proof programming language?
	How it can help as a proof assistant, and maybe how you started using it, and how it has helped you.
	Lean is a computer language, much like standard languages like Python and C and so forth, except that in most languages, the focus is on using executable code.
	Lines of code do things.
	They flip bits, or they make a robot move, or they deliver you text on the internet or something.
	Lean is a language that can also do that.
	It can also be run as a standard traditional language, but it can also produce certificates.
	A software language like Python might do a computation and give you that the answer is 7. It does the sum of 3 plus 4 is equal to 7, but Lean can produce not just the answer, but a proof of how it got.
	The answer of 7 as 3 plus 4 and all the steps involved.
	It creates these more complicated objects, not just statements, but statements with proofs attached to them.
	Every line of code is just a way of piecing together previous statements to create new ones.
	The idea is not new.
	These things are called proof assistants, and so they provide languages for which you can create quite complicated, intricate mathematical proofs.
	And they produce these certificates that give a 100% guarantee that your arguments are correct if you trust the compiler of Lean.
	But they made the compiler really small, and there are several different compilers available for this.
	How hard is it to formalize a statement?
	A lot of mathematicians were involved in the design of Lean.
	It's designed so that individual lines of code resemble individual lines of mathematical argument.
	You might want to introduce a variable, you might want to prove a contradiction.
	There are various standard things that you can do, and it's written so that ideally it should have a one-to-one correspondence.
	In practice, it isn't because Lean is like explaining a proof to an extremely pedantic colleague.
	Did you really mean this?
	What happens if this is zero?
	How do you justify this?
	Lean has a lot of automation in it to try to be less annoying.
	For example, every mathematical object has to come with a type.
	If I talk about X, is X a real number or a natural number or a function or something?
	If you write things informally, it's...
	Clearly, let X be the sum of Y and Z, and Y and Z are already real numbers, so X should also be a real number.
	Lean can do a lot of that, but every so often it says, wait a minute, can you tell me more about what this object is, what type of object it is?
	You have to think more at a philosophical level, not just computations you're doing, but what each object actually is.
	Is it using something like LLMs to do the type inference?
	It's using much more traditional, good old-fashioned AI.
	You can represent all these things as trees, and there's always an algorithm to match one tree to another tree.
	So it's actually doable to figure out if something is a real number or a natural number?
	Every object comes with a history of where it came from, and you can kind of trace it.
	Oh, I see.
	It's designed for reliability.
	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 a proof in Lean, often there's a step, okay, now I want to use the fundamental thing of calculus, say, okay, to do the next step.
	So the Lean developers have built this massive project called Metholib, a collection of tens of thousands of useful facts about mathematical objects.
	And somewhere in there is the fundamental thing of calculus, but you need to find it.
	The bottleneck now is actually lemma search.
	There's a tool that you know is in there somewhere, and you need to find it.
	There are various search engines specialized for MathLib that you can do.
	But there's now these large language models that you can say, I need the fundamental calculus at this point.
	For example, when I code, I have GitHub Copilot installed as a plugin to my IDE.
	And it scans my text and it sees what I need.
	It says, you know, I might even type, okay, now I need to use the fundamental human calculus.
	Okay.
	And then it might suggest, okay, try this.
	And like maybe 25% of the time, it works exactly.
	And then another 10, 50% of the time, it doesn't quite work, but 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.
	So, but people are beginning to use AIs a little bit on top, mostly on the level of basically fancy autocomplete.
	You can type half of one line of a proof and it will tell you.
	Yeah, but a fancy, especially fancy with a capital letter F, is remove some of the friction a mathematician might feel when they move from pen and paper to formalizing.
	Yes, yeah.
	So right now, I estimate that the effort, time and effort taken to formalize a proof is about 10 times the amount taken to write it out.
	Yeah, so it's doable, but you don't, But doesn't it kill the whole vibe of being a mathematician?
	Just having a pedantic co-worker?
	Right.
	Yeah, if that was the only aspect of it.
	Okay.
	There's some cases where it's actually more pleasant to do things formally.
	So there was a theorem I formalized, and there was a certain constant 12 that came out in the final statement, and so this 12 had to be carried all through the proof.
	And everything had to be checked.
	All these other numbers had to be consistent with this final number 12. And so we wrote a paper through this theorem with this number 12. And then a few weeks later, someone said, oh, we can actually improve this 12 to an 11 by reworking some of these steps.
	And when this happens with pen and paper, every time you change a parameter, you have to check line by line that every single line of your proof still works.
	And there can be subtle things that you didn't quite realize.
	Some properties on number 12 that you didn't even realize that you were taking advantage of.
	So a proof can break down at a subtle place.
	So we had formalized the proof of this constant 12, and then when this new paper came out, we said, oh, so that took like three weeks to formalize, and like 20 people to formalize this original proof.
	I said, oh, but now let's update the 12 to 11. And what you can do with Lean, in your headline theorem, you change your 12 to 11, you run the compiler, and 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 steps, but it immediately isolates which steps you need to change.
	But you can skip over everything which works just fine.
	And if you program things correctly, with good programming practices, most of your lines will not be read.
	And there'll just be a few places where you, I mean, if you don't hard-code your constants, but you sort of use smart tactics and so forth, you can localize the things you need to change to a very small period of time.
	So it's like within a day or two, we had updated our proof.
	This is a very quick process.
	You make a change.
	There are 10 things now that don't work.
	For each one, you make a change, and now there's five more things that don't work.
	But 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, what's the versus paper?
	Yeah, so the proofs are longer, but each individual piece is easier to read.
	So if you I often can't read it immediately, because it assumes various definitions, which I have to go back, and maybe on 10 pages earlier this was defined, and the proof is scattered all over the place, and you basically are forced to read fairly sequentially.
	It's not like, say, a novel, where in theory you could open up a novel halfway through and start reading.
	There's a lot of context.
	But when 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 say what it is, where it came from, where the stuff is justified.
	You can trace things back much easier than flipping through a math paper.
	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 pen and paper um when you want to collaborate with another mathematician um either you do it at a blackboard where you um But if you're doing it by email or something, basically you have to segment it.
	I'm going to finish section three, you do section four.
	But you can't really work on the same thing collaboratively at the same time.
	But with Lean, you can be trying to formalize some portion of the proof and say, I got stuck at line 67 here.
	I need to prove this thing, but it doesn't quite work.
	Here's the three lines of code I'm having trouble with.
	But because all the context is there, someone else can say, oh, okay, I recognize what you need to do.
	You need 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, So I can do trustless mathematics.
	So there's so many interesting questions.
	So one, you're known for being a great collaborator.
	So what is the right way to approach...
	Are you doing a divide and conquer type of thing?
	Or are you focused on a particular part and you're brainstorming?
	There's always a brainstorming process first.
	Math research projects, by their nature, when you start, you don't really know how to do the problem.
	It's not like an engineering project where somehow the theory has been established for decades and its implementation is the main difficulty.
	You have to figure out even what is the right path.
	This is what I said about cheating first.
	It's like, to go back to the bridge-building analogy, first assume you have an infinite budget and unlimited amounts of workforce and so forth.
	Now can you build this bridge?
	Now have an infinite budget but only a finite workforce.
	Now can you do that?
	Of course, no engineer can actually do this because they have fixed requirements.
	Yes, there's this sort of jam sessions always at the beginning where you try all kinds of crazy things and you make all these assumptions that are unrealistic but you plan to fix later.
	And you try to see if there's even some skeleton of an approach that might work.
	And then hopefully that breaks up the problem into smaller sub-problems, which you don't know how to do, but then you focus on the sub-ones.
	And sometimes different collaborators are better at working on certain things.
	One of my theorems I'm known for is the theorem of Ben Green, which is now called the Green-Tow theorem.
	It's a statement that the primes contain algorithmic progressions of any length.
	So it was a modification of this theorem already.
	And the way we collaborated was that Ben had already proven a similar result for progressions of length 3. He showed that sets like the primes contain lots and lots of progressions of length 3, and even subsets of the primes, certain subsets do.
	But his techniques only worked for length 3 progressions.
	They didn't work for longer progressions.
	But I had these techniques coming from a gothic theory, which is something that I had been playing with and I knew better than Ben at the time.
	And so if I could justify certain randomness properties of some set relating to primes, there's a certain technical condition which if I could have it, if Ben could supply me this fact, I could conclude the theorem.
	But what I asked was a really difficult question in number theory, which he said, there's no way we can prove this.
	So he said, can you Prove your part of the theorem using a weaker hypothesis that I have a chance to prove it.
	And he proposed something which he could prove, but it was too weak for me.
	I can't use this.
	So there was this conversation going back and forth.
	Different cheats, too.
	Yeah, yeah.
	I want to cheat more.
	He wants to cheat less.
	But eventually we found a property which A, he could prove, and B, I could use, and then we could prove our theorem.
	There are all kinds of dynamics.
	I mean, every collaboration has some story.
	No two are the same.
	And then on the flip side of that, like you mentioned, with Lean programming, now that's almost like a different story because you can do, you can create, I think you've mentioned, a kind of a blueprint 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 trustable.
	Currently, only a few mathematical projects can be cut up in this way.
	At the current state of the art, most of the lean activity is on formalizing proofs that have already been proven by humans.
	Math paper basically is a blueprint, in a sense.
	It is taking a difficult statement, like a big theorem, and breaking it up into maybe a hundred little numbers.
	But often not all written with enough detail that each one can be directly formalized.
	A blueprint is like a really pedantically written version of a paper where every step is explained to as much detail as possible.
