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

The following is a conversation with

Terrence Tao. Widely considered to be

one of the greatest mathematicians in

history. Often referred to as the Mozart

of math, he won the Fields Medal and the

Breakthrough Prize in mathematics and

has contributed groundbreaking work to a

truly astonishing range of fields in

mathematics and physics.

This was a huge honor for me for many

reasons, including the humility and

kindness that Terry showed to me

throughout all our interactions. It

means the world. This is the Lex

Freedman podcast. To support it, please

check out our sponsors in the

description or at

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And now, dear friends, here's Terren

Tao.

What was the first really difficult

research level math problem that you

encountered? One that gave you pause

maybe. Well, I mean in your

undergraduate um education, you learn

about the really hard impossible

problems like the reman hypothesis, the

twin primes conjecture. You can make

problems arbitrarily difficult. That's

not really a problem. In fact, there's

even problems that we know to be

unsolvable. What's really interesting

are the problems just at the on the

boundary between what we can do

relatively easily and what are hopeless.

Um but what are problems where like

existing techniques can do like 90% of

the job and then you just need that

remaining 10%. Um I think as a PhD

student the CA problem certainly caught

my eye and it just got solved actually.

It's a problem I've worked on a lot in

my early research. Historically it came

from a little puzzle by the Japanese

mathematician Soji Kaya uh in like 1918

or so. Um, so the puzzle is that you you

you have um a needle um in on the plane.

Um think like like a like driving like

on on on a road something and you you

want it to execute a U-turn. You want to

turn the needle around. Um but you want

to do it in as little space as possible.

So you want to use as little area in

order to turn it around. So um but the

needle is infinitely maneuverable.

So you can imagine just spinning it

around its um as a unit needle. You can

spin it around its center. Um, and I

think, um, that gives you a disc of of

area, I think pi over four. Um, or you

can do a three-point U-turn, which is

what they we teach people in in the

driving schools to do. Uh, and that

actually takes area pi over 8. So, it's

it's a little bit more efficient than um

a rotation. And so, for a while, people

thought that was the most efficient uh

way to turn things around. But,

Mazikovich uh showed that in fact, you

could actually uh turn the needle around

using as little area as you wanted. So

0001 there was some really fancy multi-

um u back and forth U-turn thing that

you could you could do that that you

could turn a needle around and in so

doing it would pass through every

intermediate direction. Is this in the

two dimensional plane? This is in the

two dimensional plane. Yeah. So we

understand everything in two dimensions.

So the next question is what happens in

three dimensions. So suppose like the

Hubble space telescope is tube in space

and you want to observe every single

star in the universe. So you want to

rotate the telescope to reach every

single direction. And here's unrealistic

part. Suppose that space is at a

premium, which it totally is not. Uh you

want to occupy as little volume as

possible in order to rotate your your

needle around in order to see every

single star in the sky. Um how small a

volume do you need to do that? And so

you can modify basic construction. And

so if your telescope has zero thickness,

then you can use as little volume as you

need. That's a simple modification of

the two dimensional construction. But

the question is that if your telescope

is not zero thickness but but just very

very thin some thickness delta what is

the minimum volume needed to be able to

see every single direction as a function

of delta. So as delta gets smaller as

you need gets thinner the volume should

go down but but how fast does it go

down? Um and the conjecture was that it

goes down very very slowly um like

logarithmically um uh roughly speaking

and that was proved after a lot of work.

So this seems like a puzzle. Why is it

interesting? So it turns out to be

surprisingly connected to a lot of

problems in partial differential

equations, in number theory, in

geometry, comics. For example, in in

wave propagation, you splash some some

water around um you create water waves

and they they travel in various

directions. Um but waves exhibit both

both particle and wave type behavior. So

you can have what's called a wave

packet, which is like a a very localized

wave that is localized in space and

moving a certain direction in time. And

so if you plot it in both space and

time, it occupies a region which looks

like a tube. And so what can happen is

that you can have a wave which initially

is very dispersed but it all comes it

all focuses at a single point later in

time. Like you can imagine dropping a

pebble into a pond and ripples spread

out. But then if you time reverse that

that um that scenario and the equations

of wave motion are time reversible. You

can imagine ripples that are converging

um to a single point and then a big

splash occurs um maybe even a

singularity.

Um and so it's possible to do that. Uh

and geometrically what's going on is

that there's always s of light rays. Um

so like if if if this wave represents

light for example um you can imagine

this wave as a superp position of

photons um all traveling at the speed of

light. They all travel on these light

rays and they're all focusing at this

one point. So you can have a very

dispersed wave focus into a very

concentrated wave at one point in space

and time, but then it defocuses again

and it separates. But potentially if the

conjecture had a negative solution. So

what that meant is that there's there's

a very efficient way to pack um tubes

pointing different directions into a

very very narrow region of of of very

narrow volume. Then you would also be

able to create waves that start out some

there'll be some arrangement of waves

that start out very very dispersed but

they would concentrate not just at a

single point but um um there'll be a

large um there'll be a lot of

concentrations in space and time and uh

um and you could create what's called a

blowup where these waves their amplitude

becomes so great that the laws of

physics that they're governed by are no

longer wave equations but something more

complicated and nonlinear. Um and so in

mathematical physics we care a lot about

whether certain equations in in wave

equations are stable or not whether they

can create um these singularities.

There's a famous unsolved problem called

the Navia Stokes regularity problem. So

the Navia Stokes equations equations

that govern the fluid flow for

incompressible fluids like water. The

question asks if you start with a smooth

velocity field of water can it ever

concentrate so much that like the

velocity becomes infinite at some point

that's called a singularity. We don't

see that um in real life. You know, if

you splash around water on the bathtub,

it won't explode on you. Um or or have

have water leaving at the speed of

light, I think. But potentially, it is

possible. Um and in fact, in recent

years, the the consensus has has drifted

towards the uh the belief that uh that

in fact for certain very special initial

configurations of of say water that

singularities can form. But people have

not yet been able to uh to actually

establish this. The clay foundation has

these seven millennium prize problems

has a million dollar prize for solving

one of these problems that this is one

of them. Of these seven only one of them

has been solved the point conjecture by

Pelman. So the Ka conjecture is not

directly directly related to the Navis

Stokes problem but understanding it

would help us understand some aspects of

things like wave concentration which

would indirectly probably help us

understand the Navis problem better. Can

you speak to the neighbors? 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 threedimensional

navia stoke equation right

so we're trying to figure out if this

thing usually doesn't blow up right but

can we say for sure it never blows up

right yeah so yeah that is literally the

the million- dollar question yeah so

this is what distinguishes

mathematicians from pretty much

everybody else like it

If something holds 99.99% of the time,

um that's good enough for most, you

know, uh for for most things, but

mathematicians are one of the few people

who really care about whether every like

100% really 100% of all um situations

are covered by by um yeah, so most fluid

most of the time um water that does not

blow up. But could you design a very

special initial state that does this?

And maybe we should say that this is a

this is a set of equations that govern

in the field of fluid dynamics. Trying

to understand how fluid behaves and it's

actually turns out to be a really comp

you know fluid is yeah extremely

complicated thing to try to model. Yeah.

So it has practical importance. So this

clay price problem concerns what's

called the incompressible navio stokes

which governs things like water. There's

something called the compressible navio

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 ny stokes equations

as best they can. Um also gathering a

lot of data so that they can get they

can in initialize the equation. There's

a lot of moving parts. So it's very

important practically. Why is it

difficult to prove general things

about the set of equations like it not

not blowing up? Short answer is

Maxwell's demon. Um so exos demon is a

concept in thermodynamics like if you

have a box of two gases and oxygen and

hydrogen uh and maybe you start with all

the oxygen one side and nitrogen the

other side but there's no barrier

between them right then they will mix um

and they should stay mixed right there

there's no reason why they should unmix

but in principle because of all the

collisions between them there could be

some sort of weird conspiracy that that

um like maybe there's a microscopic

demon called Maxwell's demon that will

um every time a oxygen and nitrogen atom

collide they will bounce off in such a

way that the oxygen sort of drifts onto

one side and then goes to the other and

uh you could have an extremely

improbable configuration emerge. Uh

which we never see. Um and and we

statistically it's extremely unlikely

but mathematically it's possible that

this can happen and we can't rule it

out. Um and this is a situation that

shows up a lot in mathematics. Um a

basic example is the digits of pi

3.14159 and so forth. The digits look

like they have no pattern and we believe

they have no pattern. On the long term,

you should see as many ones and twos and

threes as fours and fives and sixes.

