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

following is a conversation with Terence Tao, widely considered to be one of the greatest mathematicians in history, often referred to as the Mozart of math.

He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics.

This was a huge honor for me for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions.

It means the world.

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Alright, let's go.

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Our future, friends, has a lot of robots in it.

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All of that.

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You know, I run along the river often and get to meet some really interesting people.

One of the people I met was preparing for his first ultra marathon.

I believe he said it was 100 miles.

And that of course sparked in me the thought that I need for sure to

do one myself.

Some time ago now, I was planning to do something with David Goggins, and I think that's still on the sort of to-do list between the two of us, to do some crazy physical feat.

Of course, the thing that is crazy for me is a daily activity for Goggins.

But nevertheless, I think it's important in the physical domain, the mental domain, and

all domains of life to challenge yourself.

And athletic endeavors is one of the most sort of crisp, clear,

well-structured ways of challenging yourself.

But there's all kinds of things.

Writing a book,

to be honest, having kids and marriage and relationships and friendships, all of those, if you take it seriously, if you go all in and do it right, I think that's a serious challenge.

Because most of us are not prepared for it.

And you learn along the way.

And if you have the rigorous feedback loop of improving constantly and growing as a person and really doing a great job of the thing, I think

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I'm preparing for a conversation on

drugs in the Third Reich.

And funny enough, it's a kind of way to analyze Hitler's biography.

It's to look at what he consumed throughout.

And Norman Ohler does a great job of analyzing all of that and tells the story of Hitler and the Third Reich in a way that hasn't really been touched by historians before.

It's always nice to look at key moments in history through a perspective that's not often taken.

Anyway, I mentioned that because I think Hitler had a lot of stomach problems and so that was the motivation for getting a doctor, the doctor that eventually

would fill him up with all kinds of drugs.

But the doctor earned Hitler's trust by giving him probiotics, which is a kind of revolutionary thing at the time.

And so that really helped deal with whatever stomach issues that Hitler was having.

All of that is a reminder that war is waged by humans and humans are biological systems and biological systems require fuel and supplements and all of that kind of stuff.

And depending on what you put in your body, will affect your performance in the short term, in the long term.

With meth, that's true with Hitler to his last days in the bunker in Berlin.

All the cocktail of drugs that he was taking.

So

I think I got myself somewhere deep, and I'm not sure how to get out

of this.

It deserves a multi-hour conversation versus a few seconds of mention.

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

What was the first really difficult research level math problem that you encountered?

One that gave you pause, maybe?

Well, I mean, in your undergraduate education, you learn about the really hard impossible problems, like the Riemann hypothesis, between Prime's 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 relatively easily and what are hopeless.

But what are problems where

existing techniques can do like 90% of the job and then you just need that remaining 10%?

I think as a PhD student, the KAR 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, Soichi Kakea, in like 1918 or so.

So the puzzle is that you have

a needle

on the plane, or think like driving

on a road, something.

And 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 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 its as the unit needle.

You can spin it around its center.

And I think that gives you a disc of area, I think, pi over four.

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 eight.

So it's a little bit more efficient than a rotation.

And so for a while, people thought that was the most efficient way to turn things around.

But Vazikovich showed that, in fact, you could actually turn the needle around using as little area 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.

Yeah.

So we understand everything in two dimensions.

So the next question is what happens in three dimensions?

So suppose like the Hubble Space Telescope is tube in space and you want to observe every single star in the universe.

So you want to rotate the telescope to reach every single direction.

And here's 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 Besakovich'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 logarithmically,

roughly speaking.

And that was proved after a lot of work.

So this seems like a puzzle.

Why is it interesting?

So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, combinatorics.

For example, in wave propagation, if 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 in both space and time, it occupies a region which looks like a tube.

And so what can happen is that you can have a wave which initially is very dispersed, but it all focuses at a single point later in time.

Like you can imagine dropping a pebble into a pond and the 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 sort of light rays.

So like if this wave represents light, for example, you can imagine this wave as the 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 the conjecture had a negative solution, 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

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 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 non-linear.

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, 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 on the bathtub, it won't explode on you

or have water leaving at the speed of light or anything.

But potentially, it is possible.

And 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 as 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.

They have the Poincaré conjecture at Perlman.

So the Kiker conjecture is not directly, directly related to the Navi-Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navier Stokes problem better.

Can you speak to the Navier-Stokes?

So, the existence and smoothness, like you said, millennial prize problem.

Right.

You've made a lot of progress on this one.

In 2016, you published a paper, Finite Time Blow-Up for an Averaged Three-Dimensional Navier-Stokes equation.

Right.

So, we're trying to figure out if this thing usually doesn't blow up.

Right.

But can we say for sure it never blows up?

Right.

Yeah.

So, yeah, that is literally the million-dollar question.

Yeah.

So, this is what distinguishes mathematicians from pretty much everybody else.

Like if

something holds 99.99% of the time, that's good enough for most, you know, for

most things.

But mathematicians are one of the few people who really care about whether

really 100% of all situations are covered by.

Yeah.

So most fluids, 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 this is

a set of equations that govern in the field of fluid dynamics.

Yes.

Trying to understand how fluid behaves.

And it's actually turns out to be a really complicated, you know, fluid is

an extremely complicated thing to try to model.

Yeah, so it has practical importance.

So this high-price problem concerns what's called the incompressible Napier-Stokes, which governs things like water.

There's something called the compressible Napier-Stokes, which governs things like air.

And that's particularly important for weather prediction.

Weather prediction, it does a lot of computational fluid dynamics.

A lot of it is actually just trying to solve the Napier-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.

Why is it difficult to prove general things

about the set of equations like it not blowing up?

Short answer is Maxwell's demon.

So Maxwell's demon is a concept in thermodynamics.

Like if you have a box of two gases and 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, right?

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 that

maybe there's a microscopic demon called Maxwell's demon that will, every time an oxygen and nitrogen atom collide, they will bounce up in such a way that the oxygen sort of drifts onto one side and the nitrogen goes to the other.

And you could have an extremely improbable configuration emerge, which we never see.

And

statistically, it's extremely unlikely.

But mathematically, it's possible that this can happen.

And we can't rule that 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 ones and twos and threes as fours and fives and sixes.

There should be no preference.

in the digits of pi to favor, let's say, seven over eight.

But maybe there's some demon in the the digits of pi that, like, every time you compute more and more digits, it sort of biases one digit to another.

And

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.

Okay, so getting back to Nabi-Stokes, a fluid has a certain amount of energy.

And 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.

And it will move faster and faster.

And at faster speeds, the effect of viscosity is relatively less.

And so it could happen that it creates some sort of

what's called a self-similar blow-up scenario, where

the energy of the fluid starts off at some large scale, and then it all sort of transfers its energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth.

And each time it does this, it takes maybe half as long as the previous one.

And then

you could actually converge to all the energy concentrating in one point in a finite amount of time.

And

that scenario is called Fannerheim blow-up.

So in practice, this doesn't happen.

So water is what's called turbulent.

So 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 must split up into maybe three or four small eddies of their own.

And so the energy gets dispersed to the point where the viscosity can 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 the splow-up can occur.

So there are 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 in 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 Pinotheim Blower, 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 Pinotheim Blower.

I couldn't do it for the actual equations of fluors, which were too complicated.

But if I could average the equations of motion of an everyday

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 transfers 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, 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 the 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

in finite time.

So 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.

So it rules out

certain approaches.

So

the thing about math is it's not just about finding, you know, 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.

How deeply connected is it to 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 if viscosity was all there was then then nothing bad would ever happen.

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 non-linear effect, and that causes all the problems.

So there are these two competing terms in the Davios-Hokes 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

we don't know what's going on.

It's a very non-linear situation.

It's unpredictable.

It's turbulent.

So sometimes these forces are in balance at small scales, but not in balance at large scales, or vice versa.

So Navier-Sokes is what's called supercritical.

So at smaller and smaller scales, the transport terms are much stronger than the viscosity terms.

So the viscosity terms are the things that calm things down.

And so

this is why the problem is hard.

In two dimensions, so the Soviet mathematician Ladyshinskaya, she in the 60s shows in two dimensions there is is no blow-up.

And in two dimensions, the Navier Sukhus equations is what's called critical.

The effect of transport and the effect of viscosity about the same strength, even at very, very small scales.

And we have a lot of technology to handle critical and also subcritical equations and 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 from being sort of nice and predictable.

And, you know, like planetary motion.

And 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 a supercritical equation.

Lots of really strange things are going on at very fine scales.

So whenever there is some huge source of nonlinearity,

that can create a huge problem for predicting what's going to happen.

Yeah, and if the nonlinearity is somehow more and more featured and interesting at 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, for 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 just basically, you can almost approximate these planets by point masses.

And it's 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 finance calculum 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

sort of construct a liquid, a kind of liquid computer.

Right.

And then show that the halting problem from computation theory has consequences for fluid dynamics.

So

show it in that way.

Can you describe this?

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 sort of this naive way to do it.

You just keep pushing,

every time you get energy at one scale, you push it immediately to the next scale, as fast as possible.

This is sort of the naive way

to force blow up.

It turns out in five and high dimensions, this works.

But in three dimensions, there was this funny phenomenon that I discovered that if you keep

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 direct approach 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.

So

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 sort of opened the next gate and then you push that in as well.

So 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.

And 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

you know, like it was constructing like an electronic circuit.

So I actually thanked my wife for this because she was trained as an electrical engineer.

And,

she talked about

she had to design circuits and so forth.

And 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.

And you have to build a diagram.

And these diagrams, you can sort of follow it 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'll be a clock that would, and then once the clock hits a certain threshold, it would close it.

Kind of a Rude Goldberg type machine, but described mathematically.

And this ended up working.

So what I realized is that if you could pull the same thing off for the actual equations, so if the equations of water supported computation, so

like if you can imagine kind of a steampunk, but it's really water punk type of thing where,

you know, so modern computers are electronic, you know, 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, and maybe they're two different configurations corresponding to a bit being up or down.

Probably, if you had two of these moving bodies of water collide,

it would come out with some new configuration, which is which would be something like an AND gate or OR gate.

The output would depend in a very predictable way on the inputs.

And 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 of 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.

So if you could build a fluid machine, which

yeah, so

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 configuration water would transfer all its energy into the smaller configuration and then power down.

And then

clean itself up.

And then what's left is this newest state, which would then turn on and do the same thing, but smaller and faster.

And then the equation has a certain scaling symmetry.

Once you do that, it can just keep iterating.

So this, in principle, would create a 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,

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 Naviaswicks 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 Nario-Stokes to this Touring machine.

So it goes from what, the self-similar blob scenario that you're trying to get the smaller and smaller blob to now having a liquid Touring machine gets smaller and smaller and 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 think of, what you might think of as completely different

problems.

But if the mathematical form is the same,

you can draw a connection.

So

there's a lot of work previously on what's called cellular automata.

The most famous of which is Conway's Game of Life.

There's this infinite discrete grid, and at any given time, the grid is either occupied by a cell or it's empty.

And there's a very simple rule that tells you how these cells evolve.

So sometimes cells live, and sometimes they die.

And

when I was a student, it was a very popular screensaver to actually just have these animations going.

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 at a certain direction.

And that's like this vortex rings.

Yeah, so this is an analogy.

The game of life is kind of like a discrete equation and

the Navier Stokes is a continuous equation.

But mathematically, they have some similar features.

And so 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 like 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 or spit out gliders

one at a time.

And then 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 glider coming out.

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

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 polynomial 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, which could replicate.

It's so incredible.

A lot of this was like community crowdsourced by amateur mathematicians, actually.

So I knew about that work.

And so 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 in the game of life and plump them in.

But again, it shows it's possible.

You know, there's a kind of 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 we have the tools to say something profound about that?

The thing is, you can get this emergent very complicated structures, but only with very carefully prepared initial conditions.

So

these glider guns and gates and software machines, if you just plunk on randomly some cells and you unlink it in the left, you will not see any of these.