	And you're trying to make each step self-contained, depending on only a very specific number of previous statements that have been proven, so that each node of this I think that's
	a really exciting possibility because if you can find problems that could be Right.
	So I told you before about the 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 other software tools, so GitHub and things like that, allow experimental mathematics to scale up to a much greater degree than we can do now.
	So right now, if you want to Do any mathematical exploration of some mathematical pattern or something.
	You need some code to write out the pattern.
	Sometimes there are some computer algebra packages that help, but often it's just one mathematician coding lots and lots of Python or whatever.
	Because coding is such an error-prone activity, it's not practical to allow other people to collaborate with you on writing modules for your code because if one of the modules has a bug in it, the whole thing is unreliable.
	You get these bespoke Spaghetti code written by non-professional programmers, mathematicians.
	They're clunky and slow.
	Because of that, it's hard to really mass-produce experimental results.
	I'm already starting some projects where we are not just experimenting with data, but experimenting with proofs.
	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 tell you what the project is.
	So abstract algebra studies operations like multiplication and addition and their abstract properties.
	So multiplication, for example, is commutative.
	x times y is always y times x, at least for numbers.
	And it's also associative.
	x times y times z is the same as x times y times z.
	So these operations obey some laws that don't obey others.
	For example, x times x is not always equal to x.
	So that law is not always true.
	Given any operation, it obeys some laws and not others.
	And so we generated about 4,000 of these possible laws of algebra that certain operations can satisfy.
	And our question is, which laws imply which other ones?
	So, for example, does commutativity imply associativity?
	And the answer is no, because it turns out you can describe an operation which obeys the commutative law but doesn't obey the associative law.
	So by producing an example, you can show that commutativity does not imply associativity.
	But some other laws do imply other laws by substitution and so forth.
	And you can write down some algebraic proof.
	So we look at all the pairs between these 4,000 laws, and there's about 22 million of these pairs.
	And for each pair, we ask, does this law imply this law?
	If so, give a proof.
	If not, give a counterexample.
	So 22 million problems, each one of which you could give to an undergraduate algebra student, and they had a decent chance of solving the problem.
	Although there are a few, at least 22 million, there are like 100 or so that are really quite hard, but a lot are easy.
	The project was just to work out, to determine the entire graph, like which ones imply which other ones.
	That's an incredible project, by the way.
	Such a good idea, such a good test of the very thing we've been talking about on a scale that's remarkable.
	Yeah, so it would not have been feasible.
	I mean, the state of the art in the literature was like 15 equations and sort of highly implied.
	That's sort of at the limit of what a human pen and paper can do.
	So you need to scale it up.
	So you need to crowdsource, but you also need to trust.
	I mean, no one person can check 22 million of these proofs.
	You need to be computerized.
	And so it only became possible with Lean.
	We were hoping to use a lot of AI as well.
	So the project is almost complete.
	So of these 22 million, all but two had been settled.
	Wow.
	And of those two, we have a pen and paper proof of the two, and we're formalizing it.
	In fact, this morning I was working on finishing it.
	So we're almost done on this.
	It's incredible.
	How many people were able to get?
	About 50. Which, in mathematics, is considered a huge number.
	It's a huge number.
	That's crazy.
	Yeah.
	So we're going to have a paper of 50 authors and a big appendix of who contributed to what.
	Here's an interesting question, not to maybe speak even more generally about it.
	When you have this pool of people Now, okay, I'm asking a lot of pothead questions here, but I'm imagining a bunch of humans and maybe in the future some AIs.
	Can there be like an ELO rating type of situation where like a gamification of this?
	The beauty of these lean projects is that automatically you get all this data.
	So like everything's uploaded to this GitHub and GitHub tracks who contributed what.
	So you could generate statistics at any later point in time.
	You could say, oh, this person contributed this many lines of code or whatever.
	These are very crude metrics.
	I would definitely not want this to become part of your 10-year review or something.
	But I think already in enterprise computing, people do use some of these metrics.
	As part of the assessment of performance of an employee.
	Again, this is the direction which is a bit scary for academics to go down.
	We don't like metrics so much.
	And yet, academics use metrics.
	They just use old ones.
	Number of papers.
	Yeah, it's true that, yeah.
	It's going more in the right direction, right?
	Yeah.
	At least it's a very interesting metric.
	Yeah, I think it's interesting to study.
	I think you can do studies of whether these are better predictors.
	There's this problem called Goodhart's Law.
	If a statistic is actually used to incentivize performance, it becomes gamed.
	And then it is no longer a useful measure.
	Oh, humans always gamed.
	Yeah, I know.
	It's rational.
	So what we've done for this project is self-report.
	So there are actually these standard categories from the sciences of what types of contributions people give.
	So there's concept and validation and resources and coding and so forth.
	So there's a standard list of 12 categories.
	And we just ask each contributor to this big matrix of all the authors and all the categories just to tick the boxes where they think that they contributed.
	And just to give a rough idea, you know, like, oh, so you did some coding and you provided some compute, but you didn't do any of the pen and paper verification or whatever.
	And I think that that works out.
	Traditionally, mathematicians just order alphabetically by surname.
	So we don't have this tradition as in the sciences of, you know, lead author and second author and so forth, which we're proud of.
	You know, we make all the authors equal status, but it doesn't quite scale to this size.
	So a decade ago, I was involved in these things called polymath projects.
	It was the crowdsourcing mathematics, but without the lean component.
	So it was limited by, you needed a human moderator to actually check that all the contributions coming in were actually valid.
	And this was a huge bottleneck, actually.
	But still, we had projects with 10 authors or so.
	But we had decided at the time not to try to decide who did what, but to have a single pseudonym.
	So we created this fictional character called DHJ Polymath.
	In the spirit of Boracchi, Boracchi is the pseudonym for a famous group of mathematicians in the 20th century.
	The paper was authored on the pseudonym, so none of us got the author credit.
	This actually turned out to be not so great for a couple of reasons.
	One is that if you actually wanted to be considered for tenure or whatever, you could not use this paper as you submitted on your publications because you didn't have the formal author credit.
	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, so this was Tim Gower's project.
	This was Terence Tao's project.
	And not mention the other 19 or whatever people that were involved.
	So we're trying something different this time around where everyone's an author, but we will have an appendix of this matrix.
	And we'll see how that works.
	So both projects are incredible, just the fact that you're involved in such huge collaborations.
	But I think I saw a talk from Kevin Buzzard about the Lean Programming Language just a few years ago, and he was saying that this might be the future of mathematics.
	And so it's also exciting that you're embracing one of the greatest mathematicians in the world embracing this, what seems like the paving of the future of mathematics.
	So I have to ask you here about the integration of AI into this whole process.
	So DeepMind's alpha proof 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 Yeah, the difficulty increases exponentially with the number of steps involved in the proof.
	It's a combinatorial explosion.
	So the thing of large language models is that they make mistakes.
	And so if a proof has got 20 steps and your large language model has a 10% failure rate at each step of going in the wrong direction, like...
	Actually, just to take a small tangent here, how hard is the problem of mapping from natural language to the formal program?
	Oh, yeah, it's extremely hard, actually.
	Natural language, it's very fault-tolerant.
	You can make a few minor grammatical errors and a speaker in the second language can get some idea of what you're saying.
	But formal language, if you get one little thing wrong, I think the whole thing is nonsense.
	There are different incompatible preferences in languages.
	There's Lean, but also Koch, and Isabel, and so forth.
	Even converting from a formal language to a formal language is an unsolved problem.
	That is fascinating.
	But once you have an informal language, they're using their RL-trained model.
	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 what do you think is the gap?
	We talked earlier about things that are amazing over time become kind of normalized.
	So now somehow it's, of course geometry is a silverware problem.
	Right, that's true, that's true.
	I mean, it's still beautiful.
	Yeah, yeah, no, these are great works.
	It shows what's possible.
	I mean, the approach doesn't scale currently.
	Three days of Google's server time to sort of one high school math problem.
	This is not a scalable prospect, especially with the exponential increase as the complexity follows.
	We should mention that they got a silver medal performance.
	The equivalent of.
	The equivalent of a silver medal performance.
	First of all, they took way more time than was allotted, and they had this assistance where the humans helped by formalizing.
	Also, they're giving themselves full marks for the solution, which I guess is formally verified, so I guess that's fair.
	There will be a proposal at some point to actually have it.
	An AI math Olympiad where at the same time as the human contestants get the actual Olympiad problems, the AIs will also be given the same problems, the same time period, and the outputs will have to be graded by the same judges, which means that it will have to be written in natural language.
	Rather than formal language.
	Oh, I hope that happens.
	I hope this IMO happens.
	I hope next one.
	It won't happen this IMO.
	The performance is not good enough in the time period.
	But there are smaller competitions.
	There are competitions where the answer is a number rather than a long-form proof.
	And that's...
	Because it's easy to do reinforcement learning on it.
	Yeah, you got the right answer, you got the wrong answer.
	It's a very clear signal.
	But a long-form proof either has to be formal, and then the lean can give it thumbs up, thumbs down, or it's informal.
	But then you need a human to grade it.
	And if you're trying to do billions of reinforcement learning runs, you can't hire enough humans to grade those.
	It's already hard enough for the last language to do reinforcement learning on just the regular text that people get.
	But now you actually hire people and not just give thumbs up, thumbs down, but actually check the output.
	Not cracking for a while.
	So, inventing new theories.
	So, coming up with new conjectures versus proving the conjectures.
	Building new abstractions, new representations, maybe an AI-turner style with seeing new connections between disparate fields.
	That's a good question.