There should be no preference in the

digits of pi to favor let's say 7 over

8. Um, but maybe there's some demon in

the digits of pi that that like every

time you compute more digits, it sort of

biases one digit to another. Um, and

this is a conspiracy that should not

happen. There's no reason it should

happen, but um there's there's there's

no way to prove it.

uh with our current technology. Okay. So

getting back to Nabia Stokes, a fluid

has a certain amount of energy and

because a fluid is in motion, the energy

gets transported around and water is

also viscous. So if the energy is spread

out over many different locations, the

natural viscosity of the fluid will just

damp out the energy and will it will go

to zero. Um and this is what happens um

in um uh when we actually experiment

with water like you splash around there.

there's some turbulence and waves and so

forth. But eventually it it settles down

and and and the the lower the amplitude,

the smaller the velocity, the the more

calm it gets. Um but potentially there

is some sort of a demon that keeps

pushing the uh the energy of the fluid

into a smaller and smaller scale and it

will move faster and faster and at

faster speeds the effective viscosity is

relatively less. And so it could happen

that that it it creates a some sort of

um um what's called a self similar

blowup scenario where you know um the

energy of fluid starts off at some um

large scale and then it all sort of um

transfers it energy into a smaller um

region of of of the fluid which then at

a much faster rate um moves into um an

even smaller region and so forth. Um and

and each time it does this uh it takes

maybe half as as long as as the previous

one and then you you could you could

actually uh converge to all the energy

concentrating in one point in a finite

amount of time. Um and that that's uh

that scenario is called finite blow up.

Um so in practice this doesn't happen.

Um so water is what's called turbulent.

Um so it is true that um if you have a

big eddy of water it will tend to break

up into smaller eddies but it won't

transfer all the the energy from one big

eddy into one smaller eddy. It will

transfer into maybe three or four and

then those must split up into maybe

three or four small edies of their own

and so the energy gets dispersed to the

point where the viscosity can can then

keep that thing under control. Um but if

it can somehow um concentrate um all the

energy keep it all together um and do it

fast enough that the viscous effects

don't have enough time to calm

everything down then this blob can

occur. So there were papers who had

claimed that oh you just need to take

into account conservation energy and

just carefully use the viscosity and you

can keep everything under control for

not just Navia 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 Navio Stokes which is the

opposite of final time blow up that

velocity say smooth and it all failed

there was always some sign error or some

subtle mistake and and it couldn't be

salvaged. Um so what I was interested in

doing was trying to explain why we were

not able to disprove um planet time blow

up. I couldn't do it for the actual

equations of fluids which were too

complicated. But if I could average the

equations of motion of naval basically

if if um if I could turn off certain

types of of ways in which water

interacts and only keep the ones that I

want. Um, so in particular, um, if, um,

if there's a fluid and it could transfer

energy from a large Eddie into this

small Eddie or this other small Eddie, I

would turn off the energy channel that

would transfer energy to this this one

and and direct it only into um, this

smaller Eddie while still preserving the

law of conservation of energy. So you're

trying to make it blow up. Yeah. Yeah.

So I I I basically engineer um, a blow

up by changing the laws of physics,

which is one thing that mathematicians

are allowed to do. We can change the

equation. How does that help you get

closer to the proof of something? Right?

So, it provides what's called an

obstruction in mathematics. Um, so, so

what I did was that uh basically if I

turned off the um certain parts of the

equation, so which usually when you turn

off certain interactions make it less

nonlinear, it makes it more regular and

less likely to blow up. But I found that

by turning off a very well-designed set

of of of of interactions, I could force

all the energy to blow in finite time.

So what that means is that if you wanted

to prove um global regularity for Navia

Stokes um for the actual equation you

had you must use some feature of the

true equation which which my artificial

equation um does not satisfy. So it it

rules out certain um certain approaches.

So um the thing about math is is it's

not just about finding you know taking a

technique that is going to work and

applying it but you you need to not take

the techniques that don't work. Um and

for the problems that are really hard,

often there are dozens of ways that you

might think might apply to solve the

problem. But uh it's only after a lot of

experience that you realize there's no

way that these methods are going to

work. So having these counter examples

for nearby problems um kind of rules out

um uh it saves you a lot of time because

you you're not wasting um energy on on

things that you now know cannot possibly

ever work. How deeply connected is it to

that specific problem of fluid dynamics

or just some more general intuition you

build up about mathematics? Right. Yeah.

So the key phenomenon that uh my my

technique exploits is what's called

superc criticality. So in partial

differential equations often these

equations are like a tugof-war between

different forces. So in Navia Stokes

there's the dissipation um force coming

from viscosity and it's very well

understood. It's linear. It calms things

down. If if viscosity was all there was,

then then nothing bad would ever happen.

Um but there's also transport um that

that energy from in one location of

space can get transported because the

fluid is in motion to to other

locations. Um and that's a nonlinear

effect and that causes all the all the

problems. Um so there are these two

competing terms in the Davis equation

the dissipation term and the transport

term. If the dissipation term dominates,

if it's if it's large, then basically

you get regularity. And if um if the

transport term dominates, then uh then

we don't know what's going on. It's a

very nonlinear situation. It's

unpredictable. It's turbulent. So

sometimes these forces are in balance at

small scales, but not in balance at

large scales or or vice versa. Um so

Navis Stokes is what's called

supercritical. So at at smaller and

smaller scales, the transport terms are

much stronger than the viscosity terms.

So the viscosity are the things that

calm things down. Um and so this is um

um this is why the problem is hard in

two dimensions. So the Soviet

mathematician ladish skaya she in the

60s shows in two dimensions there is no

blow up and in two dimensions the nav

equations is what's called critical the

effect of transport and the effect of

viscosity about the same strength even

at very very small scales and we have a

lot of technology to handle critical and

also subcritical equations and proof um

regularity but for superc critical

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 superc critical

equations you create all kinds of blow

up examples. Once the nonlinear effects

dominate the linear effects at small

scales, you can have all kinds of bad

things happen. So this is sort of one of

the main insights of this this line of

work is that superc criticality versus

criticality and subcriticality. This

this makes a big difference. I mean

that's a key qualitative feature that

distinguishes some equations for being

sort of nice and predictable and you

know like like planetary motion and I

mean there are certain equations that

that you can predict for millions of

years and or thousands at least. Again,

it's not really a problem, but but

there's a reason why we can't predict

the weather past 2 weeks into the future

because it's a super critical equation.

Lots of really strange things are going

on at very fine scales. So, whenever

there is some huge source of

nonlinearity,

yeah, that can create a huge problem for

predicting what's going to happen. Yeah.