And that's the analogous situation with Navier-Stokes again, that with typical initial conditions, you will not have any of this weird computation going on.

But basically through through engineering,

by specially designing things in a very special way, you can pick clever constructions.

I wonder if it's possible to prove the sort of the negative of like,

basically prove that only through engineering can you ever create

something interesting.

This is a recurring challenge in mathematics that

I call it the dichotomy between structure and randomness.

That most objects that you can generate in mathematics are random.

They look like random, like the digits of pi, well, we believe is a good example.

But there's a very small number of things that have patterns.

But now, you can prove something has a pattern by just constructing, you know, like if something has a simple pattern and you have a proof that it does something like repeat itself every so often, you can do that.

But

and you can prove that that, for example, you can prove that most sequences of digits have no pattern.

So like if you just pick digits randomly, there's something called the low large numbers.

It tells you you're going to get as many ones as twos in the long run.

But

we have a lot fewer tools to, if I give you a specific pattern like the digits of pi, how can I show that this doesn't have some weird pattern to it?

Some other work that I spend a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is very structured.

So some functions are what's called additive.

Like if you have a function that maps the natural numbers, they're natural numbers.

So maybe

two maps to four, three maps to six, and so forth.

Some functions are what's called additive, which means that

if you add two inputs together, the output gets added as well.

For example, I'm multiplying 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 then adding them together.

So some

functions are additive.

Some functions are kind of additive, but not completely additive.

So for example, if I take a number n, I multiply by the square root of 2, and I take the integer part of that.

So 10 by square root of 2 is like 14 point something, so 10 up to 14.

20 went 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 of 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 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 it's basically

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 sort of partial pattern that you have.

And so, if you have these sort inverse theorems,

it creates this sort of dichotomy that either

the objects that you study are either 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 Zemerdi's theorem, proven in the 1970s.

It concerns trying to find a certain type of pattern in a set of numbers.

The pattern is arithmetic progression.

Things like 3, 5, and 7, or 10, 15, and 20.

And Zemeri, Andre Zamerdi, proved that any set of numbers that are sufficiently big,

what's called positive density, has arithmetic progressions in it of any length you wish.

So for example, the odd numbers have a set of density one half,

and they contain arithmetic progressions of any length.

So 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 Zemoni theme also applies to random sets.

If I take the set of odd 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 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

arithmetic progressions in that set.

Can you prove that

there's arithmetic progressions of arbitrary length within a random...

Yes.

Have you heard of the infinite monkey theorem?

Usually mathematicians give boring names to theorists, but occasionally they give colorful names.

The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each with a typewriter, they type out text randomly, almost surely one of them is going to generate the entire score 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.

So

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, athletic progressions of any length will eventually happen.

Okay, but you need that you but you need an extremely long random sequence for this to happen.

I suppose that's intuitive.

It's just infinity.

Yeah, infinity absorbs a lot of sins.

Yeah.

How are we humans supposed to deal with infinity?

Well, you can think of infinity as an abstraction of

a finite number of which you do not have a bound for.

So nothing in real life is truly infinite.

But, you know, you can,

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.

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.

So, you know, we spend a lot of time in our undergraduate math classes teaching analysis.

And analysis is often about how to take limits

and whether you, you know, 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 finitizing them.

So you know that something's true eventually, but you don't know when.

Now give me your rate.

Okay, so such that 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?

And that's a more quantitative question.

And this is something that you can attack by purely finite methods and you can use your finite intuition.

And 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'd 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 finitize 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 finitize so that you can have some intuition.

Yeah.

The downside is that the finitized groups are just much, much messier.

And

yeah, so the infinite ones are found first, usually, like decades earlier.

And then later on, people finetize them.

So since we mentioned a lot of math and a lot of physics, what do you use the difference between mathematics and physics as disciplines, as ways of understanding, of seeing the world?

Maybe we can throw an engineering in there.

You mentioned your wife is an engineer, give it new perspective on circuits.

So this different way of looking at the world, given that you've done mathematical physics, so

you'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 of the real world, our observations, and then our mental models as to how we think the world works.

So

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 is

we stay within the model and we ask what are the consequences of that model?

What predictions would the model make of future observations or past observations just does it fit observed data um so there's definitely a symbiosis um

it's mathemat i guess mathematics is is unusual among other disciplines is that we start from hypotheses like the axioms of a model and ask what conclusions come up from that model um in almost any other discipline uh you start with the conclusions you know i want to do this i want to build a bridge you know i want to to make money i want to do this okay and then you you you find the paths to get there.

There's a lot less sort of speculation about suppose I did this, what would happen?

You know, 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.

I mean, they want to know, you know, where is this asteroid going to go?

You know,

what is the weather going to be tomorrow?

But

mathetics also has this other direction of going from the axioms.

What do you think?

There is this tension in physics between theory and experiment.

What do you think is the more powerful way of discovering truly novel ideas about reality?

Well, you need both, top-down and bottom-up.

Yeah,

it's a real interaction between all these things.

So over time, the observations and the theory and the modeling should both get closer to reality.

But initially, and

I mean,

this is always the case.

They're always far apart to begin with.

But you need one to figure out where to push the other.

So 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.

So

it goes back and forth.

Within mathematics itself, there's also a theory and experimental component.

It's just that until very recently, theory has dominated almost completely.

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, well, like Gauss, for example, he discovered, he conjectured the most basic theorem in number theory to call the prime number theorem, which predicts how many primes are up to a million, up to a trillion.

It's not an obvious question.

And basically what he did was that he computed,

I mean, mostly

by himself, but also hired human computers, people whose professional job it was to do arithmetic,

to compute the first 100,000 fragments 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 to study, for example, Zambia's theorem, you want to study all possible subsets of the numbers 1 to 1,000.

There's only 1,000 numbers.

How bad could it be?

It turns out the number of different subsets of 1 to 1,000 is 2 to the power 1,000, which is way bigger than any computer can currently

ever

enumerate.

So you have to be...

there are certain math problems that very quickly become just intractable to attack by direct brute force computation.

Chess is another famous example.

There's a 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.

So like we can empirically solve chess now, for example.

we have very, very good AIs that can, they don't explore every single position in the game tree, but they have found some very good approximation.

And people are using actually these chess engines to make, to do experimental chess.

They're revisiting old chess theories about, oh, you know, when you, 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

commercial 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, and you can intuit as a human being.

And then from that, we humans can construct a theory of the matter.

You've mentioned the Plato's cave allegory.

So in case people don't know, it's where people are observing shadows of reality, not reality itself, and they believe what they're observing to be reality.

Is that in some sense what mathematicians and maybe all humans are doing is

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.

And the process of getting closer often means that you have to discard your initial intuitions.

So

astronomy provides great examples, you know, like, you know, like

an initial model of the world is flat because it looks flat, you you know and um

and that it's and it's big you know and the rest of the universe the skies is not you know like the sun for example looks really tiny

and so you start off with a model which is actually really far from reality

but it fits kind of the observations that you have um you know so you know so things look good you know 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, solar system goes around the galaxy, and so on and so forth.

And the gas part of the universe, you know, expanding,

the expansions are self-expanding, accelerating.

And in fact, very recently, in this year, so this even the acceleration of the universe itself is this evidence now is non-constant.

And the explanation behind why that is

catching up.

It's catching up.

I mean, it's still, you know, the dark matter, dark energy, this kind of thing.

Yes.

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.

But so people say, oh, that's fudge factors.

With enough fudge factors, you can explain anything.

But the mathematical point with 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 if you have a model with

two parameters and it explains a trillion observations, which is basically, so yeah, the dark matter model, I think it has like 14 parameters and it explains petabytes of data

that the astronomers have.

You can think of a theory, like one way to think about a physical mathematical theory, a theory, is it's a compression of the universe, and a data compression.

So you have these petabytes of observations, you'd like to compress it to a model which you can describe in five pages and specify a certain number of parameters, and if it can fit to reasonable accuracy, you know, almost all of your observations.

I mean, the more compression that you make, the better your theory.

In fact, one of the great surprises of our universe and of everything in it is that it's compressible at all.

That's the unreasonable effectiveness of mathematics.

Einstein had a quote like that.

The most incompressible thing about the universe is that it is comprehensible.

Right.

And not just comprehensible.

You can do an equation like E equals M C squared.

There is actually some mathematical possible explanation for that.

So there's this phenomenon in mathematics called universality.

So many complex systems at the macro scale are coming out of lots of tiny interactions at the macro scale.

And normally because of the common law of explosion, you would think that the macro scale equations must be like infinitely, exponentially more complicated than

the micro scale ones.

And they are if you want to solve them completely exactly.

Like if you want to model

all the atoms in a box of air, That's like Abergadro's number is humongous, right?

There's a huge number of particles.

If you actually have to track each one, it'll be ridiculous.

But certain laws emerge at the microscopic scale that almost don't depend on what's going on at the micro scale, or only depend on a very small number of parameters.

So if you want to model a gas of

quintillion particles in a box, you just need to know its temperature and pressure and volume and a few parameters, like five or six, and it models almost everything you need to know about these 10 to the 23 or whatever particles.

We don't understand universality anyway 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 now even a meme with this curve.

And even the meme applies broadly.

There's universality to the meme.

Yes, you can go meta if you like.

But there are many, many processes.

For example, you can take 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 bow curves, these Gaussians, 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 bow curve show up.

And this is also important to know when the situation fails.

So universality is not a 100% reliable thing to rely on.

That

The global financial crisis was a famous example of this.

People thought that mortgage defaults

had this sort of Gaussian type behavior, that if you ask, if you have a population of

100,000 Americans with mortgages, you ask what proportion of them will default in the mortgages.

If everything was decorrelated, it could be a nasty bill curve and

you can manage risk with options and derivatives and so forth.

And

it is a very beautiful theory.

But if there are systemic shocks in the economy that can push everybody to default at the same time, that's very non-gausing behavior.

And this wasn't fully accounted for in 2008.

Now I think there's some more awareness that this is a systemic risk is actually a much bigger issue.

And just because the model is pretty and nice, it may not match reality.

So the mathematics of working out what models do is 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 limit theorems, it told you that if you have certain axioms like non-correlation, that if all the inputs were not correlated to each other,

then you have these Gaussian behavior that things are fine.

It tells you where to look for weaknesses in the model.

So

if you have a mathematical understanding of the central limit theorem and someone proposes to use these Gaussian copyrows 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?

And so

then you can ask the economists, you know,

how much of a risk is that?

And then you can go look for that.

So 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.

Good friend.

But anyway, so you are known for this ability to go both deep and broad in mathematics.

So you're the perfect person to ask: do you think there are threads that connect all the disparate areas of mathematics?

Is there a kind of deep underlying structure

to all of mathematics?

There's certainly a lot of connecting threads and a lot of the progress of mathematics can be represented by taking by stories of two fields of mathematics that were previously not connected and finding connections.

An ancient example is

geometry and number theory.

So in the times of the ancient Greeks, these were considered different subjects.

I mean, mathematicians worked on both.

You could

work both on geometry most famously, but also on numbers.

But they were not really considered related.

I mean, a little bit like, you know, you could say that this length was five times this length because you could take five copies of this length and so forth.

But it wasn't until Descartes, who really realized that, who developed what we now call analytic geometry, that

you can parameterize the plane, a geometric object, by

two real numbers.

Every point can be.

And so geometric problems can be turned into problems about numbers.

And

today, this feels almost trivial.

There's no content to this.

Like, 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.

And this process has just gone on throughout mathematics over and over again.

Algebra and geometry were separated, and now we have a spirit algebraic geometry that connects them and over and over again.

And that's certainly the type of mathematics that I enjoy the most.

So I think there's sort of different styles to being a mathematician.

I think hedgehogs and fox, 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 are kind of who can play both roles.

And I think ideal collaboration between mathematicians involves a very, you need some diversity, like

a fox working with many hedgehogs, or vice versa.

So, yeah, but I identify mostly as a fox, certainly.

I like

arbitrage somehow, like

learning how one field works, learning the tricks of that wheel, and then going to another field, which people don't think is related, but I can adapt the tricks.