	I think the nature of what mathematicians do over time has changed a lot.
	You know, so a thousand years ago, mathematicians had to compute the date of Easter.
	And those really complicated calculations, but it's all automated for centuries.
	We don't need that anymore.
	They used to navigate to do spherical trigonometry to navigate how to get from the old world to the new.
	Very complicated calculations, again, being automated.
	Even a lot of undergraduate mathematics, even before AI, like Wolfram Alpha, for example, is not a language model, but it can solve a lot of undergraduate-level math tasks.
	On the computational side, verifying routine things like having a problem and saying, here's a problem in partial differential equations.
	Could you solve it using any of the 20 standard techniques?
	Yes, I've tried all 20 and here are the 100 different permutations and here's my results.
	That type of thing I think will work very well.
	The type of scaling once you solve one problem to make the AI attack 100 adjacent problems.
	The things that humans do Where the AI really struggles right now is knowing when it's made a wrong turn.
	It can say, oh, I'm going to solve this problem.
	I'm going to split up this problem into these two cases.
	I'm going to try this technique.
	Sometimes, if you're lucky, and it's a simple problem, it's the right technique, and you solve the problem, sometimes it would propose an approach which is just complete nonsense.
	But it looks like a proof.
	So this is one annoying thing about LLM-generated mathematics.
	We've had human-generated mathematics that's very low quality, like submissions with 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.
	And it's partly because that's what the reinforcement learning has actually trained them to do.
	To produce text that looks like what is correct, which for many applications is good enough.
	So the error is often really subtle, and then when you spot them, they're really stupid.
	No human would have actually made that mistake.
	Yeah, it's actually really frustrating in the programming context because I program a lot.
	Yeah, when a human makes...
	You can tell.
	You can tell.
	Immediately, like, okay, there's signs.
	But with AI-generate code, and then you're right, eventually you find an obvious, dumb thing that just looks like good code.
	It's very tricky, too, and frustrating for some reason.
	Yeah, so the sense of smell.
	This is one thing that humans have.
	And there's a metaphorical mathematical smell that it's not clear how to get the AIs to duplicate that.
	Eventually, I mean, so the way AlphaZero and so forth make progress on Go and chess and so forth is, in some sense, they have developed a sense of smell for Go and chess positions, that this position is good for white, it's good for black.
	They can't initiate why, but just having 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 you can say, I'm going to try to break up this problem into two small subtasks, and they can say, oh, this looks...
	The two tasks look like they're simpler tasks than your main task, and they've still got a good chance of being true.
	So this is good to try.
	Or you've 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 thing to try.
	It's very easy to transform a problem into an even harder problem.
	Very rarely do you transform it into a simpler problem.
	If they can pick up a sense of smell, then they could maybe start Competing with a human-level mathematician.
	So this is a hard question, but not competing, but collaborating.
	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, what would you like that oracle to be able to do?
	Would you like it to maybe be a verifier, like check, do the codes?
	Yes.
	Professor Tao, this is a promising, fruitful direction.
	Yeah, yeah, yeah.
	Or would you like it to generate possible proofs and then you see which one is the right one?
	Or would you like it to maybe generate different representation, totally different ways of seeing this problem?
	Yeah, I think all of the above.
	A lot of it is, we don't know how to use these tools because it's a paradigm that it's not, It's a bit unreliable in subtle ways, whilst providing sufficiently good output.
	It's an interesting combination.
	You have graduate students that you work with who are kind of like this, but not at scale.
	previous software tools that can work at scale, but very narrow.
	So we have to figure out how to use...
	Yeah, he wrote in his article a hypothetical conversation between a mathematical assistant of the future and himself trying to solve a problem, and they would have a conversation Sometimes the human would propose an idea, and the AI would evaluate it.
	Sometimes the AI would propose an idea, and sometimes a competition was required, and the AI would just go and say, okay, I've checked the 100 cases needed here.
	Or you said this is true for all N, I've checked N up to 100, and it looks good so far.
	hang on, there's a problem that N equals 46. And so just a free-form conversation where you don't know in advance where things are going to go, but just based on, I think ideas could be proposed on both sides, calculations could be proposed on both sides.
	And it's a problem that I already know a solution to.
	So I try to prompt it.
	Okay, so here's the problem.
	I suggest using this tool.
	And it'll find this lovely argument using a completely different tool, which eventually goes into the weeds.
	They say, no, no, no, try using this.
	Okay, and it might start using this.
	And then it'll go back to the tool that it wanted to do before.
	You have to keep railroading it onto the path you want.
	I could eventually force it to give the proof I wanted, but it was like herding cats.
	The amount of personal effort I had to take to not just prompt it, but also check its output, because a lot of what it looked like was going to work, I know there was a problem on 9-17, and basically arguing with it, it was more exhausting than doing it unassisted.
	But that's the current state of the art.
	I wonder if there's a phase shift that happens to where it no longer feels like herding cats, and maybe you'll surprise us how quickly that comes.
	I believe so.
	In formalization, I mentioned before that it takes 10 times longer to formalize a proof than to write it by hand.
	With these modern AI tools, and also just better tooling, the Lean developers are doing a great job.
	Adding more and more features and making it user-friendly.
	It's going from nine to eight to seven.
	Okay, no big deal.
	But one day it will drop a little one.
	And that's the phase shift.
	Because suddenly it makes sense when you write a paper to write it in Lean first.
	Or through a conversation with AI which is generally on the fly with you.
	And it becomes natural for journals to accept.
	Maybe they'll offer expedite refereeing.
	If a paper has already been formalized in Lean, they'll just ask the referee to comment on the significance of the results and how it connects to literature, and not worry so much about the correctness, because that's been certified.
	Papers are getting longer and longer in mathematics, and it's harder and harder to get good refereeing for the really long ones, unless they're really important.
	It is actually an issue, and the formalization is coming at just the right time.
	And the easier and easier it gets because of the tooling and all the other factors, then you're going to see much more math lib will grow potentially exponentially.
	It's a virtuous cycle.
	One facet of this type that happened in the past was the adoption of LaTeX.
	LaTeX is this typesetting language that all mathematicians use now.
	In the past, people used all kinds of word processors and typewriters and whatever.
	At some point, LaTeX became easier to use than all other competitors, and people switched within a few years.
	It was just a dramatic phase shift.
	It's a wild-out-there question, but what year?
	How far away are we from an AI system being a collaborator on a proof that wins the Fields Medal?
	So that level.
	Okay.
	Well, it depends on the level of collaboration.
	No, like it deserves to be, to get the field's medal.
	Like, so half and half.
	Already, like, I can imagine if it was a medal winning paper, having some AI systems in writing it, you know, just, you know, like the old complete alone.
	It's already, I use it, like it speeds up my own writing.
	Like, you know, you can have a theorem and you have a proof and the proof has three cases.
	And I write down the proof of the first case and the autocomplete just suggests that now here's how the proof of the second case could work.
	And like, it was exactly correct.
	That was great.
	Saved me like five, ten minutes of typing.
	But in that case, the AI system doesn't get the Fields Medal.
	No.
	Are we talking 20 years, 50 years, 100 years?
	What do you think?
	Okay.
	So I gave a So not fields metal-winning, but actual research-level math papers.
	Like published ideas that are in part generated by AI?
	Maybe not the ideas, but at least some of the computations, the verifications.
	Has that already happened?
	Has it already happened?
	Yeah, there are problems that were solved.
	By a complicated process, conversing with AI to propose things and the human goes and tries it and the contract doesn't work, but it might pose a different idea.
	It's hard to disentangle exactly.
	There are certainly math results which could only have been accomplished because there was a human mathematician and an AI involved.
	But it's hard to disentangle credit.
	I mean, these tools, They do not replicate all the skills needed to do mathematics, but they can replicate some non-trivial percentage of them, 30-40%.
	They can fill in gaps.
	Coding is a good example.
	It's annoying for me to code in Python.
	I'm not a professional programmer, but with AI, the friction cost of doing it is much reduced.
	So it fills in that gap for me.
	AI is getting quite good at literature review.
	I mean, there's still a problem with hallucinating references that don't exist.
	But this, I think, is a silverware problem.
	If you train in the right way and so forth, you can verify using the internet.
	You should, in a few years, get to the point where you have a And we'll do basically a fancy web search AI assistant and say, yeah, there are these six papers where something similar has happened.
	I mean, you can ask it right now and it'll give you six papers of which maybe one is legitimate and relevant.
	One exists but is not relevant, and four are hallucinated.
	It has a non-zero success rate right now, but there's so much garbage.
	The signal-to-noise ratio is so poor that it's...
	Versus helping you discover new you were not even aware of, but is the correct citation.
	Yeah, that it can sometimes do, but when it does, it's buried in a list of options.
	They're bad.
	Yeah.
	I mean, being able to automatically generate a related work section that is correct.
	Yeah.
	That's actually a beautiful thing that might be another phase shift because it assigns credit correctly.
	It breaks you out of the silos of thought.
	There's a big hump to overcome right now.
	It's like self-driving cars.
	The safety margin has to be really high for it to be feasible.
	There's a last mile problem with a lot of AI applications.
	You know, they can develop tools that work 20%, 80% of the time, but it's still not good enough.
	And in fact, even worse than good in 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.
	You know, real breakthrough, something.
	It doesn't, you know, like, feels metal, even hypothesis.
	It could be, like, really just this alpha zero moment would go, that kind of thing.
	Right, right.
	Yeah, this decade, I can see it, like, making a conjecture.
	Between two things that people thought was unrelated.
	Oh, interesting.