And if the nonlinearity is somehow more

and more featured and interesting at at

small scales. Um I mean there's there's

many equations that are nonlinear but um

in in many equations you can approximate

things by the bulk. Um so for example

planetary motion you know if you want to

understand the orbit of the moon or Mars

or something you don't really need the

micro structure of like the seismology

of the moon or or like exactly how the

mass is distributed. um you just

basically you can almost approximate

these planets by point masses and just

the aggregate behavior is important um

but if you want to model a fluid um like

like the weather you can't just say in

Los Angeles the temperature is this the

wind speed is this for super critical

equations the finance confirmation is is

really important if we can just linger

on the narto's uh equations a little bit

so you've suggested maybe you can

describe it that one of the ways to uh

solve it or to negatively resolve it

would be to

sort of to construct a liquid a kind of

liquid computer, right? And then show

that the halting problem from

computation theory has consequences for

fluid dynamics. So uh show it in that

way. Can you describe this this Yeah. So

this came out of of this work of

constructing this this this average

equation that that blew up. Um so one um

as as part of how I had to do this. So

there this naive way to do it. You you

just keep pushing um um every time you

you get energy at one scale you you push

it immediately to the next scale as as

fast as possible. This is sort of the

naive way to to to to force blow up. Um

it turns out in five and high dimensions

this works. Um but in three dimensions

there was this funny phenomenon that I

discovered that if you if you keep if if

you change the laws of physics you just

always keep trying to push um the energy

into smaller smaller scales. Um what

happens is that the energy starts

getting spread out into multi many

scales at once. Um so that you you have

energy at one scale you're pushing it

into the next scale and then um as soon

as it enters that scale you also push it

to the next scale but there's still some

energy left over from the previous

scale. um you're trying to do everything

at once. Um and this spreads out the

energy too much. Um and then it turns

out that that um it makes it vulnerable

for viscosity to come in and actually

just damp out everything. So um so it

turns out this this direct bush doesn't

doesn't actually work. There was a

separate paper by some other authors

that actually showed this um in three

dimensions. Um so what I needed was to

program a delay. Um so kind of like air

locks. So um I needed an equation which

would start with a fluid doing something

at one scale. It would push this energy

into the next scale but it would stay

there until all the energy from the from

the larger scale got transferred and

only after you pushed all the energy in

then you sort of open the next gate and

and then you you push that in as well.

So um by doing that it kind of the

energy inches forward scale by scale in

such a way that it's always um localized

at one scale at a time. Um and then it

can resist the effects of viscosity

because it's not dispersed. Um so in

order to make that happen um yeah I had

to construct a rather complicated

nonlinearity. Um and it was basically

like um you know like was constructed

like electronic circuit. So I I actually

thank my wife for this because she was

trained as a electrical engineer. Um and

um you know he talked about um uh you

know he had to design circuits and so

forth. And you know if if you want a

circuit that does a certain thing like

maybe have a light that that flashes on

and then turns off and then on and then

off. You can build it from from more

primitive components you know capacitors

and resistors and so forth and you have

to build a diagram and you um and these

diagrams you can you can sort of follow

your eyeballs and say oh yeah the the

current will build up here and then it

will stop and then it will do that. So I

knew how to build the analog of basic

electronic components, you know, like

resistors and capacitors and so forth.

And and I would I would stack them

together um in in such a way that that I

would create something that would open

one gate and then there'll be a clock

that would and then once the clock hits

a certain threshold it would close it

kind of a rude Goldberg type machine but

described mathematically and this ended

up working. So what I realized is that

if you could pull the same thing off for

the actual equations. So if the

equations of water support a computation

so um like if you can imagine kind of a

steampunk but really water punk uh type

of thing where um you know so modern

computers are electronic you know they

they they're powered by by electrons

passing through very tiny wires and

interacting with other electrons and so

forth. But instead of electrons, you can

imagine these pulses of of water moving

at certain velocity and maybe it's

they're two different configurations

corresponding to a bit being up or down.

Probably if you had two of these moving

bodies of water collide, it would come

out with some new configuration which is

which would be something like an ANDgate

or orgate. you know that if the the the

output would depend in a very

predictable way on on the inputs and

like you could chain these together and

maybe create a touring machine and and

then you could you have computers which

are made completely out of water um and

if you have computers then maybe you can

do robotics so I you know hydraulics and

so forth um and so you could create some

machine which is basically a fluid

analog what's called a vonomian machine

so vonomian proposed if you want to

colonize Mars. The sheer cost of

transporting people machines to Mars is

just ridiculous. But if you could

transport one machine to Mars and this

machine had the ability to mine the

planet, create some more materials to

smelt them and build more copies of the

same machine. Um, then you could

colonize a whole planet um over time.

Um, so uh if you could build a fluid

machine, which uh yeah, so it's it's

it's a it's a robot. Okay. And what it

would do it its purpose in life, it's

programmed so that it would create a

smaller version of itself in some sort

of cold state. It wouldn't start just

yet. Once it's ready, the big robot

configuration water would transfer all

his energy into the smaller

configuration and then power down. Okay?

And then like I clean itself up. And

then what's left is this newest state

which would then turn on and do the same

thing but smaller and faster. And then

the equation has a certain scaling

symmetry. Once you do that, it can just

keep iterating. So this in principle

would create a blow up uh for the actual

Navia Stokes and this is what I managed

to accomplish for this average Navia

Stokes. So it provided the sort of road

map to solve the problem. Now this is uh

a pipe dream because uh there are so

many things that are missing for this to

actually be a reality. Um so um I I I

can't create these basic logic gates. Um

I I don't I don't have these in these

special configurations of water. Um, I

mean there's candidates there things

called vortex rings that might possibly

work but um um but also you know analog

computing is really nasty um compared to

digital computing. I mean because

there's always errors um you you have to

you have to do a lot of error correction

along the way. I don't know how to

completely power down the big machine so

that it doesn't interfere with the the

running of the smaller machine but

everything in principle can happen like

it doesn't contradict any of the laws of

physics. Um so it's sort of evidence

that this thing is possible. Um there

are other groups who are now pursuing

ways to make navis blow up which are

nowhere near as ridiculously complicated

as this. Um um they they actually are

pursuing much closer to the the direct

self similar model which can it doesn't

quite work as is but there could be some

simpler scheme than what I just

described to make this work. There is a

real leap of genius here to go from

Navia Stokes to this touring machine. So

it goes from what the self similar blob

scenario that you're trying to get the

smaller and smaller blob to now having a

liquid toying machine gets smaller and

smaller and smaller and somehow seeing

how that

could be used

to say something about a blowup. I mean

that's a big leap. So there's precedent.

I mean um so the the thing about

mathematics is that it's really good at

um spotting connections between what you

think of what you might think of as

completely different um problems. Um but

if if the mathematical form is the same

you you can you you can you can draw a

connection um so um there's a lot of

work previously on what called cellular

automator um the most famous of which is

Conway's game of life. there's this

infinite discrete grid and at any given

time the grid is either occupied by a

cell or it's empty and there's a very

simple rule that uh tells you how these

cells evolve. So sometimes cells live

and sometimes they die. Um and this um

you know um when I was a a student it

was a very popular screen saver to

actually just have these these

animations going and and they look very

chaotic. In fact they look a little bit

like turbulent float sometimes. But at

some point people discovered more and

more interesting structures within this

game of life. Um so for example they

discovered this thing called a glider.

So a glider is a very tiny configuration

of like four or five cells which evolves

and it just moves at a certain direction

and that's like this this vortex rings

this um yeah so this is an analogy the

game of life is kind of like a discrete

equation and and um the flu navis is a

continuous equation but mathematically

they have some similar features um and

um so over time people discovered more

and more interesting things you could

build within the game of life. The game

life is a very simple system. It only

has like three or four rules um to to do

it, but but you can design all kinds of

interesting configurations inside it. Um

there's something called a glider gun

that does nothing to spit out gliders

one at a one one at a time. Um and then

after a lot of effort, people managed to

to create um and gates and or gates for

gliders. Like there's this massive

ridiculous structure which if you if a

if you have a stream of gliders um

coming in here and a stream of gliders

coming in here then you may produce a

stream of gliders coming out. If so

maybe if both of of the um streams um

have gliders then there'll be an output

stream but if only one of them does then

nothing comes out. Mhm. So they could

build something like that. And once you

could build and um these basic gates

then just from software engineering you

can build almost anything. Um you can

build a touring machine. I mean it's

like an enormous steampunk type things.

They look ridiculous. But then people

also generated self-replicating objects

in the game of life. A massive machine a

bon machine which over a huge period of

time and it always look like glider guns

inside doing these very steampunk

calculations. it would create another

version of itself which could replicate.

It's so incredible. A lot of this was

like community crowdsourced by like

amateur mathematicians actually. Um so I

knew about that that that work and so

that is part of what inspired me to

propose the same thing with Navia

Stokes. Um which is a much as I said

analog is much worse than digital like

it's going to be um you can't just

directly take the constructions in the

game of life and plunk them in. But

again it just it shows it's possible.

You know, there's a kind of emergence

that happens with these cellular automa.

Local rules.

Maybe it's similar to fluids. I don't

know. But local rules operating at scale

can create these incredibly complex

dynamic structures. Do you think any of

that is amendable to mathematical

analysis?