So, see the connections between the fields.

Yeah.

So, there are other mathematicians who are far deeper than I am, like who are really, 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 the hedgehog and the fox, depending on the context, depending on the collaboration.

So what can you, if it's at all possible, speak to the difference between those two ways of thinking about a problem?

Say you're encountering a new problem, you know, searching for the connections versus like very singular focus.

I'm much more comfortable with the

fox paradigm.

I like looking for analogies, narratives.

I spend a lot of time, if there's a result I see in one field, and I like the result, it's a cool result, but I don't like the proof.

It uses types of mathematics that I'm not super familiar with.

I often try to reprove 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.

And from that, I can get some understanding of the tools that are used in that field.

So it's very exploratory, very doing crazy things in crazy fields and reinventing the wheel a lot.

Whereas sort of the hedgehog style is, I think, much more scholarly.

You're very knowledge-based.

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

Yeah, I think you'd rely a lot more on sort of calculation than sort of trying to find narratives.

So, yeah, I mean, 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.

I think you 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, so John Conway was there at the time.

He passed away a few years ago.

But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof.

So Conway just had this amazing way of thinking about all kinds of things in a way that you would normally think of.

So

he thought of proofs themselves as occupying some sort of space.

So if you want to prove something, let's say that there's infinitely many primes, you've all got different proofs, but you could rank them in different axes.

Like some proofs are elegant, some proofs are long, some proofs are

elementary and so forth.

And so there's this cloud, so the space of all proofs itself has some sort of shape.

And so he was interested in extreme points of this shape.

Like out of all these proofs, what is one that is the shortest at the expense of everything else, or the most elementary, or whatever.

And so he gave some examples of well-known theorems, and then he would give what he thought was the extreme proof in these different aspects.

And I just found that really eye-opening that

it's not just getting a proof for a result 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 informed writing style that

when you do your math assignments and undergraduate, your homework and so forth, you're sort of encouraged to just write down any proof that works.

Okay, 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,

be adaptable to generalized other things.

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.

But

you can code something spaghetti code 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, and has fewer bugs and whatever.

And there's a similar things with mathematics.

So yeah,

first of all, there's so many beautiful things there.

And Kama is one of the great minds in mathematics ever, in 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.

Oh, yeah, yeah, yeah.

Which I also find beautiful and fun, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task.

And I believe there's even competitions on this.

And it's also a nice way to stress test not just

the

sort of the programs, or in this case, the proofs, but also the different languages.

Maybe that's a different notation or whatever to use to 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.

Trevor Burrus, Jr.: 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 is 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, I mean, what I find most appealing is connections between different things.

So

e to the pi i equals minus one.

So yeah, people are, oh, this is all the fundamental constants.

Okay,

that's cute.

But to me, so the exponential function was introduced by Euler to measure exponential growth.

So compound interest or decay, anything which is continuously growing, continuously decreasing growth and decay, or dilation or contraction, is modeled by the exponential function.

Whereas pi comes around from circles and rotation.

If you want to rotate a needle, for example, 180 degrees, you need to rotate by pi radians.

And i, complex numbers, represents the swapping between a measuring axis, so a 90 degree rotation, so a change in direction.

So the exponential function represents growth and decay in the direction that you already are.

When you stick an i in the exponential,

now it's instead of motion in the same direction as your current position, it's motion as a right angles to your current position, so rotation.

And then so e to the pi i goes minus one tells you that if you rotate for time pi, you end up at the other direction.

So it unifies geometry through dilation and exponential growth or dynamics through this act of complexification, rotation by i.

So it connects together all these tools, mathematics,

dynamics, geometry, and complex and complex and the complex numbers.

They're all considered almost, yeah, they were all next-door neighbors in mathematics because of this identity.

Do you think the thing you mentioned is Q, the

collision of notations from these disparate fields

is just a frivolous side effect?

Or do you think there is legitimate value in when the notation, all our old friends come together

at night?

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.

And 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 equals mc squared, okay, so one of the big things was the E.

So when When Aristotle first came up with his laws of motion and then Galileo and Newton and so forth,

They saw the things they could measure.

They could measure mass and acceleration and force and so forth.

And so Newtonian mechanics, for example, Effigo's MA was the famous Newton's second law of motion.

So those were the primary objects.

So they gave them the central billing in the theory.

It was only later after people started analyzing these equations that there always seemed to be these quantities that were conserved.

So in particular momentum and energy.

And it's not obvious that things happen energy.

Like it's not something you can directly measure the same way you can measure mass and velocity and so forth.

But over time, people realized that this was actually a really fundamental concept.

Hamilton, eventually, in the 19th century, reformulated Newton's laws of physics into what's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object.

Once you know how to measure the Hamiltonian of any system, you can describe completely the dynamics, like what happens to it or to all the states.

It really was a central actor, which was not obvious initially.

And this

helped actually, this change of perspective really helped when quantum mechanics came along.

Because

the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking, because everything was a particle and so forth,

to quantum mechanics, because everything was a wave.

It just looked really, really weird.

Like you ask, what is the quantum version of F equals MA?

And 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 someone called Schrödinger's equation that tells you exactly how quantum systems evolve once you have a Hamiltonian.

side by side they look completely different objects you know like one involves particles one involves waves and so forth but with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.

So for example, in classical mechanics, there's this thing called Nerd's theorem.

Every time there's a symmetry in a physical system, there is a conservation law.

So the laws of physics are translation invariant.

Like if I move 10 steps to the left, I experience the same laws of physics as if I was here.

And that corresponds to conservation momentum.

If I turn around by some angle, again, I experience the same laws of physics.

This corresponds to conservation of angle and momentum.

If I I wait for 10 minutes,

I still have the same law as a 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.

So

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.

Like, for example, we have to give up the notion of space and time being these almost Euclidean depth spaces.

And it has to be,

you know, and

we kind of know that at very tiny scales,

there's going to be quantum fluctuations, there's a space-time foam.

And trying to use Cartesian quotas X, Y, Z is going to be, it's just, it's a non-starter.

But we don't know

what to replace it it with.

We don't actually have the mathematical

concepts.

The analog of a 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 motion of the heavens with the motions of objects on the earth and so forth.

So it should happen.

It's just that the,

again, to go back to this model of the observations and theory, part of our problem is that physics is a victim of its own success.

That our two big theories of physics, general relativity and quantum mechanics, are so good now.

So together, they cover 99.9% of sort of all the observations we can make.

And 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.

But I have faith that

we've been doing this for centuries.

We've made progress before.

There's no reason why we should stop.

Do you think you will be a mathematician that develops a theory of everything?

What often happens is that when the physicists need

some theory of mathematics, there's often some precursor that the mathematicians worked out earlier.

So when Einstein started realizing that space was curved, 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 is going back to Dubna's unreasonable effectiveness of mathematics.

I think the theories that work well to explain the universe tend to also involve the same mathematical objects that work well to solve mathematical problems.

Ultimately, they're just sort of both ways of organizing data

in useful ways.

It just feels like you might need to go some weird land that's very hard to

to intuit.

You have like 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, is experiment is very tough.

Yes.

Because of how effective both theories are.

But the other is like

just, you know, you're talking about

you're not just deviating from space-time.

You're going into like some crazy number of dimensions.

You're doing all kinds of weird stuff that to us, we've gone so far from this flat earth that we started.

Is that like

we're just it's it's very hard to use our limited ape descendants of uh uh cognition to intuit what that reality really is like.

This is why analogies are so important, you know.

I mean, so yeah, the round earth is not intuitive because we're stuck on it, um, but you know, but

you know, but round objects in general, we have pretty good intuition a little bit and we have interest about how light works and so forth.

And like, it's it's actually a good exercise to actually 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

and 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 intellectually for us to go from flat to round earth because

our life is mostly lived in flat land.

Yeah.

To load that information and we're all like to take it for granted.

We take so many things for granted because science has established a lot of evidence for this kind of thing.

But you know, we're on a

round rock

flying through space.

Yeah.

Yeah.

And it's a big leap.

And you have to take a chain of those leaps the more and more and more we progress.

Right.

Yeah.

So modern science is maybe, again, a victim of its own success is that

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, you know,

we need more grounding.

I think, I mean,

there are scientists who do excellent outreach, but

there's lots of science things that you can do at home.

Lots of YouTube videos, I did a YouTube video recently with Grant Sanderson, and we talked about this earlier, that

how the ancient Greeks were able to measure things like the distance to the moon, distance to the earth, and using techniques that you could also replicate yourself.

It doesn't all have to be like fancy space telescopes and really intimidating mathematics.

Yeah, that's 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.

Right.

You know, you're like on this planet.

You don't know the shape of it, the size of it.

You see some stars,

you see some things, and you try to like localize yourself in this world and try to make some kind of general statements about distance to places.

Change of perspective is really important.

You say travel borders the mind.

This is intellectual travel.

Put yourself in the mind of the ancient Greeks or some other person of some other time period.

Make hypotheses, spherical cows, whatever.

Speculate.

And

this is what mathematicians do and

what artists do, actually.

It's just incredible that given the extreme constraints, you could still say very powerful things.

That's why it's inspiring.

Looking back in history, how much can be figured out when you don't have much to figure out stuff?

If you propose axioms, then the mathematics lets you follow those axioms to it to their conclusions.

And sometimes you can get quite a long way from

initial hypotheses.

If we can 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?

And from a sort of 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 Einstein's equation of relativity just describes 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 Young-Middle fields.

And there's this whole hierarchy of different equations, of which Einstein is considered considered one of the most non-linear and difficult.

But relatively low in the hierarchy was this thing called the wave maps equation.

So it's a wave which at any given point is fixed to be like on a sphere.

So I 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 kind of like sheaves of wheat in the wheat field.

And I was interested in the global regularity problem again for this question.

Like, is it possible for all the energy here to collect at a point?

So the equation I considered was actually what's called a critical equation, where it's actually the behavior at all scales is roughly the same.

And I was able barely to show that

you couldn't actually force a scenario where all the energy concentrated at one point, that the energy had to disperse a little bit.

And the moment it dispersed a little bit,

it would stay regular.

Yeah, this was back in 2000.

That was part of why I got interested in navigators afterwards, actually.

Yeah, so I developed some techniques to solve that problem.

So part of it is

this problem is really non-linear because of the curvature of the sphere.

There was a certain non-linear 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 sheaves of wheat, and they're all bending back and forth, and 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 sort of stabilized flow, the flow becomes a lot more linear.

I discovered a way to transform the equation to reduce the amount of non-linear effects.

And then I was able

to solve the equation.

I found this transformation while visiting my aunt in Australia.

And I was trying trying to understand the dynamics of all these fields, and I couldn't do it with pen and paper.

And I had not enough facility of computers to do any computer simulations.

So I ended up closing my eyes on the floor and just imagining myself to actually be this vector 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.

Yeah, yeah.

And And

okay, fine.

You know, 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?

What, if it's possible

to go inside your mind when you're thinking, are you visualizing

in your mind the mathematical objects, symbols, maybe?

What are you visualizing in your mind usually when you're thinking?

A lot of pen and paper.

One thing you pick up as a mathematic mathematician is sort of, I call it cheating strategically.

So

the beauty of mathematics is that you get to change the rules, change the problem, change the rules as you wish.

You don't get to do this for any other field.

Like, you know, 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 a 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 can set this, so there's a dimension that's too large.

I'll set it to one.

I'd solve the one dimensional problem first.

So there's a main term and an error term.

I'm going to make a spherical cow assumption and I'll assume the error term is zero.

And so the way you should solve these problems is not in sort of 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 10 things that are making your life difficult, find find a version of the problem that turns off nine of the difficulties but only keeps one of them

and solve that.

And then that just figures.

So

you install nine cheats.

Okay, if you install 10 cheats, then the game is trivial.

But you install 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 someone else, something else on, and then you solve that one.

And after you know how to solve the 10 problems, 10 difficulties separately, then you have to start merging them a few at a time.

As a kid, I watched a lot of of these Hong Kong action movies

from a culture.