	Generating a conjecture that's a beautiful conjecture.
	Yeah, and actually has a real chance of being correct and meaningful.
	Because that's actually kind of doable, I suppose.
	But where the data is, yeah.
	No, that would be truly amazing.
	The current models struggle a lot.
	I mean, so a version of this is, I mean, the physicists have a dream of getting the AIs to discover new laws of physics.
	The dream is you just feed it all this data, and here is a new pattern that we didn't see before.
	But the current state of the art even struggles to discover old laws of physics from the data.
	Or if it does, there's a big concern about contamination.
	It did it only because somewhere in its training it already somehow knew Boyle's Law or whatever you're trying to reconstruct.
	Part of it is that we don't have the right type of training data for this.
	So for laws of physics, we don't have a million different universes with a million different laws of nature.
	A lot of what we're missing in math is actually the negative space.
	So we have published things of things that people have been able to prove and conjectures that end up being verified or maybe counterexamples produced, but we don't have data on
	There's a trial and error process, which is a real integral part of human mathematical discovery, which we don't record because it's embarrassing.
	We make mistakes and we only like to publish our wins.
	The AI has no access to this data to train on.
	I sometimes joke that basically AI has to go through grad school and actually go to grad courses, do the assignments, go to office hours, make mistakes, get advice on how to correct the mistakes, and learn from that.
	Let me ask you, if I may, about Grigori Perlman.
	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 hasted to add that sometimes this approach actually can be very successful.
	An example you gave is Gregorio Perlman, who proved the Poincaré conjecture and did so by working alone for seven years with basically little contact with the outside world.
	Can you explain this one millennial prize problem that's been solved, Poincare conjecture, and maybe speak to the journey that Gregorio Perlman's been on?
	All right.
	So it's a question about curved spaces.
	Earth is a good example.
	So Earth, you can think of a 2D surface.
	In just being round, there could maybe be a torus with a hole in it, or it could have many holes.
	And there are many different topologies, a priori, that a surface could have, even if you assume that it's bounded and smooth and so forth.
	So we have figured out how to classify surfaces.
	As a first approximation, everything is determined by something called the genus, how many holes it has.
	So a sphere has genus 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 closed loop of rope, you can contract it to a point while staying on the surface.
	And the sphere has this property, but a torus doesn't.
	If you're on a torus and you take a rope that goes around, say, the outer diameter.
	It can't get through the hole.
	There's no way to contract it to a point.
	So it turns out that the sphere is the only surface with this property of contractibility, up to continuous deformations of the sphere.
	So things that are what I call topologically equivalent of the sphere.
	Poincaré asks the same question in higher dimensions.
	It becomes hard to visualize.
	A surface you can think of as embedded in three dimensions, but a curved three space, we don't have good intuition of 4D space to live in.
	And then there are also 3D spaces that can't even fit into four dimensions.
	You need five or six or higher.
	But anyway, 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 was actually easier.
	It was solved first in higher dimensions.
	There's somehow more room to do the deformation.
	It's easier to move things around to a sphere.
	But three was really hard.
	People tried many approaches.
	There's sort of commentary approaches where you chop up the surface into little triangles or tetrahedra, and you just try to argue based on how the faces interact with each other.
	Algebraic approaches, there's various algebraic objects called the fundamental group that you can attach to these homology and chromology and all these very fancy tools.
	They also didn't quite work.
	But Richard Hamilton's proposed a partial differential equations approach.
	The problem is that you have this object which is secretly a sphere, but it's given to you in a really Think of a ball that's been crumpled up and twisted, and it's not obvious that it's a ball.
	If you have some sort of surface which is a deformed sphere, you could think of it as the surface of a balloon.
	You could try to inflate it, blow it up, and naturally, as you fill the air, the wrinkles will smooth out, and it will turn into A nice round sphere.
	Unless, of course, it was a torus or something, in which case it would get stuck at some point.
	If you inflate a torus, there would be a point in the middle.
	When the inner ring shrinks to zero, you get a singularity, and you can't flow any further.
	He created this flow, which is now called Ricci flow, which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder, to make it look like a sphere.
	He wanted to show that either this process would give you a sphere, or it would create a singularity.
	It's very much like how PDEs have either global regularity or finite and blow up.
	Basically, it's almost exactly the same thing.
	It's all connected.
	He showed that for two-dimensional surfaces, if you start to simply connect, no singularity is ever formed.
	You never ran into trouble.
	And you could flow, and it would give you a sphere.
	So he got a new proof of the two-dimensional result.
	Well, by the way, that's a beautiful explanation where we should flow and its application in this context.
	How difficult is the mathematics here, like for the 2D case?
	Yeah, these are quite sophisticated equations on par with the Einstein equations.
	It's slightly simpler, but they were considered hard nonlinear equations to solve.
	And there's lots of special tricks in 2D that helped.
	But in 3D, The problem was that this equation was actually supercritical.
	It's the same problem as Navier-Stokes.
	As you blow up, maybe the curvature could get concentrated in smaller and smaller regions, and it looked more and more nonlinear, and things just looked worse and worse.
	there could be all kinds of singularities that showed up some singularities like if there's these things called neck pinches where the surface sort of Some singularities are simple enough that you can see what to do next.
	You just make a snip, and then you can turn one surface into two, and you build them separately.
	But there was the prospect that some really nasty, knotted singularities showed up that you couldn't see how to resolve in any way, that you couldn't do any surgery to.
	So you need to classify all the singularities.
	What are all the possible ways that things can go wrong?
	First of all, he turned the problem from a supercritical problem to a critical problem.
	I said before about how the invention of energy, the Hamiltonian, really clarified Newtonian mechanics.
	He introduced something which is now called Perlman's reduced volume and Perlman's entropy.
	He introduced new quantities, kind of like energy, that look the same at every single scale.
	And turn the problem into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did before.
	And then he had to solve, he still had to analyze the singularities of this critical problem.
	And that itself was a problem similar to this wave map thing I worked on, actually.
	So 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 resolve the Poincare conjecture.
	A lot of really ambitious steps, and nothing that a large language model today, for example, could.
	At best, I could imagine Mod proposing this idea as one of hundreds of different things to try.
	But the other 99 would be complete dead ends, but you'd only find out after months of work.
	He must have had some sense that this was the right track to pursue because it takes years to get from A to B. So you've done, like you said, actually, even strictly mathematically, but...
	What can you infer from the process he was going through?
	Because he was doing it alone.
	What are some low points in a process like that?
	You've mentioned AI doesn't know.
	When it's failing, what happens to you, you're sitting in your office, when you realize the thing you did for the last few days, maybe weeks, is a failure?
	Well, for me, I switched to a different problem.
	I'm a fox.
	I'm not a hedgehog.
	But you legitimately, that is a break that you can take, is to step away and look at a different problem.
	You can modify the problem, too.
	I mean, you can ask if there's a specific thing that's blocking you.
	Bad case keeps showing up for which your tool doesn't work.
	You can just assume by fiat that this bad case doesn't occur.
	So you do some magical thinking, but strategically to see if the rest of the argument goes through.
	If there's multiple problems with your approach, then maybe you just give up.
	But if this is the only problem and everything else checks out, then it's still worth fighting.
	So yeah, you have to do some forward reconnaissance sometimes.
	And that is sometimes productive?
	To assume like, okay, we'll figure it out eventually.
	Sometimes actually it's even productive to make mistakes.
	There was a project which actually we won some prizes for for other people.
	We worked on this PDE problem.
	Again, actually this blow-off regularity type problem.
	And it was considered very hard.
	Jean Bourguin, who was another field's methodist who worked on a special case of this, but he could not solve the general case.
	We worked on this problem for two months, and we thought we solved it.
	We had this cute argument that if anything fit, we were excited.
	We were planning celebration to all get together and have champagne or something.
	We started writing it up, and one of us, not me actually, but another co-author, said, in this lemma here, We have to estimate these 13 terms that show up in this expansion.
	We estimated 12 of them, but in our notes I can't find the estimation of the 13th, can you?
	Can someone supply that?
	I said, sure, I'll look at this.
	We completely omitted this term.
	This term turned out to be worse than the other 12 terms put together.
	In fact, we could not estimate this term.
	We tried for a few more months, and all different permutations, and there was always this one term that we could not control.
	This was very frustrating.
	But because we had already invested months and months of effort in this already, we stuck at this.
	We tried increasingly desperate things and crazy things.
	And after two years, we found an approach somewhat different, but quite a bit, from our initial strategy, which actually didn't generate these problematic terms and actually solved the problem.
	So we solved the problem after two years.
	But if we hadn't had that initial full storm of nearly solving a problem, we would have given up by month two or something and worked on an easier problem.
	If we had known it would take two years, not sure we would have started the project.
	Sometimes actually having the incorrect.
	It's like Columbus traveling the New World.
	They had an incorrect version of the measurement of the size of the Earth.
	He thought he was going to find a new trade route to India.
	Or at least that was how he sold it in his prospectus.
	I mean, it could be that he secretly knew.
	Just on the psychological element, This stuff, it feels like math is so engrossing that it can break you.
	When you invest so much yourself in the problem and then it turns out wrong, you can start to...
	A similar way chess has broken some people.
	Yeah, I think different...
	I think for some people, it's just a job.
	You have a problem, and if it doesn't work out, you go on the next one.
	The fact that you can always move on to another problem, it reduces the emotional connection.
	There are certain problems that are what are called mathematical diseases, where we just latch on to that one problem, and they spend years and years thinking about nothing but that one problem.
	You know, maybe their career suffers and so forth.