Do we have the tools to say something

profound about that? The thing is you

can get this emerg in very complicated

structures but only with very carefully

prepared initial conditions. Yeah. So so

these these these glider guns and and

gates and and so forth machines if you

just plunk down randomly some cells and

you and you will not see any of these.

Um and that's the analogous situation

with Navia Stokes again you know that

that with with typical initial

conditions you you will not have any of

this weird computation going on. Um but

basically through engineering you know

by by by specially designing things in a

very special way you can make clever

constructions. I wonder if it's possible

to prove the sort of the negative of

like basically prove that only through

engineering can you ever create

something interesting. This this is a

recurring challenge in mathematics that

um I call it the dichotomy between

structure and randomness. That most

objects that you can generate in

mathematics are random. They look like

rand like the digits of pi. Well, we

believe is a good example. Um, but

there's a very small number of things

that have patterns. Um, but um, now you

can prove something has a pattern by

just constructing, you know, like if

something has a simple pattern and you

have a proof that it it does something

like repeat itself every so often. You

can do that. But um, and you you can

prove that that for example, you can you

can prove that most sequences of of

digits have no pattern. Um, so like if

you just pick digits randomly, there's

something called low large numbers. It

tells you you're going to get as many

ones as as twos in the long run. Um but

um we have a lot fewer tools to to to if

I give you a specific pattern like the

digits of pi how can I show that this

doesn't have some weird pattern to it.

Some other work that I spend a lot of

time on is to prove what are called

structure theorems or inverse theorems

that give tests for when something is is

very structured. So some functions are

what's called additive like if you have

a function that maps natural numbers

with natural numbers. So maybe um you

know two maps to four three maps to six

and so forth. um some functions what's

called additive which means that if you

add if you add two inputs together the

output gets gets added as well uh for

example multiplying by a constant if you

multiply a number by 10 um if you if you

multiply a plus b by 10 that's the same

as multiplying a by 10 and b by 10 and

then adding them together so some um

functions are additive some are kind of

additive but not completely additive um

so for example if I take a number n I

multiply by the square root of two and I

take the integer part of that So 10 by

square of two is like 14 point

something. So 10 up to 14. Um 20 up to

28. Um so in that case additively is

true then. So 10 + 10 is 20 and 14 + 14

is 28. But because of this rounding

sometimes there's roundoff errors and

and sometimes when you um add a plus b

this function doesn't quite give you the

sum of of the two individual outputs but

the sum plus minus one. Um so it's

almost additive but not quite additive.

Um so there's a lot of useful results in

mathematics and I've worked a lot on

developing things like this to the

effect that if if a function exhibits

some structure like this then um it's

basically there's a reason for why it's

true and the reason is because there's

there's some other nearby function which

is actually um completely structured

which is explaining this sort of partial

pattern that you have. Um and so if you

have these so inverse theorems it um it

creates this sort of dichotomy that they

either the objects that you study are

either have no structure at all or they

are somehow related to something that is

structured. Um and in either way in

either um in either case you can make

progress. Um a good example of this is

that there's this old theorem in

mathematics called sim theorem proven in

the 1970s. It concerns trying to find a

certain type of pattern in a set of

numbers. the patterns that have make

progression things like 3 five and seven

or or or 10 15 and 20 andreli

proved that um any set of of numbers

that are sufficiently big um what's

called positive density has um

arithmetic progressions in it of of any

length you wish um so for example um the

odd numbers have a set of density 1/2 um

and they contain arithmetic progressions

of any length um so in that case it's

obvious because the the odd numbers are

really really structured I can just take

11 13 15 17 I just I can I can easily

find arithmetic progressions in in in

that set. Um but um zerminism also

applies to random sets. If I take the

set of odd numbers and I flip a coin um

and for each number and I only keep the

numbers which for which I got a heads

okay so I just flip coins. I just

randomly take out half the numbers I

keep one half. So that's a set that has

no no patterns at all. But just from

random fluctuations, you will still get

a lot of um um of arithmetic

progressions in that set. Can you prove

that

there's arithmetic progressions of

arbitrary length within a random? Yes.

Um have you heard of the infinite monkey

theorem? Usually mathematicians give

boring names to theorists, but

occasionally they they give colorful

names. Yes. The popular version of the

infinite monkey theorem is that if you

have an infinite number of monkeys in a

room with each with a typewriter they

type out uh text randomly almost surely

one of them is going to generate the

entire screw of Hamlet or any other

finite string of text. Uh it will just

take some time quite a lot of time

actually but if you have an infinite

number then it happens. Um so um

basically the the if you take an

infinite string of of digits or whatever

um eventually any finite pattern you

wish will emerge. Um it may take a long

time but it will eventually happen. Um

in particular arithmetic progressions of

any length will eventually happen. Okay.

But you need that but you need an

extremely long random sequence for this

to happen. I suppose that's intuitive.

It's just infinity. Yeah. Infinity

absorbs a lot of sins. Yeah. How are we

humans supposed to deal with infinity?

Well, you can think of infinity as as as

just an abstraction of um a finite

number for which you you do not have a

bound for um that uh you know I mean so

nothing in real life is truly infinite.

Um but you know you can um you know you

can ask yourself questions like you know

what if I had as much money as I wanted

you know or what if I could go as fast

as I wanted and a way in which

mathematicians formalize that is

mathematics has found a formalism to

idealize instead of something being

extremely large or extremely small to

actually be exactly infinite or zero. Um

and often the the mathematics becomes a

lot cleaner when you do that. I mean in

physics we we joke about uh assuming

spherical cows. um you know like real

world problems have got all kinds of

real world effects but you can idealize

send certain things to infinity send

certain things to zero um and um and the

mathematics becomes a lot simpler to

work with there. I wonder how often

using infinity

uh forces us to deviate from um the

physics of reality. Yeah. So there's a

lot of pitfalls. Um so you know we we

spend a lot of time in undergraduate

math classes teaching analysis. Um and

analysis is often about how to take

limits and and and and whether you you

know so for example a plus b is always b

plus a. Um so when you have a finite

number of terms you add them you can

swap them and there there's no problem.

But when you have infinite number of

terms there these sort of shell games

you can play where you can have a series

which converges to one value but you

rearrange it and it suddenly converges

to another value. And so you can make

mistakes. You have to know what you're

doing when you allow infinity. Um you

have to introduce these epsilons and

deltas and and this there's a certain

type of way of reasoning that helps you

avoid mistakes. Um

in more recent years um people have

started taking results that are true in

infinite limits and what's called

finetizing them. Um so you know that

something's true eventually but um you

don't know when. Now give me a rate.

Okay. Okay, so it's such a if I have

don't have an infinite number of monkeys

but but a large finite number of

monkeys, how long do I have to wait for

H to come out? Um and that's a more

quantitative question. Um and this is

something that you can you can um attack

by purely finite methods and you can use

your finite intuition. Um and in this

case it turns out to be exponential in

the length of the text that you're

you're trying to generate. Um so um and

so this is why you never see the monkeys

create Hamilton. you can maybe see them

create a four-letter word, but nothing

that big. And so I personally find once

you finitize an infinite statement, it's

it does become much more intuitive and

it's no longer so so weird. Um so even

if you're working with infinity, it's

good to finitize so that you can have

some intuition. Yeah. The downside is

that the finite groups are just much

much messier and and uh yeah. So so the

infinite ones are found first usually

like decades earlier and then later on

people finize them. So since we

mentioned a lot of math and a lot of

physics uh what is the difference

between mathematics and physics as

disciplines as ways of understanding of

seeing the world maybe we can throw in

engineering in there you mentioned your

wife is an engineer give it new

perspective on circuits right so this

different way of looking at the world

given that you've done mathematical

physics so you you've you've worn all

the hats right so I think science in

general is interaction between three

things um there's the real world um

there's is what we observe of the

reward, our observations and then our

mental models as to how we think the

world works. Um so um we can't directly

access reality. Okay. Uh all we have are

the observations which are incomplete

and they they have errors. Um and um

there are many many cases where we would

um uh we want to know for example what

is the weather like tomorrow and we

don't yet have the observation we'd like

to a prediction. Um and then we have

these simplified models sometimes making

unrealistic assumptions you know

spherical cow type things. Those are the

mathematical models. Mathematics is

concerned with the models. Science

collects the observations and it

proposes the models that might explain

these observations. What mathematics

does we we stay within the model and we

ask what are the consequences of that

model? what observations would what

predictions would the model make of the

of future observations um or past

observations does it fit observed data

um so there's definitely a symbiosis um

it's ma I guess mathematics is is

unusual among other disciplines is that

we start from hypothesis like the axims

of a model and ask what conclusions come

up from that that model um in almost any

other discipline uh you start with the

conclusions you know I want to do this I

want to build a bridge, you know, I I

want to to make money. I want to do

this. Okay. And then you you you find

the path to get there. Um

a lot there there's a lot less sort of

speculation about suppose I did this,

what would happen? Um you know, planning

and and and modeling um uh speculative

fiction maybe is one other place. Uh but

uh that's about it actually. Most of

things we do in life is conclusions

driven including physics and science.