And

one thing is that every time it's a fight scene, so 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 worse choreography cinema,

but they would win.

Are you usually pen and paper?

Are you working with computer and LaTeX?

Mostly pen and paper.

Actually, so 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.

I mean, and this beautiful blackboard is erase, and

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, has some iteration or something, I'd have to

remember how to set up a Python program

and how does a for loop work and debug it and it would have taken two hours and so forth.

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 so um maybe a good entry point is just talking about computer assisted proofs in general can you describe the lean formal proof programming language and how it can help as a proof assistant and maybe how you started using it and how uh it has helped you

so um lean is a computer language um much like sort of standard languages like Python and C and so forth, except that in most languages, the focus is on producing executable code.

Lines of code do things.

They flip bits or they make a robot move or they deliver you text on the internet or something.

So 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.

So a software language like Python might do a computation and give you that the answer is seven.

Okay, that it does the sum of three plus four is equal to seven.

But lean Lean can produce not just the answer, but a proof that how it got the answer of seven as three plus four

and all the steps involved.

So

it creates these more complicated objects, not just statements, but statements with proofs attached to them.

And

every line of code is just a way of piecing together previous statements to create new ones.

So 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 the same for the.

Can you give people some intuition about the difference between writing on pen and paper versus using lean programming language?

How hard is it to formalize

a statement?

So lean, a lot of mathematicians were involved in the design of lean.

So it's designed so that um individual lines of code resemble individual lines of mathematical argument.

Like 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 it's written so that ideally it should be like a one-to-one correspondence.

In practice it isn't because Lean is like explaining a proof to an extremely pedantic colleague who will will point out, okay, did you really mean this?

Like what what happens if this is zero?

Okay, did you how do you justify this?

So lean has a lot of automation in it to try to be less annoying.

So for example, every mathematical object has to come of a type.

Like 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 often in terms of context.

You say, you know, clearly x is equal to, let x be the sum of y and z, and y and z were already real numbers, so x should also be a real number.

So 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 sort of the computations that you're doing, but sort of what each object actually is in some sense.

Is it using something like LLMs to do the type inference, or like you match over the real number?

It's using much more traditional, what's called 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.

Yeah, every object comes with a history of where it came from, and you can kind of trace it.

Oh, I see.

Yeah, so it's designed for reliability.

So 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 with 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 theorem calculus, but you need to find it.

So a lot of the bottleneck now is actually lemma search.

There's a tool that you know is in there somewhere, and you need to find it.

And so you can, there are various search engines specialized for Matholib that you can do.

But there's now these large language models that you can say, I need the fundamental theorem calculus at this point.

And it was like, okay,

for example, when I code, I have GitHub Copilot installed as a plug-in 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 thing calculus.

Okay.

And then it might suggest, okay, try this.

And like maybe 25% of the time, it works exactly.

And then another 10, 15% of the time, it doesn't quite work, but it's close enough that I can say, oh, if I just change it here and here, it will work.

And then like half the time, it gives me complete rubbish.

But people are beginning to use AIs a little bit on top,

mostly on the level of basically fancy autocomplete.

That you can type half of one line of a proof and it will find, it will tell you.

Yeah, but a fancy, especially fancy with the sort of capital letter F, is

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,

it's annoying.

But doesn't it kill the whole vibe of being a mathematician?

Yeah, just so I mean, having a pedantic co-worker.

Right, yeah, if that was the only aspect of it.

Okay, but

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 to be checked that it goes,

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,

like every time you change a parameter, you have to check line by line that every single line of your proof still works.

And there can be subtle things that you didn't quite realize, some properties of 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 the proof with this constant 12.

And then when this new paper came out, we said, okay, let's...

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 an 11, you run the compiler, and like of the thousands of lines of code you have, 90% of them still work.

And there's a couple that are lined in red.

Now I can't justify 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 sort of good programming practices, most of your lines will not be red.

And there'll just be a few places where you, I mean, if you don't hard code your constants, but you sort of

you use smart tactics and so forth,

you can localize the things you need to change to a very small period of time.

So like within a day or two, we had updated our proof.

Because this is a very quick process here.

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 has a proof, are you able to like

what's what's the uh versus paper?

And yeah, so the proofs are longer, but each individual piece is easier to read.

So if you take a math paper and you jump to page 27 and you look at paragraph six and you have a line of text of math.

I often can't read it immediately because it assumes various definitions, which I had 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 like, you know, 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 would say what it is, where it came from, where this stuff is justified.

You can trace things back much easier than sort of flipping through a math paper.

So one thing that Lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn't do in the past.

So traditionally with pen and paper, when you want to collaborate with another mathematician,

either you do it at a blackboard where you can really interact, but if you're doing it sort of 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 sort of work on the same thing, collaboratively at the same time.

But with Lean, you can be trying to formalize some portion of the proof and say, oh, I got stuck at line 67 here.

I need to prove this thing, but it doesn't quite work.

Here's like 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

with dozens of people across the world, most of whom I don't, have never met in person.

And I may not know actually even whether they're how reliable they are

in the proof they could figure it, but Lean gives me a certificate of trust.

So I can do I can do trustless mathematics.

So there's so many interesting questions.

There's so so one, you're you're known for being a great collaborator.

So what is the right way to approach

solving a difficult problem in mathematics when you're collaborating?

Are you doing a divide and conquer type of thing or

are you focusing on a particular part and you're brainstorming?

There's always a brainstorming process first.

Yeah, so math research projects sort of by their nature, when you start, you don't really know how to do the problem.

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.

So this is what I said about cheating first.

It's like

to go back to the bridge building analogy.

So first assume you have infinite budget and unlimited amounts of workforce and so forth.

Now can you build this bridge?

Okay.

Now have infinite budget, but only finite workforce.

Now can you do that and so forth?

So, I mean, of course,

no engineer can actually do this.

Like I say, they have fixed requirements.

Yes, there's this sort of jam sessions always at the beginning where you try all kinds of crazy things and you 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.

So one of my theorems I'm known for is a theorem of Ben Green, for example, called the Green-Tau theorem.

It's a statement that the primes contain arithmetic progressions of any length.

So it was a modification of this theorem with some already.

And the way we collaborated was that Ben had already proven a similar result for progressions of length three.

He showed that sets like the primes contain lots and lots of progressions of length three,

and even subsets of the primes, certain subsets do.

But his techniques only worked for length three progressions.

They didn't work for longer progressions.

But I had these techniques coming from a goddamn 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 the primes, there's a certain technical condition which, if I could have it, if Ben could supply me this fact, I could give 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.

So the Hyperin 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 I do.

And

yeah, so

there are all kinds of dynamics, you know.

I mean, it's every

collaboration

has some story, and 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 right for a problem and then you can really do a divide and conquer with lean

where you're working on separate parts right and they're using the computer system proof checker essentially yeah to make sure that everything is correct along the way yeah so it makes everything compatible and yeah and trustable

yeah so Currently, only a few mathematical projects can be cut up in this way.

At the current state of the art, most of the lean activity is on formalizing boosts that have already been proven by humans.

A math paper basically is

a blueprint in a sense.

It is taking a difficult statement, like a big theorem, and breaking up into maybe a hundred little lemmas,

but often not all written with enough detail that each one can be sort of directly formalized.

A blueprint is like a really pedantically written version of a paper where every step is explained to as much detail as possible and trying to make each step kind of self-contained

and or depending on only a very specific number of previous statements that have been proven so that each node of this blueprint graph that gets generated can be tackled independently of the others.

And you don't even need to know how the whole thing works.

So it's like a modern supply chain.

If you want to create an iPhone or some other complicated object,

no one person can build a single object, but you can have specialists who just, if they're given some widgets from some other company, they can combine them together to form a slightly bigger widget.

I think that's a really exciting possibility because you can have, if you can find problems that could be

broken down in this way, then you could have thousands of contributors, 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 of experimental.

I think the platform that Lean and other software tools, so GitHub and things like that,

will allow experimental mathematics to be to scale up to a much greater degree than we can do now.

So right now, if you want to

do any mathematical exploration of some mathematical pattern or something, you need some code to write out the pattern.

And I mean, sometimes there are some computer algebra packages that help, but often it's just one mathematician coding lots and lots of Python or whatever.

And because coding is such an error-prone activity, it's not practical to allow other people to collaborate with you on writing modules for your code.

Because if one of the modules has a bug in it, the whole thing is unreliable.

So you get these bespoke

spaghetti code written by not professional programmers, but by mathematicians, you know, and they're clunky and slow.

And

so because of that, it's hard to really mass-produce experimental results.

But

yeah, but I think with Lean, I mean, so I'm already starting some projects where we are not just experimenting with data, but experimenting with proofs.

So I have this project called the 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.

Okay, 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 and

don't obey others.

For example, x times x is not always equal to x.

So that law is not always true.

So given any operation, it obeys some 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 list up at 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, of these 22 million, there are like 100 or so that are really quite hard.

Okay, but a lot are easy.

And the project was just to work out, to determine the entire graph, like which ones imply which other ones.

That's an incredible project, by the way.

Such a good idea, such a good test of the very thing we've been talking about at a scale that's remarkable.

Yeah, so it would not have been feasible.

I mean, the state of the art in the literature was like, you know, 15 equations and sort of how they imply it.

That's sort of at the limit of what a human with 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 have been settled.

Wow.

And well, actually, and of those two, we have a pen and paper proof of the two.

And we're formalizing it.

In fact, this morning I was working on finishing it.

So we're almost done on this.

It's incredible.

Yeah, fantastic.

How many people were able to get

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 with 50 authors

and a big appendix of who contributed what.

Here's an interesting question.

Now, to maybe speak even more generally about it.

When you have this pool of people,

is there a way to organize the contributions by level of expertise of the people, all the contributors?

Now, okay.

I'm asking you 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 to be uploaded for this GitHub and GitHub tracks who contributed what.

So you could generate statistics from at any at any later point in time.

You can say, oh, this person contributed this many lines of code or whatever.

I mean, these are very crude metrics.

I would definitely not want this to become like, you know, part of your 10-year review or something.

But I mean, 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 a direction which is a bit scary for academics to go down.

We don't like metrics so much.

And yet academics use metrics.

They just use

old ones.

Number of papers.

Yeah,

it's true that, yeah, I mean, it feels like this is a metric while flawed is going more in the right direction, right?

Yeah.

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

Yeah, I think it's interesting to study.

I mean, I think you can 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, yeah, I know.

I mean, it's rational.

So, what we've done for this project is self-report.

So,

there are actually 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 or so categories.

And we just ask each contributor to, there's a big matrix of all the authors in all the categories, just to tick the boxes where they think that they contributed.

And just give a rough idea.

So you did some coding

and you provided some compute, but you didn't do any for 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 lead author and second author and so forth,

which we're proud of.

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 a 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 rather.

And this was a huge bottleneck, actually.

But still, we had projects that were, you know, 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 Bobaki, Bobaki is the pseudonym for a famous group of mathematicians in the 20th century.

And so 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.

So one is that if you actually wanted to be considered for tenure or whatever, you could not use this paper in your

as your submitted as 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 primary project, or This was Terrence Tower's Point My Project, and not mention the other 19 or whatever people that were involved.

So we're trying something different this time around where we have everyone's an author,

but we will have an appendix with this matrix, and we'll see how that works.

I mean, 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 languages a few years ago, and you're 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 deep minds alpha proof was trained using reinforcement learning on both failed and successful formal lean proofs of i am o problems so this is sort of high level high school.

Yes.

Very high level high school level mathematics problems.

What do you think about the system and maybe what is the gap between this system that is able to prove the high school level problems

versus gradual level

problems?

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 last 20 model has a 10% failure rate at each step

of going in the wrong direction,

it's just extremely unlikely to actually reach the end.

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, you know, it's very fault-tolerant.

Like, you can make a few minor grammatical errors, and a speaker in the second language can get some idea of what you're saying.

But formal language,

if you get one little thing wrong,

I do that the whole thing is nonsense.

Even formal to formal is very hard.