	They say, oh, but I've got this big win.
	Once I finish this problem, that will make up for all the years of lost opportunity.
	Occasionally, occasionally it works, but I really don't recommend it for people who have the right fortitude.
	I've never been super invested in any one problem.
	One thing that helps is that we don't need to call our problems in advance.
	When we do grant proposals, we say we will study this set of problems.
	But even though we don't promise, definitely by five years, I will supply a proof of all these things.
	You promise to make some progress or discover some interesting phenomena.
	Maybe you don't solve the problem, but you find some related problem that you can say something new about.
	That's a much more feasible task.
	But I'm sure for you there's problems like this.
	You have made so much progress towards the hardest problems in the history of mathematics.
	So is there a problem that just haunts you?
	sits there in the dark corners, you know, twin prime conjecture, Riemann hypothesis, Goldbach conjecture.
	Twin prime, that sounds...
	Do you think so?
	Yeah, there's no even viable strategy.
	Even if I activate all the cheats that I know of in this problem, there's still no way to get made a beat.
	I think it needs a breakthrough in another area of mathematics to happen first, and for someone to recognize that it would be useful.
	I think to transport into this problem.
	So we should maybe step back for a little bit and just talk about prime numbers.
	Okay.
	So they're often referred to as the atoms of mathematics.
	Can you just speak to the structure that these atoms provide?
	So the natural numbers have two basic operations attached to them, addition and multiplication.
	So if you want to generate the natural numbers, you can do one of two things.
	You can just start with one and add one to itself over and over again, and that generates you the natural numbers.
	So additively, they're very easy to generate.
	One, two, three, four, five.
	Or you can take the prime numbers, if you want to generate multiplicatively, you can take all the prime numbers, 2, 3, 5, 7, and multiply them all together.
	And together, that gives you all the natural numbers, except maybe for one.
	So there are these two separate ways of thinking about the natural numbers, from an added point of view and a multiplicative point of view.
	And separately, they're not so bad.
	So any question about natural numbers that only involves addition is relatively easy to solve.
	And any question that only involves multiplication is a little bit easier to solve.
	But what has been frustrating is that you combine the two together and suddenly you get an extremely rich There are certain polynomials in some number of variables.
	Is there a solution in the natural numbers?
	And the answer depends on an undecidable statement, like whether the axioms of mathematics are consistent or not.
	But even the simplest problems that combine something more applicative, such as the primes, with something additive, such as shifting by two, separately we understand both of them well.
	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...
	It proves that prime numbers contain arithmetic progressions of any length.
	Which is mind-blowing that you can prove something like that.
	Right.
	So what we've realized because of this type of research is that different patterns have different levels of interstructibility.
	What makes the twin prime problem hard is that if you take all the primes in the world, 3, 5, 7, 11, and so forth, there are some twins in there.
	11 and 13 is a pair of twin primes and so forth.
	But you could easily, if you wanted to, redact the primes to get rid of these twins.
	The twins, they show up, and there are infinitely many of them, but they're actually reasonably sparse.
	Initially, there's quite a few, but once you go out of the millions and trillions, they become rarer and rarer.
	If someone was given access to the database of primes and just edited out a few primes here and there, they could make the Trim Prime Conjecture false by just removing 0.01% of the primes or something.
	Just well chosen to do this.
	And so you could present a censored database of the primes which passes all of the statistical tests of the primes.
	It obeys things like the Paranormal Theorem and other things about the primes, but doesn't contain any Trim Primes anymore.
	And this is a real obstacle for the twin prime conjecture.
	it means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes and so it must be some very um subtle delicate feature of the Okay, so that's out.
	Yeah.
	On the other hand, athletic progressions has turned out to be much more robust.
	You can take the primes, and you can eliminate 99% of the primes, actually.
	And you can take any 99% you want.
	And it turns out, and another thing we proved, is that you still get athletic progressions.
	Athletic progressions are much, you know, they're like cockroaches.
	Of arbitrary length, though.
	Yes, yes.
	That's crazy!
	For people who don't know, arithmetic progressions is a sequence of numbers that differ by some fixed amount.
	It's an infinite monkey type phenomenon.
	For any fixed length of your set, you don't get arbitrary lengths of progressions.
	You only get quite short progressions.
	But you're saying twin prime is not an infinite monkey phenomenon.
	I mean, it's a very subtle monkey.
	It's still an infinite monkey phenomenon.
	Right, yeah.
	If the primes were really genuinely random, if the primes were generated by monkeys, then yes, in fact, the infinite monkey theorem would...
	It doesn't appear random almost.
	Well, we don't know.
	We believe the primes behave like a random set.
	So the reason why we care about the twin prime conjecture is it's a test case for whether we can genuinely, confidently say we have 0% chance of error.
	That the primers behave like a random set.
	Random versions of the primes we know contain twins, at least 100% probably, or probably tending to 100% as you go out further and further.
	The primes we believe that are random.
	The reason why ethnic progressions are indestructible is that regardless of whether your set looks random or looks structured, like periodic, in both cases, ethnic progressions appear.
	But for different reasons.
	And this is basically all the ways in which there are many proofs of these sort of ethnic progression epitheliums, 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.
	But in twin primes, if the primes are random, then you're happy, you win.
	But if the primes are structured, they can be structured in a specific way that eliminates the twins.
	And we can't rule out that one conspiracy.
	And yet, you were able to make a, as I understand, progress on the k-tuple version.
	Right, yeah.
	So, the one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove.
	That, you know, if you believe the world is won by lizards, you say, here's some evidence that it's not won by lizards, but that evidence was planted by lizards.
	you may have encountered this kind of phenomenon yes like a pure like there's almost no way to definitively A conspiracy is solely devoted to eliminating twin primes.
	You have to also infiltrate other areas of mathematics, but it could be made consistent, at least as far as we know.
	But there's a weird phenomenon that you can make one conspiracy rule out other conspiracies.
	So, you know, if the world is run by lizards, they can't also be run by aliens.
	Right.
	So one unreasonable thing is hard to disprove, but more than one, there are tools.
	So, yeah, so for example, we know there's infinitely many primes that are no two, which are, so the infinitely many primes which differ by at most 246, actually, is the current.
	So there's like a bound on that.
	So, like, there's twin primes, there's things called cousin primes that differ by four.
	This thing called sexy primes that differ by six.
	What are sexy primes?
	Primes that differ by six.
	The name is much less exciting than the name suggests.
	Got it.
	So you can make a conspiracy rule out one of these, but once you have 50 of them, it turns out that you can't rule out all of them at once.
	It just requires too much energy somehow in this conspiracy space.
	How do you do the bound part?
	So it's ultimately based on what's called the pigeonhole principle.
	So the pigeonhole principle is the statement that if you have a number of pigeons and they all have to go 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 close together.
	So, for instance, if you have 100 numbers and they all range from 1 to 1,000, two of them have to be at most 10 apart.
	Because you can divide up the numbers from 1 to 100 into 100 pigeonholes.
	Let's say you have 101 numbers.
	If you have 101 numbers, then two of them have to be distanced less than 10 because two of them have to belong to the same pigeonhole.
	So it's a basic feature of a basic principle in mathematics.
	So it doesn't quite work if the primes are regular because the primes get sparser and sparser as you go out.
	that fewer and fewer numbers are private.
	But it turns out that There are numbers that are almost prime, but they don't have no factors at all other than themselves in one.
	They have very few factors.
	It turns out that we understand almost primes a lot better than primes.
	For example, it was known for a long time that there were twin almost primes.
	This has been worked out.
	Almost primes are something we can't understand.
	You can actually restrict attention to a suitable set of almost primes.
	And whereas the primes are very sparse overall, relative to the almost primes, they actually are much less sparse.
	You can set up a set of almost primes where the primes have density like, say, 1%.
	And that gives you a shot at proving, by applying some sort of original principle, that there's pairs of primes that are just only 100 apart.
	But in order to prove the training prime conjecture, you need to get the density of primes inside the almost primes up to a threshold of 50%.
	Once you get up to 50%, you would get trend primes.
	But unfortunately, there are barriers.
	We know that no matter what kind of good set of almost primes you pick, the density of primes can never get above 50%.
	is called The Parody Barrier.
	And I would love to find So one of my long-term dreams is to find a way to breach that barrier.
	Because it would open up not only the Trim-Up conjecture, the Go-Back conjecture, and many other problems in number theory are currently blocked because our current techniques would It's like going past the speed of light.
	Yeah, so we should say a twin prime conjecture.
	One of the biggest problems in the history of mathematics.
	Goldberg conjecture also.
	They feel like next door neighbors.
	Has there been days when you felt you saw the path?
	Oh, yeah.
	Yeah, sometimes you try something and it works super well.
	You again, again, the sense of mathematical smell.
	We talked about it earlier.
	You learn from experience when things are going too well.
	Because there are certain difficulties that you sort of have to encounter.
	I think the way a colleague might put it is that if you are on the streets of New York and you put in a blindfold and you put in a car and after some hours the blindfold is off and you're in Beijing.
	That was too easy somehow.
	There was no ocean being crossed.
	Even if you don't know Exactly what was done.
	You're suspecting that something wasn't right.
	But is that still in the back of your head?
	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.
	I get busy with so many things these days.
	But yeah, when I have free time and I'm too frustrated to work on my sort of real research projects, I can play with these things for fun, and usually you get nowhere.
	You have to learn to just say, okay, fine.
	Once again, nothing happened.
	I will move on.