You I mean they want to know you know

where is this asteroid going to go? What

was what what is the weather going to be

tomorrow? Um but um Bathe also has this

other direction of of going from the uh

the axioms. What do you think there is

this tension in physics between theory

and experiment? Mhm. What do you think

is the more powerful way of discovering

truly novel ideas about reality? Well,

you need both top down and bottom up. Um

yeah, it's it's a real interaction

between all these things. So over time

the observations and the theory and the

modeling should both get closer to

reality. But initially and it is I mean

this is um this is always the case. You

know they're always far apart to begin

with. Um but you need one to figure out

where to push the other you know. So um

if your model is predicting anomalies um

that are not picked up by experiment

that tells experimenters where to look

you know um to to to to find more data

to refine the models. Um yeah so it it

it goes it goes back and forth. Um

within mathematics itself there's

there's also a theory and experimental

component. It's just that until very

recently theory has dominated almost

completely like 99% of mathematics is

theoretical mathematics and there's a

very tiny amount of experimental

mathematics. Um I mean people do do it

you know like if they want to study

prime numbers or whatever they can just

generate large data sets and with a so

once we had computers um we be to do it

a little bit. Um although even before

well like Gaus for example he discovered

he conjectured the most basic theorem in

in number theory to call the prime

number theorem which predicts how many

primes that 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 um by himself but

also hired human computers um people who

whose professional job it was to do

arithmetic um to compute the first

100,000 tribes or something and made

tables and made a prediction um that was

an early example of experimental

mathematics

um but until very recently it was not um

yeah I mean theoretical mathematics was

just much more successful I mean because

doing complicated mathematical

computations is uh was just not not

feasible until very recently. Uh and

even nowadays, you know, even though we

have powerful computers, only some

mathematical things can be um explored

numerically. There's something called

the comatorial explosion. If you want us

to study, for example, Zodius the you

want to study all possible subsets of

the numbers 1 to a,000. There's only

1,000 numbers. How bad could it be? It

turns out the number of different

subsets of of 1 to a,000 is 2 to the^

1,000 which is way bigger than than that

any computer can currently can can in

fact anybody ever will ever um

enumerate. Um so you have you have to be

um there are certain math problems that

very quickly become just intractable to

attack by direct brute force

computation. Uh chess is another um

famous example. The number of chess

positions uh we can't get a computer to

fully explore.

But now we have AI um um we have tools

to explore this space not with 100%

guarantees of success but with

experiment you know so like um we can

empirically solve chess now for example

we have we have very very good AIs that

that can you know they don't explore

every single position in in the game

tree but they have found some very good

approximation um and people are using

actually these chess engines to make uh

to do experimental chess um that they're

revisiting old chess theories about, oh,

you know, when you this type of opening,

you know, this is a good, this is a good

type of move, this is not, and they can

use these chess engines to actually

refine in some case overturn um um

conventional wisdom about chess. And I

do hope that uh that mathematics will

will have a larger experimental

component in the future perhaps powered

by AI. We'll of course talk about that

but in the case of chess and there's a

similar thing in mathematics that I

don't believe it's providing a kind of

formal explanation of the different

positions. It's just saying which

position is better or not that you can

intuit it as a human being and then from

that we humans can construct a theory of

the matter. You've mentioned the Plato's

cave allegory. Mhm. So in case people

don't know, it's where people are

observing shadows of reality, not

reality itself, and they believe what

they're observing to be reality. Is that

in some sense what mathematicians and

maybe all humans are doing is um looking

at shadows

of reality? Is it possible for us to

truly access

reality? Well, there these three

onlogical things. there's actual

reality, there's our observations and

our our models. Um, and technically they

are distinct and I think they will

always be distinct. Um, but they can get

closer um over time. Um, you know, so um

and the process of getting closer often

means that you you have to discard your

initial intuitions. Um so um like

astronomy provides great examples you

know like you know like you an initial

model of the world is is flat because it

looks flat 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 um and so you start off with a

model which is actually really far from

reality um but it fits kind of the

observations that you have um you know

so you know so things look good you know

but but over time as you make more and

more observations bring it closer to to

reality Okay. Um the model gets dragged

along with it and so over time we had to

realize that the earth was round that it

spins. It goes around the solar system.

Solar system goes around the galaxy and

so on and so forth. And the guys

universe is expanding the expansion

itself expanding accelerating and in

fact very recently in this year. So this

uh even the acceleration of the universe

itself is this evidence that this

non-constant and uh the explanation

behind why that is it's catching up. Um

it's catching up. I mean it's still you

know the dark matter or dark energy this

this kind of thing. We have we have a

model that sort of explains that fits

the data really well. It just has a few

parameters that um you have to specify.

Um but so you know people say that's

fudge factors you know with with enough

fudge factors you can explain anything.

Um but uh the mathematical point of the

model is that um you want to have fewer

parameters in your model than data

points in your observational set. So if

you have a model with 10 parameters that

explains 10 10 observations that is a

completely useless model. It's what's

called overfitted. But like if you have

a model with you know two parameters and

it explains a trillion observations

which is basically uh so yeah the the

the dark matter model I think has like

14 parameters and it explains pabytes of

data um that that that the astronomers

have. Um you can think of of a theory

like one way to think about um physical

math theory theory is it's a compression

of of the universe um and data

compression. So you know you have these

pabytes of observations you'd like to

compress it to a model which you can

describe in five pages and specify a

certain number of parameters and if it

can fit to reasonable accuracy you know

almost all of your observations. I mean

the more compression that you make the

better your theory. In fact, one of the

great surprises of our universe and of

everything in it is that it's

compressible at all. It's the

unreasonable effectiveness of

mathematics. Yeah, Einstein had a quote

like that. The the most incomprehensible

thing about the universe is that it is

comprehensible, right? And not just

comprehensible. You can do an equation

like E= MC². There is actually a some

mathematical possible explanation for

that. Um, so there's this phenomenon in

mathematics called universality. So many

complex systems at the macro scale are

coming out of lots of tiny interactions

at the macro scale and normally because

of the common form of explosion you

would think that uh the macros scale

equations must be like infinitely

exponentially more complicated than than

the uh the microscale ones and they are

if you want to solve them completely

exactly like if you want to model um all

the atoms in a box of of air that's like

Avagadro's number is humongous right

there's a huge number of particles if

you actually have to track each one

it'll be ridiculous. this but certain

laws emerge at the microscopic scale

that almost don't depend on what's going

on at the micros scale or only depend on

a very small number of parameters. So if

you want to model a gas um of you know

quintilion particles in a box you just

need to know it temperature and pressure

and volume and a few parameters like

five or six and it models almost

everything you need need to know about

these 10 to 23 or whatever particles. Um

so we we have um we we don't understand

universality anywhere near as we would

like mathematically but there are much

simpler toy models where we do um have a

good understanding of why univers

universality occurs. Um um most basic

one is is the central limit theorem that

explains why the bell curve shows up

everywhere in nature that so many things

are distributed by what's called a

Gaussian distribution famous bell curve.