There are different incompatible proof assistant languages.

There's Lean, but also Koch and Isabel and so forth.

And actually, even converting from a formal language to formal language

is an unsolved, basically unsolved problem.

That is fascinating.

Okay, so but once you have an informal language, they're using

their RL train model, so something akin to alpha zero 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?

Yeah, we talked earlier about things that are amazing over time become kind of normalized.

So, yeah, now somehow it's, of course, geometry is a solvable 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 um it's it

um the approach doesn't scale currently yeah three days of google's server is server time to solve one uh high school math problem yeah this this is not a scalable uh prospect um especially with the exponential increase in um as as the complexity um increases which mentioned that they got a silver metal performance the equivalent of i mean yeah equivalent of a silver medal

so first of all they took way more time than was uh allotted um and they had this assistance with where the humans started helped by formalizing.

But

also, they're giving us those full marks for the solution, which I guess is formally verified.

So I guess that's fair.

There are efforts.

There will be a proposal at some point to actually have an AI Math Olympiad where at the same time as the human contestants get the actual Olympiad.

problems, AIs will also be given the same problems, the same time period, and the outputs will will have to be created by the same judges.

Which means that will have to be written in natural language rather than formal language.

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

AIs are actually a lot better at problems where there's a specific numerical answer.

Because it's easy to

reinforce reinforcement learning on it.

Yeah, you got the right answer, you got the wrong answer.

It's a very clear signal.

But a long-form proof either has to be formal, and then the lean can give it a thumbs up, thumbs down, or it's informal.

But then you need a human to grade it to tell.

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 nine cameras to do reinforcement learning on just the regular text that people get.

But now if you actually hire people not just give thumbs up, thumbs down, but actually check the output mathematically,

that's too expensive.

So if we just explore this possible future,

what is the thing that humans do that's most special in

mathematics?

So that you could see AI

not cracking for a while.

So, inventing new theories, so coming up with new conjectures versus proving the conjectures,

building new abstractions, new representations, maybe

an AI tern 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.

So, a thousand years ago, mathematicians had to compute the date of Easter and those really complicated calculations, you know, but it's all automated, been automated for centuries.

We don't need that anymore.

You know, they used to navigate to do spherical navigation, spherical trigonometry, to navigate how to get from

the old world to the new.

Very complicated calculations, again, we've been automated.

You know, 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.

So on the computational side, verifying routine things like having a problem and

say, here's a problem in partial factorial equations.

Could you solve it using any of the 20 standard techniques?

And they'll say, yes, I've tried all 20, and here are the 100 different permutations, and here's my results.

And that type of thing, I think it will work very well.

Type of scaling to, once you've solved one problem, to make the AI attack 100 adjacent problems.

The things that humans do.

still,

so where the AI really struggles right now

is knowing when when it's made a wrong turn.

That 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.

And sometimes if you're lucky, it's a simple problem, it's the right technique, and you solve the problem.

And sometimes

it will have a problem, 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.

So

we've had human generated mathematics that's very low quality,

like submissions for 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 that's partly because that's what the reinforcement learning has actually trained them to do,

to make things, to produce text that looks like

what is correct, which for many applications is good enough.

So the errors are often really subtle, and then when you spot them, they're really stupid.

Like, you know, like no human would have actually made that mistake.

Yeah, it's actually really frustrating in the programming context because I program a lot.

And

yeah, when a human makes

low-quality code, there's something called code smell, right?

You can tell.

You can tell.

Immediately, like

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.

Yeah, so it's very tricky too and frustrating for some reason to

work through.

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 programs on Go and chess and so forth is in some sense, they have developed a sense of smell for go and chess positions.

this position is good for white that's good for black um they can't enunciate why um but just having that that sense of smell lets them strategize so if ais gain that ability to sort of a sense of viability of certain proof strategies says so you can say i'm going to try to break up this problem into two small sub-tasks and they can say oh this looks good the two tasks look like they're simpler tasks than than your main task and they've still got a good chance of being true um so this is good to try or no you've you've you you made the problem worse because each of the two subproblems is actually harder than your original problem, which is actually what normally happens if you try a random thing to try.

Normally,

it's very easy to transform a problem into an even harder problem.

Very rarely do you transform to a simpler problem.

So if they can pick up a sense of smell, then they could maybe start competing with

human-level mathematicians.

So this is a hard question, but not competing, but collaborating.

If, okay, hypothetical, if I gave you an oracle

that was able to do some aspect of what you do, and you could just collaborate with it,

what would that oracle, what would you like that oracle to be able to do?

Would you like it to maybe be a verifier, like check, do the code smell?

Like you're, yes,

Professor Tao, this is the correct, this is a good, this is a promising, fruitful direction.

Yeah, yeah, yeah.

Or, 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?

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.

Yeah, we have not had in the past systems that are competent enough to understand complex instructions that can work at massive scale, but are also unreliable.

It's an interesting,

unreliable in subtle ways

whilst providing sufficiently good output.

It's an interesting combination.

I mean,

you have graduate students that you work with who are kind of like this, but not at scale.

And we had previous software tools that can work at scale, but very narrow.

So we have to figure out how to use.

I mean, so Tim Coward actually, imagine, he actually foresaw like in in 2000 he was envisioning what mathematics would look like in in actually two and a half decades

and that's funny yeah he he wrote in his in his article like a hypothetical conversation between a mathematical assistant of the future um and himself you know he found a sort of a problem and they would have to have a conversation that sometimes the human would propose an idea and the ai would would evaluate it and sometimes the ai would propose an idea um and uh and sometimes a computation was required and they would just go and say, okay, I've checked the 100 cases needed here.

Or

the first,

you said this is true for all n.

I've checked the n up to 100 and it looks good so far.

Or hang on, there's a problem at 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 are going to propose on both sides, calculations get proposed on both sides.

I've had conversations with AI where I say, okay, we're going to collaborate to solve this math problem.

And it's a problem that I already know the 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.

I say, no, no, no, try using this.

Okay.

And it might start using this.

And then it'll go back to the tool that I wanted to do before.

And you have to keep railroading it onto the path you want.

And I could eventually force it to give the proof I wanted.

But it was like herding cats.

And the amount of personal effort I had to take to not just sort of prompt it, but also check its output, because after a lot of what it looked like it was going to work, oh no, there's a problem online 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's no longer feels like creating cats.

Maybe you'll surprise us how quickly that comes.

I believe so.

So in formalization, I mentioned before that it takes 10 times longer to formalize a proof than 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 you'll drop a little one.

And that's the phase shift.

Because suddenly

it makes sense when you write a paper.

to to write it in lean first

or through a conversation with AI who is generating lean 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 actually it's harder and harder to get good refereeing for the really long ones, unless they're really important.

It is actually an issue which the formalization is coming in at just the right time for this to be.

And the easier and easier to guess because of the tooling and all the other factors, then you're going to see much more like math lib will grow potentially exponentially.

It's a virtuous cycle.

I mean, one face shit of this type that happened in the past was the adoption of LaTeX.

So LaTeX is this typesetting language that all mathematicians use now.

So in the past, people use all kinds of word processors and typewriters and whatever.

But at some point, LaTeX became easier to use than all other competitors.

And people just switch 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

a

AI system being a collaborator on a proof that wins the Fields Medal to that level?

Okay.

Well, it depends on the level of collaboration.

No, like it deserves to be, to get the Fields Medal.

Like, so half and half.

Already, like, I could imagine if it was

wedding paper, having some AI systems and writing it, you know, just, you know, like the order complete alone is 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 order complete just suggests, okay, 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 particular print, but by 2026, which is now next year,

there will be

math collaborations where the AI, so not Fields Medal winning, but actual research level math collections.

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 then comes back doesn't work, but

it might propose 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 authentication and an AI involved.

But

it's hard to sort of disentangle credit.

I mean, these tools, they do not replicate all the skills needed to do mathematics, but they can replicate sort of some non-trivial percentage of them, 30, 40%.

So they can fill in gaps.

So

coding is a good example.

So

it's annoying for me to code in Python.

I'm not a native,

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, it's still a problem with hallucinating references that don't exist.

But this, I think, is a civil war problem.

If you train in the right way and so forth,

and verify

using the internet,

You should, in a few years, get the point where you have

a lemma that you need.

And say, has anyone proven this lemma before?

And it will do basically a fancy web search AI assistant and say, yeah, yeah, there are these six papers where something similar has happened.

I mean, you can ask it right now, and it will 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, so much, the signal-to-noise ratio is so poor that

it's most helpful when you already somewhat know the literature.

And you just need to be prompted to be reminded of a paper that was already subconsciously in your memory.

Versus helping you discover new you were not even aware of, but is the correct citation.

Yeah, that's yeah, that it can sometimes do, but when it does, it's buried in a list of options for which the other.

I mean, being able to automatically generate a related work section that is correct,

that's actually a beautiful thing that might be another phase shift because it assigns credit correctly.

Yeah.

It does, it breaks you out of the silos of

thought.

Yeah, no,

there's a big hump to overcome right now.

I mean, it's it's like self-driving cars.

The safety margin has to be really high for it to be um uh to be feasible.

So, yeah, so there's a last mile problem um with a a lot of AI applications um that uh you know they can develop tools that work 20%, 80% of the time, but it's still not good enough.

And in fact, even worse than good in some ways.

I mean, another way of asking the Fields Metal question is,

what year do you think you'll wake up and be like real surprised?

You read the headline, the news of something happened that AI did, like,

you know, real breakthrough, something.

It doesn't, you know, like Fields Metal, even hypothesis, it could be like really just

this alpha zero moment would go that right right um

yeah this this decade i can i can see it like making a conjecture between two unrelated two two things that people thought was unrelated oh interesting generating a conjecture that's a beautiful conjecture yeah and and actually has a real chance of being correct and and and meaningful and um because that's actually kind of doable I suppose but what the data is it's yeah yeah no that would be truly amazing um

the current models struggle a 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, okay,

and

here's a new pattern that we didn't see before.

But it actually even struggles, 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 of contamination that it did only because it's like somewhere in a straining data iterative, somehow new,

you know, Boyle's law or whatever you're trying to reconstruct

part of it is we don't have the right type of training data for this

yeah so for laws of physics like we don't have like a million different universes with a million different laws of nature

and

like

a lot of what we're missing in math is actually the negative space of so we have published things of things that people have been able to prove

and conjectures that end up being verified or maybe counterexamples produced but we don't have data on things that were proposed, and they're kind of a good thing to try.

But then people quickly realize that it was the wrong conjecture, and then they said, oh, but we should actually change our claim to modify it in this way to actually make it more plausible.

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.

And

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 in love with the problem and it really cannot rest until you solve it.

But you also hasten to add that sometimes this approach actually can be very successful.

And the example you gave is Gregoria 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 Millennium Prize problem that's been solved, Poincaré conjecture, and maybe speak to the journey that Gregori Perlman's been on.

All right, so it's a question about curved spaces.

Earth is a good example.

So Earth you can think was a 2D surface.

Interesting round the Earth 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's determined by some 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 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 of a torus, there is no way, it can't get through the hole.

There is no way to contract it to a point.

So it turns out that the sphere is the only surface with this property of contractability, up to like continuous deformations of the sphere.

So

things that are what I call topologically equivalent of the sphere.

So Poincaré asks the same question in higher dimensions.

So it becomes hard to visualize because surface you can think of as embedded in three dimensions, but a curved free space,

we don't have good intuition of 4D space to live it.

And 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 Poincaré conjecture.

Weirdly, in higher dimensions, four and five was actually easier.

So

it was solved first in higher dimensions.

There's somehow more room to do the deformation.

It's easier to move things around to be a sphere.

But three was really hard.

So 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 each other.

There were

algebraic approaches.

There's various algebraic objects, like things called the fundamental group that you can attach to these homology and cohomology

and all these very fancy tools.

They also didn't quite work.

But Richard Hamilton proposed a partial differential equations approach.

So

you take...

So the problem is that you have this object which is sort of secretly as a sphere, but it's given to you in a really

in a weird way.