	Very occasionally, one of these problems I actually solved, or sometimes as you say, you think you solved it and then you throw it for maybe 15 minutes and then you think, I should check this because this is What's your gut say about when these problems will be solved?
	Twin Prime and Gobot?
	Twin Prime, I think we'll keep getting more partial results.
	It does need at least one.
	This parity barrier is the biggest remaining obstacle.
	There are simpler versions of the conjecture where we are getting really close.
	So I think we will, in 10 years, we will have it.
	Many more, much closer results.
	We may not have the whole thing.
	Yeah, so twin trimes is somewhat close.
	Riemann hypothesis, I have no clue.
	I mean, it has to happen by accident, I think.
	So the Riemann hypothesis is a kind of more general conjecture about the distribution of prime numbers, right?
	Right, yeah.
	It states that sort of viewed multiplicatively, like for questions only involving multiplication, no addition, the primes really do behave as randomly as you could hope.
	So there's a phenomenon in probability called square root cancellation.
	If you want to poll, say, America on some issue, and you ask one or two voters, you may have sampled a bad sample, and you get a really imprecise measurement of the full average.
	But if you sample more and more people, the accuracy gets better and better.
	The accuracy improves the square root of the number of people you sampled.
	If you sample 1,000 people, you can get a 2-3% margin error.
	In the same sense, if you measure the primes in a certain multiplicative sense, there's a certain type of statistic you can measure, and it's called the Riemann's data function, and it fluctuates up and down.
	But in some sense, as you keep averaging more and more, if you stumble 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 as with many others in mathematics, we have very few tools to show that something really genuinely behaves.
	Really random.
	And this is actually not just a little bit random, but it's asking that it behaves as random as an actually random set, this square root cancellation.
	And we know, because of things related to the parity problem, that most of us' usual techniques cannot hope to settle this question.
	The proof has to come out of left field.
	But what that is, no one has any serious proposal.
	And there's various ways to sort of, as I said, you can modify the primes a little bit and you can destroy the Riemann hypothesis.
	So it has to be very delicate.
	You can't apply something that has huge margins of error.
	It has to just barely work.
	And there's all these pitfalls that you dodge very adeptly.
	The prime numbers is just fascinating.
	Yeah, yeah, yeah.
	What to you is most mysterious about the prime numbers?
	That's a good question.
	Conjecturally, we have a good model of them.
	As I said, they have certain patterns.
	The primes are usually odd, for instance.
	Apart from these obvious patterns, they behave very randomly.
	There's something called the Kramer random model of the primes.
	After a certain point, primes just behave like a random set.
	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.
	I can tell you with complete certainty that Trin-Pack-Conject is true.
	The random model gives overwhelming odds it is true.
	I just can't prove it.
	Most of our mathematics is optimized for solving things with patterns in them.
	The primes have this anti-pattern, as do almost everything, really.
	But we can't prove that.
	I guess it's not mysterious that the primes are kind of random because there's no reason for them to be To have any kind of secret pattern.
	But what is mysterious is what is the mechanism that really forces the randomness to happen.
	This is just absent.
	Another incredibly surprisingly difficult problem is the collage conjecture.
	Oh yes.
	Simple to state, beautiful to visualize in its simplicity, and yet extremely, Paul Erdar said about the Colossus conjecture that mathematics may not be ready for such problems.
	Others have stated that it is an extraordinarily difficult problem, completely out of reach.
	This is in 2010.
	Out of reach of present-day mathematics, and yet you have made some progress.
	Why is it so difficult to make?
	Can you actually even explain what it is?
	Oh, yeah.
	So it's a problem that you can explain.
	It helps with some visual aids.
	So you take any natural number, like say 13, and you apply the following procedure to it.
	So if it's even, you divide it by 2. And if it's odd, you multiply it by 3 and add 1. So even numbers get smaller, odd numbers get bigger.
	So 13 will become 40, because 13 times 3 is 39, add 1 you get 40. So it's a simple process.
	For odd numbers and even numbers, they're both very easy operations.
	And then you put it together, it's still reasonably simple.
	But then you ask what happens when you iterate it.
	You take the output that you just got and feed it back in.
	So 13 becomes 40. 40 is now even.
	Divide by 2 is 20. 20 is still even.
	Divide by 2 is 10. 5. And then 5 times 3 plus 1 is 16. And then 8, 4, 2, 1. And then from 1 it goes 1, 4, 2, 1, 4, 2, 1. It cycles forever.
	So the sequence I just described, 13, 40, 20, 10, and so forth, these are also called hailstone sequences because There's an oversimplified model of hailstorm formation, which is not actually quite correct, but is somehow taught to high school students as a first approximation.
	A little nugget of ice gets a nice crystal.
	It forms in a cloud, and it goes up and down because of the wind.
	Sometimes when it's cold, it requires a bit more mass, and maybe it melts a little bit.
	This process of going up and down creates this partially melted ice, which eventually causes hellstone, and eventually it falls down to the Earth.
	The conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, this process that goes up if you're odd and down if you're even, eventually goes down to Earth.
	All the time.
	No matter where you start with this very simple algorithm, you end up at 1. And you might climb for a while.
	Right.
	If you plot these sequences, they look like Brownian motion.
	They look like the stock market.
	They just go up and down in a seemingly random pattern.
	And in fact, usually that's what happens.
	If you plug in a random number, you can actually prove, at least initially, that it would look like a random walk.
	And that's actually a random walk with a downward drift.
	It's like if you're always gambling on roulette at the casino with odds slightly weighted against you.
	So sometimes you win, sometimes you lose, but over in the long run, you lose a bit more than you win.
	And so normally your wallet will go to zero if you just keep playing over and over again.
	So statistically, it makes sense.
	Yes.
	So the result that I proved, roughly speaking, is that statistically, like 90% of all inputs would drift down to...
	It's like if I told you that if you go to a casino, most of the time, if you keep playing long enough, you end up with a smaller amount in your wallet than when you started.
	That's kind of like the result that I proved.
	So why is that result?
	Can you continue down that thread to prove the full conjecture?
	Well, the problem is that I used arguments from probability theory, and there's always this exceptional event.
	So in probability, we have these low, large numbers, which tells you things like if you play a game at a casino with a losing expectation, over time, you are guaranteed, almost surely, with probability as close to 100% as you wish, you're guaranteed to lose money.
	But there's always this exceptional outlier.
	It is mathematically possible that even when the game 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, most of the time, 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 a special number that you stick in that shoots off infinity while all other numbers crash to Earth, crash to one.
	In fact, there's some mathematicians, Alex Kontorovich, for instance, who've proposed that Actually, these Kaldats iterations are like this cellular automator.
	If you look at what happened in binary, they do actually look a little bit like these Game of Life-type patterns.
	In an analogy to how the Game of Life can create these massive self-applicating objects, possibly you could create some sort of heavier-than-air flying machine, a number which is actually encoding this machine.
	Whose job it is to encode is to create a version of a cell which is larger.
	Heavier-than-air machine encoded in a number that flies forever.
	Conway, in fact, worked on this problem as well.
	Oh, wow.
	So Conway, so similar, in fact, that was one of my inspirations for the Navi-Stokes project.
	Conway studied generalizations of the collapse problem, where instead of 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 do things like this.
	In fact, he invented a programming language for these kind of fractional linear transformations.
	He got a fact-trat as a play on Fortran, and he showed that you can program You could make a program that 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, and things like that.
	The general class of problems is really as complicated as all the mathematics.
	Some of the mystery of the cellular automata that we talked about.
	Having a mathematical framework to say anything about cellular automata may be the same kind of framework as required.
	Yeah, yeah.
	If you want to do it, not statistically, but you really want 100% of all inputs for the Earth.
	Yeah, so what might be feasible is statistically 99%, you know, go to one.
	But like everything, 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?
	Riemann is up there.
	P equals NP is a good one because that's a meta problem.
	If you solve that 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.
	A lot of stuff is built on top of them now.
	There's ripple effects.
	P equals 1P has more ripple effects than basically any other.
	Right.
	If the Riemann hypothesis is disproven, that would be a big mental shock to the number theorists.
	But it would have follow-on effects for cryptography.
	Because a lot of cryptography uses number theory.
	It uses number theory constructions involving primes and so forth.
	And it relies very much on the intuition that number 3s have built over many, many years of what operations involving crimes behave randomly and what ones don't.
	And in particular, our encryption methods are designed to turn text with information on it into text which is indistinguishable from random noise.
	And hence, we believe to be almost impossible to crack, at least mathematically.
	Something as core to our beliefs is wrong.
	It means that there are actual patterns of the primes that we're not aware of.
	And if there's one, there's probably going to be more.
	And suddenly, a lot of our crypto systems are in doubt.
	Yeah.
	But then, how do you then say stuff about the primes?
	Yeah.
	You're going towards the colex conjecture again.
	Because you want it to be random, right?
	Yes.
	You want it to be random.
	So more broadly, I'm just looking for more tools, more ways to show that things are random.
	How do you prove a conspiracy doesn't happen?
	Is there any chance to you that P equals NP?
	Can you imagine a possible universe?
	It is possible.
	I mean, there's various scenarios.
	I mean, 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 probably P is not equal to NP.
	I mean, it seems like it's one of those cases similar to Riemann hypothesis.
	I think the evidence is leaning pretty heavily on the no.
	Certainly more on the no than on the yes.
	The funny thing about P equals NP is that we have also a lot more obstructions than we do for almost any other problem.
	So while there's evidence, we also have a lot of results ruling out many, many types of approaches to the problem.
	This is the one thing that the computer scientists have actually been very good at.
	It's actually saying that certain approaches cannot work.