There's now even a meme with this curve

and even the meme applies broadly

universality to the meme. Yeah. Yes, you

can go meta if you like. But there are

many many processes for example you can

take lots and lots of independent um

random variables and average them

together um uh in in various ways. you

take a simple average or more

complicated average and we can prove in

various cases that that these these bell

curves these gaussians emerge and it is

a satisfying satisfying explanation. Um

sometimes they don't. Um so so if you

have many different inputs and they're

all correlated in some systemic way then

you can get something very far from a

bow curve show up. Uh and this is also

important to know when this system

fails. So universality is not a 100%

reliable thing to rely on that um um the

global financial crisis was a a famous

example of this. Uh people thought that

uh um mortgage defaults um had this sort

of um Gaussian type behavior that that

if you if you ask if a population of of

of uh you know 100,000 Americans with

mortgages ask what what proportion of

them would default on the mortgages. Um

if everything was decorated it would be

an asset bell curve and and like you can

you can manage risk with options and

derivatives and so forth and um and it

there's a very beautiful theory um but

if there are systemic shocks in the

economy uh that can push everybody to

default at the same time that's very

non-gian behavior um and uh this wasn't

fully accounted for in 2008

now I think there's some more awareness

that this is systemic risk is actually a

much bigger issue and uh just because

the model is pretty uh and nice uh it

may not match reality. Right. So, so the

mathematics of working out what models

do is really important. Um, but um also

the science of validating when the

models fit reality and when they don't.

Um, I mean that you need both. Um, and

but mathematics can help because it it

can for example these central limit

theorems it tells you that if you have

certain aums like like non-correlation

that if all the inputs were not

correlated to each other um then you

have this kind of behavior things are

fine. it it tells you where to look for

weaknesses in the model. So if you have

a mathematical understanding of central

limit theorem and someone proposes use

these Gaussian copy or whatever to to

model um default risk um if you're

mathematically um trained you would say

okay but what if this systemic

correlation between all your inputs and

so then then you can ask the economists

you know how how how much of a risk is

that um and then you can you can you can

go look for that. So there's always this

this this synergy between science and

and mathematics. A little bit on the

topic of universality. Mhm.

You're known and celebrated for working

across an incredible breadth of

mathematics reminiscent of Hilbert a

century ago. In fact, the great Fields

Medal winning mathematician Tim Gow has

said that you are the closest thing we

get to Hilbert.

He's a colleague of yours. Oh yeah. Good

friend. But anyway, so you are known for

this ability to go both deep and broad

in mathematics. So you're the perfect

person to ask, do you think there are

threads that connect all the disparate

areas of mathematics? Is there a kind of

deep underlying structure

uh to all of mathematics? There's

certainly a lot of connecting threads.

Um and a lot of the progress of

mathematics has can be represented by

taking by stories of two fields of

mathematics that were previously not

connected and finding connections. Um an

ancient example is um geometry and

number theory you know. So so in the

times of the ancient Greeks these were

considered different subjects. Um I mean

mathematicians worked on both. You know

you could work both on on geometry most

famously but also on numbers. Um but

they were not really considered related.

Um I mean a little bit like you know you

could say that that this length was five

times this length because you could take

five copies of this length and so forth.

But it wasn't until Deart who really

realized that who developed analytic

geometry that you can you can

parameterize the plane a geometric

object by um by two real numbers. Every

point can be and so geometric problems

can be turned into into problems about

numbers. Um and the the today this feels

almost trivial like like there's there's

there's no content to this like of

course uh you you know um a plane is xx

and y and because that's what we teach

and it's internalized. Um but it was an

important development that these these

two fields were unified. Um and this

process has just gone on throughout

mathematics over and over again. algebra

and geometry were separated and now we

have a student algebraic geometry that

connects them and over and over again

and that's certainly the type of

mathematics that 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. Um and in

mathematics there's definitely both

hedgehogs and foxes. Um and then there's

people who are kind of uh who can play

both roles. Um and I think like ideal

collaboration between mathematicians

involves a very you need some diversity

like um a fox working with many

hedgehogs or or vice versa. So yeah but

I identify mostly as a fox certainly I I

like uh arbitrage somehow you like like

um learning how one field works learning

the tricks of that field and then going

to another field which people don't

think is related but I can I can adapt

the tricks. So see the connections

between the fields. Yeah. So there are

other mathematicians who are far deeper

than I am. Like who really they're

really hedgehogs. They they know

everything about one field and they're

much faster and and and more effective

in that field. But I can I can give them

these extra tools. I mean you said that

you can be both the hedgehog and 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 with the uh the uh the

fox paradigm. Yeah. So, um yeah, I I

like looking for analogies, narratives.

Um I I spend a lot of time if there's a

result I see in one field and I like the

result, it's a cool result, but I don't

like the proof. like it uses types of

mathematics that I'm not super familiar

with. Um I often try to reprove it

myself using the tools that I favor. Um

often my proof is worse. Um but um by

the exercise of doing so um I can say oh

now I can see what the other proof was

trying to do. Um and from that I can get

some understanding of of the tools that

are used in in that field. So it's very

exploratory, very doing crazy things in

crazy fields and like reinventing the

wheel a lot. Yeah. Whereas the hedgehog

style is uh I think much more scholarly,

you know, you you you're very knowledge

based. You you you you stay up to speed

on like all the developments in this

field. You you know all the history. Um

you have a very good understanding of of

exactly the strengths and weaknesses of

of each particular uh technique. Um

yeah uh I think you you rely a lot more

on sort of calculation than sort of

trying to find narratives. Um so yeah I

mean I can do that too but uh there are

other people who are extremely good at

that. Let's step back and uh

uh maybe look at the the a bit of a

romanticized version of mathematics.

Mhm. So, uh I think you've said that

early on in your life, uh math was more

like a puzzle solving activity when you

were uh young. When did you first

encounter a problem or proof where you

realize math can have a kind of elegance

and beauty to it?

That's a good question. Um when I came

to graduate school uh in Princeton, um

so John Conway was there at the time. He

he passed away a few years ago. But uh I

remember one of the very first research

talks I I went to was a talk by Conway

on what he called extreme proof. So

Conway had just had this this amazing

way of of thinking about all kinds of

things in in a way that you would

normally think of. So um he thought of

proofs themselves as occupying some sort

of space, you know. So, so um if you

want to prove something, let's say that

there's infinitely many primes, okay,

you avoid different proofs, but you

could you could rank them in different

axes like some proofs are elegant, some

are long, some proofs are are um

elementary and so forth. Um and so

there's this cloud. So the space of all

proofs itself has some sort of shape. Um

and so he was interested in in extreme

points of this shape like out of all all

these proofs what is one that is the

shortest at the the extent of every

everything else or or the most

elementary or or whatever. Um and so he

gave some examples of well-known

theorems and then he would give what he

thought was was the extreme proof um in

these different aspects. Um and I I just

found that really eye opening um that

that um you know it's not just getting a

proof for a result was interesting but

but once you have that proof you know

trying to to uh to optimize it in

various ways. Um that that proof um uh

proofing itself had some craftsmanship

to it. Um it it certainly informed my

writing style. Um but you know like when

you do your your math assignments and as

undergraduate your homework and so

forth, you you're sort of encouraged to

just write down any proof that works,

okay, and hand it in and get a get as

long as it gets a tick mark, you you

move on. Um but if you want your your

results to actually be influential and

be read by people, um it can't just be

correct. It should also um be a pleasure

to read, you know, um motivated um be

adaptable to to generalize to other um

things. Um it's the same in many other

disciplines like like coding. It's a

there's a lot of analogies between math

and coding. I like analogies if you

haven't noticed. Um but um you know like

you can code something spaghetti code

that works for a certain task and it's

quick and dirty and it works. But uh

there's lots of good principles for for

um writing code well so that other

people can use it build upon it and so

on and has fewer bugs and whatever. Um

and there's similar things with mathemat

mathematics. So yeah the first of all

there's so many beautiful things there

and and is one of the great minds uh in

mathematics ever and computer science.

Uh just even considering the space of

proofs. Yeah. and saying, "Okay, what

does this space look like and what are

the extremes?" Uh, like you mentioned,

coding as an analogy is interesting

because there's also this activity

called the code golf. Oh, yeah. Yeah.