So it's like think of a ball that's been kind of crumpled up and twisted, and it's not obvious that it's a ball.

But

if you have some sort of surface which is a deformed sphere,

you could, for example, think of it as the surface of a balloon.

You could try to inflate it, you blow it up.

And naturally, as you fill it with air, the wrinkles will sort of smooth out, and it will turn into a nice round sphere.

Unless, of course, course, it was a torus or something, in which case it would get stuck at some point.

Like if you inflate a torus,

there'll be a point in the middle.

When the inner ring shrinks to zero, you get a singularity and you can't blow up any further.

You can't flow any further.

So he created this flow, which is 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.

And he wanted to show that either this process would give you a sphere or it would create a singularity.

Actually, very much like how PDs either perhaps they have global regularity regularity or Feynman blow up.

Basically, it's almost exactly

the same thing.

It's all connected.

And he showed that for two dimensions, two-dimensional surfaces,

if you started something connected, no singularity is ever formed.

You never ran into trouble, and you could flow, and it would give you a sphere.

And so he got a new proof of the two-dimensional result.

But by the way, that's a beautiful explanation of Rachel Flow and its application in this context.

How difficult is the mathematics here?

Like for the 2D case, is it?

Yeah, these are quite sophisticated equations on par with the Einstein equations.

Slightly simpler, but

yeah, but they were considered hard non-linear 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 has the same problems as Nabi-Stokes.

As you blow up, maybe the curvature could get concentrated in finer and smaller and smaller regions, and

it looked more and more non-linear, and things just looked worse and worse.

And 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

behaves like a barbell and it pinches at a point.

Some singularities are simple enough that you can sort of see what to do next.

You just make a snip and then you can turn one surface into two and evolve them separately.

But there was the prospect that there's 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, like what are all the possible ways that things can go wrong.

So what Perlman did was that, first of all, he made the problem, he turned the problem from a supercritical problem to a critical problem.

I said before about how the invention

of energy, the Hamiltonian, like really clarified

Newtonian mechanics.

So 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 at every single scale and turned the problem into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did before.

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 mapsing 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, he was able to resolve the point-quarter conjecture.

So,

quite like a lot of really ambitious steps, and like nothing that a large language model today, for example, could.

I mean,

at best, I could imagine 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 them from A to B.

So you've done, like you said, actually, you see, even strictly mathematically, but

more broadly

in terms of the process, you've done similarly difficult things.

What can you infer from the process he was going through?

Because he was doing it alone.

What are some low points in a process like that?

When you start to, like, you've mentioned hardship, like, 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.

So,

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 add some cheat.

If there's a specific thing that's blocking you, that this

some

bad case keeps showing up

for which your tool doesn't work, you can just assume by fear this this bad case doesn't occur.

So you do some magical thinking

but strategically, okay, for the point to see if the rest of the 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.

Oh, yeah, yeah, yeah.

Eventually.

Sometimes actually, it's even productive to make mistakes.

So one of the, I mean, there was a project which actually

we won some prizes for, actually,

with four 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, he worked on a special case of this, but he could not solve the general case.

And we worked on this problem for two months, and we thought we solved it.

We had this...

this cute argument that if everything fit and we were excited we were planning celebration reach to all get together and have champagne or something

And we started writing it up.

And one of us, not me, Aki, but another co-author, said, oh,

in this lemma here,

we have to estimate these 13 terms that show up in this expansion.

And we estimate 12 of them, but in our notes, I can't find the estimation of the 13th.

Can you?

Can someone supply that?

And I said, sure, I'll look at this.

And Aki is, yeah, we didn't cover that.

We completely omitted this term.

And this term turned out to be worse than the other 12 terms put together.

In fact, we could not estimate this term.

And we tried for a few more months, and all different permutations and there was always this one thing, one term that we could not control.

And so like

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 that the approach was actually somewhat different.

by quite a bit from our initial strategy, which did actually didn't generate these problematic terms and actually solve the problem.

So we solved the problem after two years.

But if we hadn't had that initial false dawn of nearly solving the problem, we would have given up by month two or something and 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 in the New World.

They had an incorrect version of a 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 saw it in his prospectus.

I mean, it could be that he secretly knew, but just on the psychological element,

do you have like emotional or

like self-doubt that just overwhelms you most like that?

You know, because

this stuff, it feels like math is so engrossing that like it can break you.

When you like invest so much yourself in the problem.

And then it turns out wrong, you could start to

similar way chess has broken some people.

Yeah,

I think different mathematicians have different levels of emotional investment in what they do.

I mean, I think for some people, it's just a job.

You know, you have a problem, and if it doesn't work out, you go on the next one.

Yeah, so the fact that you can always move on to another problem, it reduces the emotional connection.

I mean,

there are cases, you know, so there are certain problems that are what are called mathematical diseases where

people just latch onto that one problem and they spend years and years thinking about nothing but that one problem.

And, you know, maybe their career suffers and so forth.

They say, oh, but this big win, this will, you know,

once I finish this problem,

I will make up for all the years

of lost opportunity.

But that's, that's,

I mean, occasionally, occasionally it works, but

I really don't recommend it for people without the right fortitude.

Yeah, so I've never been super invested in any one problem.

One thing that helps helps is that we don't need to call our problems in advance.

Well, when we do grant proposals, we sort of say we will study this set of problems.

But even there, 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.

And maybe you don't solve the problem, but you find some related problem that you can say something new about.

And 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?

It sits there in the dark corners, you know, twin prime conjecture, Riemann hypothesis, global conjecture.

Twin prime, that sounds well.

Again, so I mean, the problem is like a Riemann hypothesis, those are so far out of reach.

How do you think so?

Yeah, there's no even viable strategy.

Like even if I activate all my all the cheats that I know of in this problem,

like this is still no way to get me to be.

I think it needs a breakthrough in another area of mathematics to happen first, and for someone to recognize that that would be a useful thing to transport into this problem.

So we should maybe step back for a little bit and just talk about prime numbers.

So they're often referred to as the atoms of mathematics.

Can you just speak to the structure that these atoms

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 number.

If you want to generate multiplicatively, you can take all the prime numbers, two, three, five, seven, 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 additive point of view and a multiplicative point of view.

And separately, they're not so bad.

So like any question about the natural numbers that only involves addition is relatively easy to solve.

And any question that only involves multiplication is relatively easy to solve.

But what has been frustrating is that you combine the two together.

And suddenly you get this extremely rich, I mean, we know that there are statements in number theory that are actually as undecidable.

There are certain polynomials in some number of variables, you know, is there a solution in the natural numbers?

And the answer depends on an undecidable statement, like whether the axioms of mathematics are consistent or not.

But

yeah, but even the simplest problems that combine something multiplicative, such as the primes, with something additive, such as shifting by two,

separately, we undecide both of them well.

But if you ask, when you shift the prime by two,

can you get a, how often can you get another prime?

It's been amazingly hard to relate the two.

And we should say that the twin prime conjecture is just that it posits that there are infinitely many pairs of prime numbers that differ by two.

Now,

the interesting thing is that you have been very successful at pushing forward the field in answering these complicated questions

of this variety.

Like you mentioned, the green tile theorem.

It proves that prime numbers contain arithmetic progressions of any length.

Right.

Which is mind-blowing that you could prove something like that.

Right.

Yeah.

So what we've realized because of this type of research is that

different patterns have different levels of

interstructability.

So what makes the twin prime problem hard is that if you take all the primes in the world, you know, three, five, seven, eleven, so forth, there are some twins in there.

11 and 13 is a twin prime, pair of twin primes and so forth.

But you could easily, if you wanted to,

redact the primes to get rid of to get rid of

these twins.

The twins, they show up and there are infinitely many of of them, but they're actually reasonably sparse.

Initially, there's quite a few, but once you got to the millions, the trillions, they become rarer and rarer.

And you could actually just, you know,

if someone was given access to the database of primes, you just edit out a few primes here and there, they could make the twin pound conjecture false by just removing like 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 prime number theorem and other facts about the primes, but doesn't contain any trim primes anymore.

And this is a real obstacle to the trim prime conjecture.

It means that any proof strategy to actually find trin primes in the actual primes must fail when applied to these slightly edited primes.

And so it must be some very subtle, delicate feature of the primes that you can't just get from

aggregate

statistical analysis.

Okay, so that's out.

Yeah.

On the other hand, athletic progressions has turned out to be much more robust.

Like you can take the primes and you can eliminate 99% of the primes, actually.

And you can take any 99% that you want.

And it turns out, and another thing we prove is that you still get arithmetic progressions.

Arithmetic progressions are much, you know, they're like cockroaches.

Of arbitrary length.

Yes.

Yes.

That's crazy.

I mean, so

for people who don't know, arithmetic progressions is a sequence of numbers that differ by some fixed amount.

Yeah.

But it's again, like, it's an infinite monkey type phenomenon.

For any fixed length of your set, you don't get arbitrary length 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, I think.

If the primes were really genuinely random, if the primes were generated by monkeys,

then yes, in fact, the infinite monkey theorem would.

Oh, but you're saying that twin prime is it doesn't you can't use the same tools like the it doesn't appear random almost well we don't know yeah we we we we believe the primes behave like a random set so the reason why we care about the trim hypotheses conjecture is it's a test case for whether we can genuinely confidently say with 0% chance of error that the primes behave like a random set okay random yeah random versions of the primes we know contain twins um at least with 100 probability uh or probably tending to 100 as you go out further and further.

Yeah, so the primes we believe that they're random.

The reason why arithmetic progressions are indestructible is that regardless of whether your set looks random or looks structured, like periodic, in both cases, arithmetic progressions appear, but for different reasons.

And this is basically all the ways in which this thing...

There are many proofs of these sort of arithmetic progression type theorems, and they're all proven by some sort of dichotomy where your set is either structured or random.

And in both cases, you can say something, and then you you put the two together.

But in twin primes if the primes are random then you're happy, you win.

But if your parameters are structured they could 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 Zane-Shan progress on the K-tuple version.

Right, yeah.

So the funny thing about conspiracies is that any one conspiracy theory is really hard to disprove.

That, you know, if you believe the world is run by lizards, you say, here's some evidence that it's not run by lizards, but that evidence was planted by the lizards.

You may have encountered this kind of phenomenon.

There's almost no way to

definitively rule out a conspiracy.

And the same is true in mathematics, that a conspiracy that's

solely devoted to eliminating twin primes,

you would have to also infiltrate other areas of mathematics to sort of, 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 lists, it can't also be run by aliens.

Right.

So, one unreasonable thing is hard to dispute.

But more than one, there are tools.

So, yeah, so for example, we know there's infinitely many primes that are

no two of which are

the infinite pairs of primes which differ by at most

246 actually

is the current mechanism.

So, there's like a bound

on the.

Right.

So, like there's twin primes, there's this thing called cousin primes that differ by four.

There's called sexy primes that differ by six.

What are sexy primes?

Primes that differ by six.

The name is much less, the concept is much less exciting than the name suggests.

Got it.

So you can make a conspiracy rule out one of these, but like once you have like 50 of them, it turns out that you can't rule out all of them at once.

It just requires too much energy somehow in this conspiracy space.

How do you do the bound part?

How do you develop a bound for the difference between the prizes that

there's an infinite number of?

So it's ultimately based on what's called the pigeonhole principle.

So the pigeonhole principle is a statement that if you have a number of pigeons and they all have to go into pigeon holes and you have more pigeons than pigeon holes, then one of the pigeon holes has to have at least two pigeons there.

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 pigeon holes.

Let's say if you have 101 numbers, if you have 101 numbers, then two of them have to be distance less than 10 apart because two of them have to belong to the same pigeon hole.

So it's a

basic feature of a basic principle in mathematics.

So it doesn't quite work with the primes directly because the primes get sparser and sparser as you go out, that fewer and fewer numbers are prime.

But it turns out that there's a way to assign weights to

numbers.

So there are numbers that are kind of almost prime, but

they don't have no factors at all other than themselves and one, but they have very few factors.

And it turns out that we understand almost primes a lot better than understand primes.