	No-go theorems.
	It could be unassailable.
	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, you know, what are you going to do now that you've won this prestigious award?
	And then you just quickly, very humbly said that, you know, this shiny medal is not going to solve any of the problems I'm currently working on.
	So I'm just going to keep working on them.
	First of all, it's funny to me that you would answer an email in that context.
	And second of all, it just shows your humility.
	But anyway, maybe you could speak to the Fields Medal, but it's another way for me to ask about Gregorio Perlman.
	What do you think about him famously declining the Fields Medal and the Millennial Prize, which came with a $1 million 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.
	I've never met him.
	I think I'd be interested to meet him one day, but I never had the chance.
	I know people who met him, but he's always had strong views about certain things.
	It's not like he was completely isolated from the math community.
	He would give talks and write papers and so forth.
	But at some point, he just decided not to engage with the rest of the community.
	He was disillusioned or something.
	I don't know.
	He decided to peace out and collect mushrooms in St. Petersburg or something.
	That's fine.
	You can do that.
	I mean, that's another sort of flip side.
	I mean, a lot of problems that we solve, some of them do have practical application.
	That's great.
	But if you stop thinking about a problem, so he hasn't published since in this field, but that's fine.
	There's many, many other people who've done so as well.
	Yeah, so I guess one thing I didn't realize initially with the Fields Medal is that it sort of makes you part of the establishment.
	Most career mathematicians, you just focus on publishing your next paper, maybe promoting one rank, and starting a few projects, maybe taking some students or something.
	But then suddenly people want your opinion on things, and you have to think a little bit about things that you might just so foolishly say because you know no one's going to listen to you.
	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 play with ideas?
	I have a lot less free time than I had previously.
	I mean, mostly by choice.
	I mean, I can always see I have the option to sort of decline.
	I could decline even more.
	Or I could acquire a reputation being so unreliable that people don't even ask anymore.
	I love the different algorithms here.
	it's always an option.
	But, you know, there are things that are like, I don't spend as much time as I do as a postdoc just working on one problem at a time or fooling around.
	I still do that a little bit.
	But yeah, as you're advancing your career, some of the more soft skills are As a postdoc is published or perish, you're incentivized to basically focus on proving very technical theorems to prove yourself as well as prove the theorems.
	But then as you get more senior, you have to start mentoring and giving interviews and trying to shape Direction in the field, both research-wise and sometimes you have to do various administrative things.
	And it's kind of the right social contract, because you need to work in the trenches to see what can help mathematicians.
	The other side of the establishment, sort of the really positive thing, is that...
	It's just how the human mind works.
	This is where I would probably say that I like the Fields Medal, that it does inspire a lot of young people somehow.
	This is just how human brains work.
	Yeah.
	At the same time, I also want to give sort of respect to somebody like Gregorio Perlman, who...
	Those are his principles.
	And any human that's able for their principles to do the thing that most humans would not be able to do.
	It's beautiful to see.
	Some recognition is necessarily important.
	But yeah, it's also important to not let these things take over your life.
	And only be concerned about getting the next big award or whatever.
	Again, you see these people try to only solve really big math problems and not work on things that are less sexy, if you wish, but actually still interesting and instructive.
	As you say, the way the human mind works, we understand things better when they're attached to humans.
	Also, if they're attached to a small number of humans, the way our human mind is wired, we can comprehend The relationship between 10 or 20 people.
	But once you get beyond 100 people, there's a limit.
	I figured there's a name for it.
	Beyond which, it just becomes the other.
	You have to simplify the pole mass.
	99.9% of humanity becomes the other.
	Often these models are incorrect and this causes all kinds of problems.
	To humanize a subject, if you identify a small number of people, Representative people of the subject, role models, for example.
	That has some role, but too much of it can be harmful.
	I'll be the first to say that my own career path is not that of a typical mathematician.
	I had a very accelerated education.
	I skipped a lot of classes.
	I think I had very fortunate mentoring opportunities, and I think I was at the right place at the right time.
	Just because someone doesn't have my trajectory doesn't mean that they can't be good mathematicians.
	They can be good mathematicians in a very different style, and we need people of a different style.
	Sometimes too much focus is given on the person who does the last step to complete a project in mathematics or elsewhere that's really taken centuries or decades with lots and lots of previous work.
	But that's a story that's difficult to tell.
	If you're not an expert, because it's easier to just say, one person did this one thing.
	It makes for a much simpler history.
	I think on the whole, it is a hugely positive thing 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 They're individual humans on a team, and there's a lot of brilliance there, but it's just a nice shorthand, like pie.
	Steve Jobs.
	Yeah, yeah.
	As a starting point, as a first approximation.
	And then read some biographies and then look into much deeper first approximation.
	Yeah.
	That's right.
	So you mentioned you were a Princeton to Andrew Wiles at that time.
	Oh, yeah.
	He 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 math history?
	I was a graduate student at the time.
	I vaguely remember there was press attention and we all had the same pigeonholes in the same mailroom.
	Suddenly, Andrew Wiles' mailbox exploded to be overflowing.
	That's a good metric.
	We all talked about it at tea and so forth.
	Most of us didn't understand the proof.
	We didn't understand high-level details.
	In fact, there's an ongoing project to formalize it in Lean.
	Can we take that small tangent?
	How difficult is that?
	As I understand, the proof for Fermat's last theorem has super complicated objects.
	It's really difficult to formalize now.
	I guess you're right.
	The objects that they use, you can define them.
	They've been defined in Lean.
	Just defining what they are.
	It can be done.
	That's really not trivial, but it's been done.
	But there's a lot of really basic facts about these objects that have taken decades to prove that they're in all these different math papers.
	And so lots of these have to be formalized as well.
	Kevin Buzzard's goal, actually, he has a five-year grant to formalize film as last year.
	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 number theorists at the time.
	And then some other work would have to be done to get from there.
	So it's a different area of mathematics than the type of mathematics I'm used to.
	In analysis, which is kind of my area, the objects we study are kind of much closer to the ground.
	I study things like prime numbers.
	And functions and things that are within the scope of a high school math education to at least define.
	But then there's this very advanced algebraic side of number theory where people have been building structures upon structures for quite a while.
	And it's a very sturdy structure.
	It's been very, at the base at least, extremely well-developed with textbooks and so forth.
	But it does get to the point where...
	What inspires you about his journey that was similar as we talked about?
	Seven years mostly working in secret.
	Yeah, that is a romantic...
	So it kind of fits with sort of the...
	So that certainly kind of accentuated that perspective.
	It is a great achievement.
	His style of solving problems is so different from my own.
	Which is great.
	We need people like that.
	Can you speak to it?
	Like what, in terms of like the, you like the collaborative...
	But you need the people who have the tenacity and the fearlessness.
	I've collaborated with people like that where I want to give up because the first approach that we tried didn't work and the second one didn't approach.
	They're convinced and they have the third, fourth, and the fifth approach works.
	And I'd have to eat my words.
	Okay, I didn't think this was going to work, but yes, you were right all along.
	And we should say, for people who don't know, not only are you known for the brilliance of your work, but the incredible productivity, just the number of papers, which are all of very high quality.
	So there's something to be said about being able to jump from topic to topic.
	Yeah, it works for me.
	Yeah, I mean, there are also people who are very productive and they focus very deeply on, yeah.
	I think everyone has to find their own workflow.
	One thing which is, A shame in mathematics is that there's a one-size-fits-all approach to teaching mathematics.
	We have a certain curriculum and so forth.
	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 native math language until very late, or usually too late, so they stop doing mathematics, and they have a bad experience with a teacher who's trying to teach them one way to do mathematics and they don't like it.
	My theory is that humans don't come...
	We have a vision center and a language center and some other centers which have evolution as honed, but we don't have an innate sense of mathematics.
	But our other centers are sophisticated enough that we can repurpose other areas of our brain to do mathematics.
	Some people have figured out how to use the visual center to do mathematics, and so they think very visually when they do mathematics.
	Some people have repurposed their language center, and they think very symbolically.
	Some people, if they are very competitive and they like gaming, there's a part of your brain that's very good at solving puzzles and games, and that can be repurposed.
	But when I talk to other mathematicians, I can tell that they're using some different styles of thinking.
	I mean, not disjoint, but they may prefer visual.
	I don't actually prefer visual so much.
	I need also visual aids myself.
	Mathematics provides a common language so we can still talk to each other even if we are thinking in different ways.
	But you can tell there's a different set of subsystems being used in the thinking process.
	They take different paths.
	They're very quick at things that I struggle with and vice versa.
	And yet they still get to the same goal.
	That's beautiful.
	Yeah, but I mean, the way we educate, unless you have a personalized tutor or something, You have to teach the 30 kids.
	If they have 30 different styles, you can't teach 30 different ways.
	On that topic, what advice would you give to students, young students, who are struggling with math but are interested in it and would like to get better?
	Is there something in this complicated educational context?
	Yeah, it's a tricky problem.
	One nice thing is that there are now lots of sources for math faculty enrichment outside the classroom.
	In my day, there were already math competitions.
	There are also popular math books in the library.
	Now you have YouTube.
	There are forums devoted to solving math puzzles.
	Math shows up in other places.
	For example, there are hobbyists who play poker for fun.
	They, for very specific reasons, are interested in very specific probability questions.
	There's a community of amateur probabilists in poker, in chess, in baseball.
	there's meth all over the place um and i'm hoping actually with the with these new sort of tools of This doesn't happen at all currently.
	In the sciences, there's some scope for citizen science.
	Astronomers are amateurs who discover comets.