Yeah. Which I also find beautiful and

fun where people use different

programming languages to try to write

the shortest possible program that

accomplishes a particular tasks. Then I

believe there's even competitions on

this. Yeah. And uh it's also a nice way

to stress test not just the

sort of the programs or in this case the

proofs but also the different languages

maybe that's the different notation or

whatever to use to to accomplish a

different task. Yeah, you learn a lot. I

mean it may seem like a frivolous

exercise but it can generate all these

insights which if you didn't have this

artificial um objective to to to pursue

you might not see. What to you is the

most beautiful or elegant equation in

mathematics? I mean one of the things

that people often look to in in beauty

is the simplicity. So if you look at E=

MC² so when when a few concepts come

together that's why the oiler identity

is often considered uh the most

beautiful equation in mathematics. Do

you do you find beauty in that one and

the oil identity? Yeah. Well, as I said,

I mean, what I find most appealing is is

connections between different things

that um so the if ei= minus one um so

yeah people oh uses all the fundamental

constants okay that that's I mean that's

cute um but but to me so the exponential

function was interested by oil to

measure exponential growth you know so

compound interest or decay anything

which is continuously growing

continuously decreasing growth and decay

or dilation or contraction is modeled by

the exponential function Um whereas pi

uh comes around from circles and

rotation right if you want to rotate a

needle for example 180° you need to

rotate by pi radians and i complex

numbers represents the swing between

imagine axis of a 90° rotation so a

change in direction so the x function

represents growth and decay in the

direction where you really are um when

you stick an i in the exponential it now

it's it's instead of motion in the same

direction as your current position it's

the motion has right angles to

composition. So rotation um and then so

e e pi equ= minus 1 tells you that if

you rotate for time pi you end up at the

other direction. So it unifies geometry

through dilation and exponential growth

or dynamics through this act of of

complexification rotation by by i. So it

connects together all these tools

mathematics. Yeah. Yeah. dynamic

structure and complex and complex and um

the complex numbers they all considered

almost yeah they were all next door

neighbors in mathematics because of this

identity. Do do you think the thing you

mentioned is cute the the the collision

of notations from these disperate

fields?

Um it's just a frivolous side effect or

do you think there is legitimate like

value in when the notation all the our

old friends come together

night? Well, it's it's it's confirmation

that you have the right concepts. Um so

when you first study anything um you you

have to measure things and give them

names. Um and initially sometimes your

because your your model is again too far

off from reality you give the wrong

things the best names and you only find

out later what's what's really important

physicists can do this sometimes I mean

but it turns out okay so actually with

physics okay so E= MC² okay so one of

the the big things was the E right so

when when Aristotle first came up with

his laws of of motion and then and then

um Galileo or Newton and so forth you

know they saw the things they could they

could measure they could measure mass

and acceleration and force and so forth

and so Newtonian mechanics for example

F= ma was the famous Newton second law

of motion so those were the the primary

objects so they gave them the central

building in the theory it was only later

after people started analyzing these

equations that there always seemed to be

these quantities that were conserved um

so momentum and energy um uh and it's

not obvious that things happen energy

like it's not something you can directly

measure the same way you can measure

mass and and and velocity so forth but

over time people realize is that this

was actually a really fundamental

concept. Hamilton eventually in 19th

century reformulated Newton's laws of

physics into what's called Hamiltonian

mechanics where the energy which is now

called the Hamiltonian was the dominant

object once you know how to measure the

Hamiltonian of any system. You can

describe completely the dynamics like

what happens to to all the states like

it's um it it really was a central actor

which was not obvious initially. Um and

this uh helped actually uh this change

of perspective really helped when

quantum mechanics came along. Uh because

um the early physicists who studied

quantum mechanics, they had a lot of

trouble trying to adapt their Newtonian

thinking because everything was a

particle and so forth to to to quantum

mechanics, you know, because I think

because it was a wave. It just looked

really really weird. Um like you ask

what is the quantum version of F equals

MA? And it's really really hard to to

give an answer to that. Um but it turns

out that the Hamiltonian which was so um

secretly behind the scenes in classical

mechanics also is the key uh object in

um um in quantum mechanics that there's

there's also an object called

Hamiltonian. It's a different type of

object. It's what's called an operator

rather than than a function. But um and

um but again once you specify it you

specify the entire dynamics. So there's

something called Shingers equation that

tells you exactly how quantum systems

evolve once you have a Hamiltonian. So

side by side they look completely

different objects you know like so one

involves particles one involves waves

and so forth but with this centrality

you could start actually transferring a

lot of intuition and facts from

classical mechanics to quantum

mechanics.

For example, in classical mechanics,

there's this thing called ner's theorem.

Every time there's a symmetry in a

physical system, there is a conservation

law. So the laws of physics are

translation invariant. Like if I move 10

steps to the left, I experience the same

laws of physics as if I was here. And

that corresponds to conservation

momentum. Um if I turn around by by some

angle again, I experience the same laws

of physics. This corresponds to

conservation angular momentum. If I wait

for 10 minutes, um I still have the same

laws of physics. Um so this time

translation variance. this corresponds

to the low conservation of energy. Um,

so there's this fundamental connection

between symmetry and conservation. Um,

and that's also true in quantum

mechanics. Even though the equations are

completely different, but because

they're both coming from the

Hamiltonian, the Hamiltonian controls

everything. Um, every time the

Hamiltonian has a symmetry, the

equations will will have a conservation

law. Um, so it's it's it's it's once you

have the right language, it actually

makes things um a lot a lot cleaner. One

of the problems why we can't unify

quantum mechanics and general relativity

yet we haven't figured out what the

fundamental objects are like for example

we have to give up the notion of space

and time being these almost uklidian

type spaces and there has to be um you

know and you know we kind of know that

at very tiny scales um there's going to

be quite fluctuations of space

space-time foam um and trying to to use

cartigian coord xyz is going to be it's

it's just it's it's a non-starter but we

don't know how to what to replace it

with um We don't actually have the

mathematical um um concepts the analog

Hamiltonian that sort of organized

everything. Does your gut say that there

is a theory of everything. So this is

even possible to unify to find this

language that unifies general relativity

and quantum mechanics. I believe so. I

mean the history of physics has been out

of unification much like mathematics um

over the years. You know electricity and

magnetism were separate theories and

then Maxwell unified them. you know,

Newton unified the the motions of the

heavens with the motions on of objects

on the earth and so forth. So, it should

happen. It's just that the um u again to

go back to this model of the

observations and and theory. Part of our

problem is that physics is a victim's

own success that our two big theories of

of of physics general relativity and

quantum mechanics are so are so good now

that together they cover 99.9% of sort

of all the observations we can make. Um,

and you have to like either go to

extremely insane particle accelerations

or or the early universe or or or things

that are really hard to measure um in

order to get any deviation from either

of these two theories to the point where

you can actually figure out how to how

to combine them together. Um, but I have

faith that we, you know, we've we've

been doing this for centuries and we've

made progress before. There's no reason

why we should stop. Do you think it will

be a mathematician that develops uh

theory of everything? What often happens

is that when the physicists need uh um

some of mathematics, there's often some

precursor that the mathematicians um

worked out earlier. So when Einstein

started realizing that space was curved,

he went to some mathematician and asked

is there is there some theory of curved

space that the mathematicians already

came up with that could be useful and he

said oh yeah there's I think Reman came

up with something um and so yeah Reman

had developed remmaning geometry um

which is precisely you know a theory of

spaces that occurred in various general

ways which turned out to be almost

exactly what was needed um for

Einstein's theory. This is going back to

Dwick's unreasonable effectiveness of

mathematics. I think the theories that

work well to explain the universe tend

to also involve the same mathematical

objects that work well to solve

mathematical problems. Ultimately,

they're just sort of both ways of

organizing data um in in in useful ways.

It just feels like you might need to go

some weird land that's very hard to to

intuit it like you know you have like

string theory. Yeah, that that's that

was that was a leading candidate for

many decades. It's I think it's slowly

falling out of fashion because it's it's

not matching experiment. So one of the

big challenges of course like you said

is experiment is very tough. Yes.