And so, for example, it was known for a long time that there were trin almost primes.

This has been worked out.

So almost primes are something we kind of understand.

So you can actually restrict attention to a suitable set of almost primes.

whereas the primes are very sparse overall, relative to the almost primes, primes, they're actually 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 distant only 100, 100 apart.

But in order to prove the trin 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 will get trin 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%.

It's called the parity barrier.

And I would love to find, yeah, so one of my long-term dreams is to find a way to breach that barrier because it would open up not only the twin-product conjecture, but the go-back conjecture and many other problems in number theory are currently blocked because our current techniques would require going beyond this theoretical parity barriers.

It's like going fast the speed of light.

Yeah, so we should say a twin prime conjecture.

One of the biggest problems in the history of mathematics, Goldbach 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 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, you know, 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,

you know, I mean, that was too easy somehow.

Like there was no ocean being crossed.

Even if you don't know exactly what how

what what was done, you're suspecting that that something w wasn't right.

But is that still in the back of your head to

do you return to these to the prime do you return to the prime numbers every once in a while to see?

Yeah, when I have nothing better to do, which is less and less and tired, which is I get busy with so many things these days.

But yeah, when I have free time and I'm not, and I'm too frustrated to work on my sort of real research projects, and I also don't want to do my administrative stuff or I don't want to do some errand for my family.

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 forred for maybe 15 minutes, and then you think, I should check this, because this is too easy to be true, and it usually is.

What's your gut say about when these problems would be solved between Prime and GoBach?

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.

So trim primes is somewhat close.

Riemann hypothesis, I have no clue.

I mean, it has happened 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's

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 that,

you know, like if you want to poll, say, America

on some issue, and you ask one or two voters, and you may have sampled a bad sample, and then you get a really imprecise measurement of the full average.

But if you sample more and more people, the accuracy gets better and better.

And the accuracy improves like the square root of the number of people you sample.

So if you sample a thousand thousand people, you can get like a two, three percent margin of error.

So in the same sense, if you measure the primes in a certain multiplicative sense, there's a certain type of statistic you can measure.

And it's called the Riemann's data function, and it fluctuates up and down.

But in some sense, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random.

And there's a very precise way to quantify that.

And the Riemann hypothesis is a very elegant way that captures this.

But

as with many other ways in mathematics, we have very few tools to show that something really genuinely behaves like really random.

And this is actually not just a little bit random, but it's asking that it behaves as random as an actually random set,

the square root cancellation.

And we know actually because of things related to the parity problem, actually, that most of us usual techniques cannot hope to settle this question.

The proof has to come out of left field.

Yeah, 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 have to dodge very adeptly.

The prime numbers are just fascinating.

Yeah, yeah, yeah.

What to you is

most mysterious about the prime numbers

that's a good good question.

So, like, conjecturally, we have a good model of them.

I mean, like, as I said, I mean, they have certain patterns, like, the primes are usually odd, for instance.

But apart from this obvious patterns, they behave very randomly.

And just assuming that they behave, so there's something called the Kramer random model of the primes, that

after a certain point, primes just behave like a random set.

And there's various slight modifications to this model, but this has been a very good model.

It matches the numerics.

It tells us what to predict.

Like, I can tell you with complete certainty the Trinity Prime Conjecture is true.

The random model gives overwhelming odds that it's true.

I just can't prove it.

Most of our mathematics is optimized for solving things with patterns in them.

And the primes have this anti-pattern,

as do almost everything, really.

But we can't prove that.

Yeah, I guess it's not mysterious that the primes be

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?

And this is just absent.

Another incredibly surprisingly difficult problem is the Collatz conjecture.

Oh, yes.

Simple to state,

beautiful to visualize in its simplicity, and yet extremely difficult to solve.

And yet you have been able to make progress.

Paul Erdar said about the Collatz conjecture that mathematics may not be ready for such problems.

Others have stated that it is an extraordinarily difficult problem, completely out of reach.

This is in 2010, out of reach of present-day mathematics.

And yet you have made some progress.

Why is it so difficult to make?

Can you actually even explain what it is?

Oh yeah, so it's a problem that you can explain.

It helps with some visual aids.

But yeah, 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 divided by 2 is 20.

20 is still even divided by 2, 10.

5, and then 5 times 3 plus 1 is 16.

And then 8, 4, 2, 1.

so and then from one it goes one four two one four two one it cycles forever so this sequence i just described um you know 13 40 20 10 so both these are also called hailstone sequences because there's an oversimplified model of of hailstone formation yeah which is not actually quite correct but it's still somehow taught to high school students as a first approximation is that um like a a little nugget of ice gets gets an ice crystal forms in a cloud and it goes up and down because of the wind and and sometimes when it's cold it acquires acquires a bit more mass and maybe it melts a little bit.

And this process of going up and down creates this sort of partially melted ice, which eventually creates this hellstone.

And eventually it falls down to the earth.

So the conjecture is that no matter how high you start up, like you take a number, which is in the millions or billions,

this process that goes up if you're odd and down if you're even, it eventually goes down to earth all the time.

No matter where you start with this very simple algorithm, you end up at one.

Right.

And you might climb for a while.

Right.

Yeah.

So it's

yeah if you plotted um these sequences they look like brownian motion um they look like the stock market you know they just go up and down in a in a seemingly random pattern and in fact usually that's what happens that if you plug in a random number you can actually prove at least initially that it would look like um random walk um and that's actually a random walk with a downward drift um it's like if you're always gambling on on a roulette at the casino with odds slightly weighted against you so sometimes you you win sometimes you lose but over in the long run you lose a bit more than you win.

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, asserts that statistically, like 99% of all inputs would drift down to,

maybe not all the way to one, but to be much, much smaller than what you started.

So

it's like if I told you that if you go to a casino, most of the time you end up, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started.

That's kind of like the result that I proved.

So why is that result?

Like, can you continue down the 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 casino with a game at a casino with a losing expectation, over time you are guaranteed, or almost surely, with

probability as close to 100% as you wish, you're guaranteed to lose money.

But there's always this exceptional outlier.

It is mathematically possible that even in 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 Nabi-Stokes, there could be, you know, most of the time your waves can disperse, there could be just one outlier choice of initial conditions that would lead you to blower.

And there could be one outlier choice of

a special number that they stick in that shoots off to infinity while all other numbers crash to earth, crash to one.

In fact,

there's some mathematicians who've Alex Kantovich, for instance, who've proposed that

actually these collapse iterations are like these cellular automata.

Actually, if you look at what they happen in binary, they do actually look a little bit like these game of life type patterns.

And in an analogy to how the game of life can create these massive, like self-replicating objects and so forth, possibly you could create some sort of heavier-than-air flying machine,

a number which is actually encoding this machine, which is just whose job it is to encode, is to create a version of itself which is larger.

Heavier-than-air machine encoded in a number

that flies forever.

Yeah, so 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 the Navy-Stokes project.

Conway studied generalizations of the collapse problem, where instead of multiplying by three and adding one or dividing by two, you have more complicated branching rules.

But instead of having two cases, maybe you have 17 cases and 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 called a factrat as a a play on Fortran, and he showed that

you can program, it was too incomplete.

You could make a program that if your number you insert in was encoded as a prime, it would sink to zero.

It would go down, otherwise it would go up, and things like that.

So 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, maybe the same kind of framework is required in the clock's injector.

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, putting it equal to one, but like everything, yeah, that book's hard.

What would you say is out of these within reach, famous problems is the hardest problem we have today.

Is there 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 cos 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 though.

There's ripple effects.

P equals NP 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, uses number theory constructions involving primes and so forth.

And it relies very much on the intuition that number theorists have built over many, many years of what operations involving primes behave randomly and what ones don't.

And in particular, our encryption

methods are designed to turn text with information on it into text which is indistinguishable from

random noise.

So, and hence we believe to be almost impossible to crack, at least mathematically.

But

if something as core to our beliefs as the Riemann hypothesis 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.

Like you're going towards the collect conjecture again.

Because

you want it to be random, right?

You want it to be random.

Yeah, 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?

Right.

Is there any chance to you that P equals NP?

Is there some, can you imagine a possible universe?

It is possible.

I mean, there's 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 undecidable.

Yeah, we don't know.

There's a funny story I read that when you won the Fields Medal, somebody from the internet wrote you

and asked,

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 metal is not going to solve any of the problems I'm currently working on.

So

I'm going to keep working on them.

It's just, first of all, it's funny to me that you would answer an email in that context.

And second of all,

it just shows your humility.

But anyway, maybe you could speak to the Fields Medal, but it's another way for me to ask

about Gregoria 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.

Yeah, no, he's somewhat of an outlier, even among mathematicians who tend to

have somewhat idealistic views.

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.

He's always had strong views about certain things.

I mean, it's not like he was completely isolated from the math community.

I mean, 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.

And he decided

to peace out and collect mushrooms in St.

Petersburg or something.

And that's fine.

You can do that.

I mean, that's another sort of flip side.

A lot of problems that we solve, some of them do have practical application, and 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 mathematicians,

career mathematicians, you just focus on publishing the next paper, maybe getting one

promoted one rank,

and starting a few projects, maybe taking some students or something.

yeah but then suddenly people want your opinion on things and uh you have to think a little bit about uh you know things that you might just so foolishly say because you know no one's going to listen to you uh

it's more important now is it constraining to you are you able to still have fun and be a rebel and try crazy stuff and well play with ideas i have a lot less free time than i had previously um i mean mostly by choice i mean i i obviously i have the option to sort of uh decline so i decline a lot of things i i I could decline even more.

Um, or I could acquire a reputation of being so unreliable that people don't even ask anymore.

Uh, this is I love the different algorithms here.

This is great.

This is it's it's always an option.

Um, but you know, um, there are things that are like

I mean, so I mean, I don't spend as much time as I do as a postdoc, you know, 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, so math somehow front loads all the technical skills to the early stages of your career.

So

yeah, so it's as a postdoctoral publisher parish,

you're incentivized to basically focus on proving very technical theorems, so 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 the direction of 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

you get to be a light that's an inspiration to a lot of young mathematicians or young people that are just interested in mathematics.

It's like,

it's just how the human mind works.

This is where I would probably

say that I like the Fields metal,

that it does inspire a lot of young people somehow.

This is just how human brains work.

At the same time, I also want to give sort of respect to somebody like Gregoria Perlman, who

is critical of awards in his mind.

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 necessary and important.

But yeah,

it's also important to not let these things take over your life and like only be concerned about getting the next big award or whatever.

I mean,

yeah, so again, you see these people try to only solve like a 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.

And also if they're attached to a small number of of humans, like

the way our humans mind is wired, we can comprehend the relationships between 10 or 20 people.

But once you get beyond that, like 100 people,

there's a limit.

I pick this name for it,

beyond which it just becomes the other.

And so you have to simplify the pole mass of, you know, 99.9% of humanity becomes the other.

And

often these models are incorrect and this causes all kinds of problems.

But

so, yeah, so to humanize a subject, you know, if you identify a small number of people, I say, you know, these are representative people of a subject, role models, for example,

that has some role.

But it can also be,

yeah, too much of it can be harmful because it's,

I'll be the first to say that my own career traffic.

path is not that of a typical mathematician.

I had a very accelerated education.

I skipped a lot of classes.

I think I was 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.

I mean, they be mathematicians in a very different style.

And we need people of a different style.

And

even if, and 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 building on lots of previous work um but that's a story that's difficult to tell um if you're not an expert because you know it's easier to just say one person did this one thing you know it makes for a much simpler history i think on the whole it um is a hugely positive thing to to talk about steve jobs as a representative of apple when i personally know and of course everybody knows the incredible design the incredible engineering teams just the individual humans on those teams they're not a team They're individual humans on a team, and there's a lot of brilliance there.

But it's just a nice shorthand, like a very, like pie.

Yeah.

Steve Jobs.

Yeah.

Yeah.

Pie.

As a starting point,

as a first approximation.

That's how you do it.

And then read some biographies and then look into much deeper first approximation.

Yeah.