	Biologists are people who identify butterflies.
	There are a small number of activities where amateur mathematicians can discover new primes.
	But previously, because we had to verify every single contribution, most mathematical research projects, it would not help to have input from the general public.
	In fact, it would just be time-consuming because of error-checking and everything.
	But one thing about these formalization projects is that they are bringing in more people.
	So I'm sure there are high school students who have already contributed to some of these formalizing projects, who contributed to MathLib.
	You know, you don't need to be a PhD holder to just work on one-atopic thing.
	There's something about the formalization here that also, as a very first step, opens it up to the programming community, too.
	Yes.
	The people who are already comfortable with programming.
	It seems like programming is somehow maybe just the feeling, but it feels more accessible to folks than math.
	Math is seen as this extreme, especially modern mathematics, seen as this extremely difficult You can execute code and you can get results.
	You can print a whole other world pretty quickly.
	If programming was taught as an almost entirely theoretical subject, where you're just taught the computer science, the theory of functions and routines and so forth, and outside of some very specialized homework assignments, you're not actually programmed on the weekend for fun.
	It would be considered as hard as math.
	There are communities of non-mathematicians where they're deploying math for some very specific purpose, like optimizing their poker game.
	And for them, then math becomes fun for them.
	What advice would you give in general to young people how to pick a career, how to find themselves?
	That's a tough, tough, tough question.
	There's a lot of certainty now in the world.
	There was this period after the war where, at least in the West, if you came from a good demographic, there was a very stable path to a good career.
	You go to college, you get an education, you pick one profession and you stick to it.
	It's becoming much more a thing of the past.
	I think you just have to be adaptable and flexible.
	I think people have to get skills that are transferable.
	Learning one specific Programming language or one specific subject of mathematics or something.
	That itself is not a super transferable skill, but knowing how to reason with abstract concepts or how to problem-solve when things go wrong, these are things which I think we will still need.
	Even as our tools get better, you'll be working with AIs more and so forth.
	Actually, you're an interesting case study.
	You're 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.
	First of all, you kept learning new fields, but you learned lean.
	That's a non-trivial thing to learn.
	For a lot of people, that's an extremely uncomfortable leap to take, right?
	A lot of mathematicians.
	First of all, I've always been interested in new ways to do mathematics.
	I feel like a lot of the ways we do things right now are Inefficient.
	Me and my colleagues 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.
	Why can't we search and get a quick response?
	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.
	At the time, Peter Schultzer had just formalized one of his new theorems.
	There were some other developments in computer-assisted proof that looked quite interesting.
	I said, oh, we should run a workshop on this.
	This would be a good idea.
	I was a bit too enthusiastic about this idea, so I got voluntold to actually run it.
	I did with a bunch of other people, Kevin Buzzard and Jordan Ellenberg and a bunch of other people.
	And it was a nice success.
	We brought together a bunch of mathematicians and computer scientists and other people, and we got up to speed on the state of the art.
	And it was really interesting developments that most mathematicians didn't know was going on.
	Lots of nice proofs of concept.
	Just hints of what was going to happen.
	This was just before ChatGBT, but even then there was one talk about language models and the potential capability of those in the future.
	So that got me...
	Then ChatGPT came out, and suddenly AI was everywhere.
	I got interviewed a lot about this topic, and in particular, the interaction between AI and formal proof assistants.
	I said, yeah, they should be combined.
	There's a perfect synergy to happen here.
	And at some point, I realized that I have to actually do not just talk the talk, but walk the walk.
	I don't work in machine learning, and I don't work in proof-formization.
	There's a limit to how much I can just rely on authority and say, I'm a warm-up mathematician.
	Just trust me when I say that this is going to change mathematics, and I don't do any of it myself.
	so I felt like I had to actually justify it.
	A lot of what I get into, actually, I don't quite see in advice as how much time I'm going to spend on it.
	And it's only after I'm sort of waist-deep in a project that I realized by that point out or have some of the sort of challenges that a beginner would, right?
	So new concepts, new ways of thinking, also, you know, sucking at a thing that others...
	I think in that talk, you could be a field-metall winning mathematician and an undergrad knows something better than you.
	Yeah, I think mathematics inherently...
	And inevitably, we make mistakes.
	You can't cover up your mistakes with just bravado, because people will ask for your proofs, and if you don't have the proofs, you don't have the proofs.
	I love math.
	It does keep us honest.
	It's not a perfect panacea, but I think we do have more of a culture of admitting error, because we're forced to.
	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.
	Who are the candidates?
	Euler, Gauss, Newton, Ramanujan, Hilbert?
	First of all, as I mentioned before, there's some time dependence.
	On the day.
	If you pop cumulatively over time, for example, Euclid is one of the contenders.
	And then maybe some unnamed anonymous mathematicians before that.
	Whoever came up with the concept of numbers.
	Do mathematicians today still feel the impact of Hilbert?
	Just directly of everything that's happened in the 20th century?
	Yeah, Hilbert spaces.
	We have lots of things that are named after him, of course.
	Just the arrangement of mathematics and just the introduction of certain concepts.
	I mean, 23 problems have been extremely influential.
	There's some strange power to the declaring which problems are hard to solve, the statement of the open problems.
	Yeah, I mean, there's this bystander effect everywhere.
	If no one says you should do X, everyone just mills around waiting for somebody else to do something, and nothing gets done.
	One thing that actually you have to teach undergraduates in mathematics is that you should always try something.
	A lot of paralysis in an undergraduate trying a math problem.
	If they recognize that there's a certain technique 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 think there's a quote from The Simpsons, I've tried nothing and I'm all out of ideas.
	So, you know, the next step then is to try anything, no matter how stupid.
	In fact, it's almost the stupider of the better.
	A technique which is almost guaranteed to fail, but the way it fails is going to be instructive.
	It fails because you're not at all taking into account this hypothesis.
	Oh, this hypothesis must be useful.
	That's a clue.
	I think you also suggested somewhere this fascinating approach, which really stuck with me.
	I started using it and it really works.
	I think you said it's called structured procrastination.
	No, yes.
	It's when you really don't want to do a thing.
	Do you imagine a thing you don't want to do more?
	Yes.
	That's worse than that.
	And then in that way, you procrastinate by not doing the thing that's worse.
	Yeah, yeah.
	It's a nice hack.
	It actually works.
	Yeah, yeah.
	I mean, with anything.
	Psychology is really important.
	You talk to athletes like marathon runners and so forth, and they talk about what's the most important thing.
	Is it the training regimen or the diet?
	So much of it is psychology.
	Just tricking yourself to think that the problem is feasible so that you're motivated to do it.
	Is there something our human mind will never be able to comprehend?
	Well, as a mathematician, I mean, it's a reduction.
	There must be a large number that you can't understand.
	That was the first thing that came to mind.
	So that, but even broadly, is there something about our mind that we're going to be limited, even with the help of mathematics?
	Well, okay.
	How much augmentation are you willing?
	For example, if I didn't even have pen and paper, If I had no technology whatsoever, so I'm not allowed blackboard, pen and paper.
	You're already much more limited than you would be.
	Incredibly limited.
	Even language, the English language is a technology.
	It's one that's been very internalized.
	So you're right, the formulation of the problem is incorrect because there really is no longer just a solo human.
	We're already augmented and extremely complex Is it really like a collective intelligence?
	Yes, I guess.
	So humanity, plural, has much more intelligence, in principle, on its good days, than the individual humans put together.
	It can all have less.
	The mathematical community, plural, is an incredibly super-intelligent entity.
	That no single human mathematician can come close to replicating.
	You see it a little bit on these question-and-answer sites.
	So there's Math Overflow, which is the math version of Stack Overflow.
	And sometimes you get very quick responses to very difficult questions from the community.
	And it's a pleasure to watch, actually, as an expert.
	I'm a fan spectator of that site, just seeing the brilliance of the different people, the depth of knowledge that some people have.
	And the willingness to engage in the rigor and the nuance of the particular question is pretty cool to watch.
	It's almost like just fun to watch.
	What gives you hope about this whole thing we have going on, human civilization?
	I think the younger generation is always really creative and enthusiastic and inventive.
	It's a pleasure working with young students.
	The progress of science tells us that the problems that used to be really difficult can become trivial to solve.
	It was like navigation.
	Just knowing where you were on the planet was this horrendous problem.
	People died or lost fortunes because they couldn't navigate.
	We have devices in our pockets that do this automatically for us.
	It's a completely solved problem.
	Seem unfeasible for us now.
	Could be maybe just homework exercises.
	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?
	Because in the next 200 years, just imagine showing up in 200 years.
	Yeah, well, already plenty has happened.
	If you could go back in time and talk to your teenage self or something, you know what I mean?
	Just the internet and now AI.
	I mean, again, they're beginning to be internalized.
	And so, yeah, of course an AI can understand our voice and give reasonable answers.
	slightly incorrect answers to any question, but this was mind-blowing even two years ago.
	And in the moment, it's hilarious to watch on the internet and so on, the drama, people take everything for granted very quickly, Out of anything that's created, somebody needs to take one opinion, another person needs to take an opposite opinion and argue with each other about it.
	But when you look at the arc of things, I mean, it's just even in 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 the culture is healthy, the community of humans can be so much better 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, of course, for putting those ideas out on the blog.
	And I can't tell you how honored I am that you would spend your time with me today.
	I was looking forward to this for a long time.
	Terry, I'm a huge fan.
	You inspire me.
	You inspire millions of people.
	Thank you so much for talking.
	Thank you.
	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 lexfriedman.com slash 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.

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