That's right.

So you mentioned you were a person 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 Fermas last year.

What did you think?

Maybe looking back now with more context about that moment in math history.

Yeah, so I was a graduate student at the time.

I mean, I vaguely remember, you know, there was press attention, and

we all had the same,

we all had pigeonholes in the same mail room, you know, so we all pitched out mail.

And like, suddenly, Andrew Wilse's mailbox exploded to be overflowing.

That's a good metric.

Yeah.

You know, so yeah, we all talked about it at tea and so forth.

I mean, we didn't understand, most of us sort of understand the proof.

We understand sort of high-level details.

In fact, there's an ongoing project to formalize it in lean.

Kevin Buzzard is actually.

Yeah, can we take that small tangent?

How difficult does that?

Because as I understand,

the proof for Fermas-Last's theorem has super complicated objects.

Yeah,

really difficult to formalize, no?

Yeah, I guess you're right.

The objects that they use,

you can define them.

So they've been defined in Lean.

So just defining what they are can be done.

That's already 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 in that

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 Fermis Last theorem.

And his aim is that he doesn't think he will be able to get all the way down to the basic axioms, but he wants to formalize it to the point where the only things that he needs to rely on as black boxes are things that were known by 1980 to number theorists at the time.

And then some other person or 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 scope of a high school math education to at least define.

Yeah, but then there's this very advanced algebraic side of number theory where people have been building structures upon structures for quite a while.

And it's a very sturdy structure.

It's been very,

at the base, at least it's extremely well developed in the textbooks and so forth.

But

it does get to the point where

if you haven't taken these years of study and you want to ask about what is going on at like level six of this tower.

You have to spend quite a bit of time before they can even get to the point where you can see you see something you recognize.

What inspires you about his journey?

That was similar as we talked about, seven years

mostly working in secret.

Yeah,

that is a romantic, yeah.

So it kind of fits with sort of the

romantic image that I think people have of mathematicians to the extent that they think of them at all as these kind of eccentric

wizards or something.

So that certainly kind of

accentuated that perspective.

I mean, it is a great achievement.

His style of solving problems is so different from my own,

which is great.

I mean, we need people like that.

In terms of like the, you like the collaborative.

I like moving on from a problem if it's giving too much difficulty.

Got it.

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.

But they're convinced and they have the third, fourth, and the fifth approach works.

And I have to use 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.

Like one thing which is a shame in mathematics is that we have mathematics, there's sort of a one-size-fits-all approach to teaching mathematics.

And, you know, so we have a certain curriculum and so forth.

I mean, you know, maybe like if you do math competitions or something, you get a slightly different experience.

But

I think many people

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, they don't like it.

My theory is that humans don't come, evolution has not given us a math center of a brain directly.

We have a vision center and a language center and some other centers which have evolution has honed, but

we don't have an innate sense of mathematics.

But our other centers are sophisticated enough that different people

we we can repurpose other areas of our brain to do mathematics.

So, some people have figured out how to use the visual center to do mathematics, and so they think think very visually when they do mathematics.

Some people have repurposed the language center and they think very symbolically.

Um, you know, um,

some people like if they are very competitive and they like gaming,

there's a type of there's a part of your brain that's very good at

solving puzzles and games, and and and and that can be repurposed but

like when i talk to other mathematicians you know they don't quite think that i can tell that they're using somehow different styles of of thinking than i am i mean not not disjoint but they they may prefer visual like i'm i i i don't actually prefer visual so much i need lots of visual aids myself um

you know Mathematics provides a common language so we can still talk to each other, even if we are thinking in 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.

And yeah, but I mean, the way we educate, unless you have like a personalized tutor or something, I mean, education sort of just by nature scale has to be mass-produced.

You know, you have to teach to 30 kids.

You know, 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?

What would you give?

Yeah, it's a tricky problem.

One nice thing is that there are now lots of sources for math faculty enrichment outside the classroom.

So in my day, already there are math competitions.

And there are also like popular math books in the library.

But now you have YouTube,

there are forums just devoted to solving math puzzles.

And math shows up in other places.

For example,

there are hobbyists who play poker for fun.

And

they're for very specific reasons are interested in very specific probability questions.

And they actually, there's a community of amateur probabilists in poker.

in chess, in baseball.

I mean,

there's math all over the place.

And

I'm hoping actually with these new sort of tools for lean and so forth that actually we can incorporate the broader public into math research projects.

Like this

doesn't happen at all currently.

So in the sciences, there is some scope for citizen science.

Like astronomers,

there are amateurs who would discover comets and there's biologists, there are people who could identify butterflies and so forth.

And in method, there are a small number of activities where amateur mathematicians can discover new primes and so forth.

But previously, because we have 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 just 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've already contributed to some of these, these formalizing projects, who contributed to Matholib,

you don't need to be a PhD holder to just work on one atomic thing.

There's something about the formalization here that also,

as a very first step, opens it up to the programming community to the people who are already comfortable with programming.

It seems like programming is somehow maybe just the feeling, but it feels more accessible to folks than math.

Math is seen as this like extreme, especially modern mathematics, seen as this extremely difficult to enter area.

And programming is not.

So that could be just an entry point.

You can execute code and you can get results.

You can print out another 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 would not actually program like on the weekend for fun.

Yeah, it would be as considered as hard as math.

So as I said, there are

communities of non-matheticians 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.

Yeah.

So

there's a lot less certainty now in the world.

I mean, there was this period after the war where,

at least in the West, if you came from a good demographic, you,

you know, like you, there was a very stable path to it 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.

So I think you just have to be adaptable and flexible.

I think people have to get skills that are transferable.

You know, like learning one specific programming language or one specific stuff with mathematics or something.

That itself is not a super transferable skill, but sort of knowing how to

reason with abstract concepts or how to problem solve and things go wrong.

These are things which I think we will still need, even as our tools get better.

And

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

But actually, you're an interesting case study.

I mean, you're like

one of the great living mathematicians,

right?

And then you had a way of doing things, and then all of a sudden you start learning.

First of all, you kept learning new fields,

but you learn lean.

That's a non-trivial thing to learn.

Like that's a,

that's a, for a lot of people, that's an extremely uncomfortable leap to take, right?

Yeah.

A lot of mathematicians.

First of all, I've always been interested in new ways to do mathematics.

I, I, I feel like a lot of the ways we do things right now are inefficient.

Um, like, I, I, I spend like many of my colleagues who 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.

and why can't we search and get a quick response and so on so that's why i've always been interested in exploring new workflows about four or five years ago i was on a committee where we had to ask for ideas for interesting workshops to run at a math institute and at the time peter schulzer had just uh formalized one of his his new theorems and um there's some other developments in computer assisted proof that

look quite interesting and I said oh we should we should uh um we should run a workshop on this this would be a good idea

And then I was a bit too enthusiastic about this idea.

So I got volunteered to actually run it.

So 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 the speed and state of the art.

And it was really interesting developments that most mathematicians didn't know what was going on.

That lots of nice proofs of of concept, you know, just sort of hints of what was going to happen.

This was just before ChatGPT, but there was even then there was one talk about language models and the potential capability of those in the future.

So that got me excited about the subject.

So I started giving talks about this is something we should, more of us should start looking at.

Now that I arranged to run this conference.

And then ChatGPT came out and suddenly AI was everywhere.

And so I got interviewed a lot about this topic

and in particular

the interaction between AI and formal proof assistance.

And I said, yeah, they should be combined.

This is

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 formalization.

And there's a limit to how much I can just rely on authority and saying, you know,

I'm a warned mathematician.

Just trust me.

When I say that this is going to change mathematics, when I'm not doing it any, when I don't do any of it myself.

So I felt like I had to actually

uh

justify it yeah

a lot of what i get into actually um i don't quite see an advice as how much time i'm going to spend on it and it's only after i'm sort of waist deep in in in in a project that i i realize by that point i'm committed well that's deeply admirable that you're willing to go into the fray be in some small way a beginner right

or have some of the sort of challenges that a beginner would right yeah new concepts new ways of thinking also,

you know, sucking at a thing that others, I think, I think in that talk,

you could be a Fields Medal winning mathematician and an undergrad knows something better than you.

Yeah.

I think mathematics inherently, I mean, mathematics is so huge these days that nobody knows all of modern mathematics.

And inevitably, we make mistakes.

And,

you know,

you can't cover up your mistakes with just sort of bravado and

I mean, because people will ask for your your proofs and if you don't have the proofs you don't have the proofs um i love math yeah so it does keep us honest i mean not i mean you can still uh it's not a perfect uh panacea but i think uh we do have more of a culture of admitting error than because we're forced to all the time big ridiculous question i'm sorry for it once again who is the greatest mathematician of all time maybe one who's no longer with us uh who are the candidates euler gauss Newton, Ramonagen, Hilbert.

So first of all, as mentioned before, there's some time dependence.

On the day.

Yeah, like if you if you if you plot cumulatively over time, for example, Euclid, like sort of, like, is, is, is one of the lead contenders.

And then maybe some unnamed, anonymous messages before that.

You know, whoever came up with the concept of numbers.

You know.

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, God for us.

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,

this is 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.

And

it's

one thing that actually you have to teach undergraduates in mathematics is that you should always try something.

So

you see 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 to 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, like the next step then is to try anything, like no matter how stupid.

And in fact, almost the stupider the better,

which, you know, I think it could just almost guarantee to fail, but the way it fails is going to be instructive.

Like 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, that you imagine a thing you don't want to do more.

Yes.

Because 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, like, you know, I mean, like,

psychology is really important.

Like, you, you, you, you, you talk to athletes like marathon runners and so forth, and

they talk about what's the most important thing?

Is it their training regimen or the diet?

And so,

so much of it is actually psychology.

You know, 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, you know, like

by induction,

there must be some

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.

I mean,

how much augmentation are you willing?

For example, if I didn't even have pen and paper,

like if I had no technology whatsoever, okay, 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 in extremely complicated, intricate ways, right?

Yeah, yeah, yeah.

So we're already like a collective intelligence.

Yes, yeah, yes.

So humanity plural has much more intelligence in principle on its good days

than the individual humans put together.

It can all have less.

Okay, but

yeah, so yeah,

the mathematical community plural is an incredibly super intelligent

entity

that no single human mathematician can come closer to replicating.

You see it a little bit on these question analysis sites.

So this Math Overflow, which is the math version of Stack Overflow.

And sometimes you get this very quick responses to very difficult questions from the community.

And it's a pleasure to watch, actually,

as an an expert.

I'm a fan spectator of

that site, just seeing the brilliance of the different people there,

the depth of knowledge that some people have and the the willingness to engage in the in the rigor and the nuance of a particular question.

It's pretty cool to watch.

It's fun.

It's almost like just fun to watch.

What gives you hope about this whole thing we have going on, human civilization?

I think, yeah, the um the younger generation is always like like 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.

Navigation, just knowing where you were on the planet was this horrendous problem.

People died

or lost fortunes because they couldn't navigate.

And we have devices in our pockets that do this automatically for us.

It's a completely solved problem.

So things that seem unfeasible for us now could be maybe just homework exercises for me.

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.

Because in the next hundred years, 200 years, just imagine showing up in 200 years.

Yeah.

Well, already plenty has happened.

If you could go back in time and and talk to your teenage self or something,

just the internet and our AI, I mean,

they're getting to be internalized.

And so, yeah, of course, an AI can understand our voice and give reasonable,

slightly incorrect answers to any question.

But yeah,

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.

And then

we humans seem to entertain ourselves with drama.

Out of anything that's created, somebody needs to take one opinion, another person needs to take an opposite opinion, argue with each other about it.

But when you look at the arc of things, I mean, just even in progress of robotics,

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 more intelligent and mature and rational than the individuals within it.

Well, one place I can always count on rationality is the comment section of your blog, which I'm a fan of.

There's a lot of really smart people there.

And thank you, 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.

Oh, thank you.

It was a pleasure.

Thanks for listening to this conversation with Terrence Tao.

To support this podcast, please check out our sponsors in the description or at 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 next time.