Because of the how effective both

theories are. But the other is like just

you know you're talking about you're not

just deviating from spaceime. You're

going into like some crazy number of

dimensions. You're doing all kinds of

weird stuff that to us we've gone so far

from this flat earth that we started at

like now we're just it's it's very hard

to use our limited ape descendants of uh

uh cognition to intuitit what that

reality really is like. This is why

analogies are so important, you know. I

mean, so yeah, the round earth is not

intuitive because we're stuck on it, but

you know, but you know, but round

objects in general, we have pretty good

intuition over uh and we have intuition

about light works and so forth. And like

it's it's actually a good exercise to

actually work out how eclipses and

phases of of the sun and the moon and so

forth can be really easily explained by

by by by round earth and round moon, you

know, um and models. Um and and you can

just take you know a basketball and a

golf ball and and and a light source and

actually do these things yourself. Um so

the intuition is there. Um but yeah you

have to transfer it. That is a big leap

intellectually for us to go from flat to

round earth because you know our life is

mostly lived in flat land. Yeah. To load

that information and we all like take it

for granted. We take so many things for

granted because science has established

a lot of evidence for this kind of

thing. But you know, we're on a round

rock. Yeah. Flying through space. Yeah.

Yeah. And it's a big leap and you have

to take a chain of those leaps the more

and more and more we progress. Right.

Yeah. So modern science is maybe again a

victim of its own success is that you

know in order to be more accurate it has

to to move further and further away from

your initial intuition. And so um for

someone who hasn't gone through the

whole process of science education it

looks more more suspicious because of

that. So, you know, we we need we need

more grounding. I mean, I I think um I

mean, you know, there are there are

scientists who do excellent outreach. Um

but there's this there this there's

there there's lots of science things

that you can do at home. There's lots of

YouTube videos. I did a YouTube video

recent of Grant Sanderson. We talked

about this earlier that uh you know how

the ancient Greeks were able to measure

things like the distance to the moon,

distance to the earth, and you know,

using techniques that you you could also

replicate yourself. Um it doesn't all

have to be like fancy space telescopes

and and very intimidating mathematics.

Yeah, that's uh I highly recommend that.

I believe you give a lecture and you

also did an incredible video with Grant.

It's a beautiful experience to try to

put yourself in the mind of a person

from that time. Mhm. Shrouded in

mystery, right? You know, you're like on

this planet, you don't know the shape of

it, the size of it. You see some stars,

you see some you see some things and you

try to like localize yourself in this

world. Yeah. Yeah. And try to make some

kind of general statements about

distance to places. Change your

perspective is really important. You say

travel bordens the mind. This is

intellectual travel. You know put

yourself in the mind of the ancient

Greeks or or some other person some

other time period. Make hypothesis

spherical cows whatever you know

speculate. Um and you know this is this

is what mathematicians do and some what

artists do actually. It's just

incredible that given the extreme

constraints, you could still say very

powerful things. That's why it's

inspiring looking back in history. How

much can be figured out right when you

don't have much to figure out stuff like

if you propose axioms then the

mathematics lets you follow those a to

their conclusions and sometimes you can

get quite a quite a long way from you

know initial hypothesis. If we can stay

in the land of the weird, you mentioned

general relativity. You've uh you've

contributed uh to the mathematical

understanding of Einstein's field

equations. Can you explain this work and

from a sort of mathematical standpoint

uh what aspects of general relativity

are intriguing to you, challenging to

you? I have worked on some equations.

There's something called the the wave

maps equation or the sigma field model

which is not quite the equation of

space-time gravity itself but of certain

fields that might exist on top of

spaceime. Um so Einstein's equations of

relativity just describes space and time

itself. Um but then there's other fields

that live on top of that. There's the

electromagnetic field. Um there's

control fields and there's this whole

hierarchy of different equations of

which Einstein is considered one of the

most nonlinear and difficult. But

relatively low in the hierarchy was this

thing called the wave maps equation. So

it's a wave which at any given point uh

is fixed to be like on a sphere. Um so

uh I can think of a bunch of arrows in

space and time and and the arrows

pointing in in different directions. Um

but they propagate like waves. If you

wiggle an arrow it was it will propagate

and make all the arrows move kind of

like sheets of wheat in the wheat field.

And I was interested in the global

regularity problem again for this

question like is it possible for for all

the energy here to collect at a point.

So the equation I considered was

actually what's called a critical

equation where it's actually the

behavior at all scales is roughly the

same. Um and I was able barely to show

that um that you couldn't actually force

a scenario where all the energy

concentrated at one point that the

energy had to disperse a little bit and

the moment it dis little bit it it would

it would stay regular. Yeah. This was

back in 2000. That was part of why I got

interested in narrows afterwards

actually. Yeah. So I developed some

techniques to um solve that problem. So

part of it is it was um this problem is

really nonlinear uh because of the

curvature of the sphere. Um this there

was a certain nonlinear effect which was

a non-perturbative effect. It was when

you sort of looked at it normally it

looked larger than the linear effects of

the wave equation. Um and so it was hard

to to keep things under control even

when the energy was small. But I

developed what's called a gauge

transformation. So the equation is kind

of like an evolution of of of heaves of

wheat and and they're all bending back

and forth and so there's a lot of

motion. Um but like if you imagine like

stabilizing the flow by attaching little

cameras at different points in space

which are trying to move in a way that

captures most of the motion and under

this stabilized flow the flow becomes a

lot more linear. I discovered a way to

transform the the equation to reduce the

amount of of nonlinear effects. Um and

then I was able to to to to solve the

equation. I found this transformation

while visiting my aunt in Australia and

I was trying to understand the dynamics

of all these fields and I I couldn't do

it with pen and paper. Um and I had not

enough facility of computers to do any

computer simulations. So I ended up

closing my eyes being on on the floor

and just imagining myself to actually be

this vector field and rolling around to

try to to see how to change coordinates

in such a way that somehow things in all

directions would behave in a reasonably

linear fashion. And yeah, my aunt walked

in on me while I was doing that and she

was asking what do I what am I doing

doing this? It's complicated is the

answer. Yeah. Yeah. And you know, okay,

fine. You know, you're a young man. I

don't ask questions. I I I have to ask

about the you know um how do you

approach solving difficult problems?

What if it's possible

to go inside your mind when you're

thinking? Are you visualizing

in your mind the mathematical objects

symbols maybe what are you visualizing

in your mind usually when you're

thinking um a lot of pen and paper one

thing you pick up as a mathematician is

sort of uh I call it cheating

strategically um so u the the beauty of

mathematics is that is that you get to

change the rule change the problem

change the rules as you wish this you

don't get to do this for any other field

like you know if if you're an engineer

and someone says build a bridge over

this this You can't say I want to build

this up bridge over here instead or I

want to build out of paper in instead of

steel. Um but a mathematician you can

you can do whatever you want. Um

it's it's like trying to solve a

computer game where you can there's

unlimited cheat codes available. Uh and

so you know you you can you can set

this. So there's a dimension that's too

large. I'll set it to one. I'd solve the

one dimension problem first. So there's

a main term and an error term. I'm going

to make a spherical car assumption. I'll

assume the error term is zero. And so

the way you should solve these problems

is is not in sort of this iron man mode

where you make things maximally

difficult. Um but actually the way you

should you should approach any

reasonable math problem is that you if

if there are 10 things that are making

your life difficult. Find a version of

the problem that turns off nine of the

difficulties but only keeps one of them.

Um and so that um and then that just so

you you you install nine cheats. Okay.

You install 10 cheats then then the game

is trivial. You saw nine cheats, you

solve one problem that that that teaches

you how how to deal with that particular

difficulty and then you turn that one

off and you turn someone else something

else else on and then you solve that one

and after you you know how to solve the

10 problems 10 difficulties separately

then you have to start merging them a

few at a time. Um I I as a kid I watched

a lot of these Hong Kong action movies.

Um it's from a culture. Um and uh one

thing is that every time there was a

fight scene, you know, so maybe the the

hero will get swarmed by a hundred bad

guy goons or whatever. But it would

always be choreographed so that he'd

always be only fighting one person at a

time and then he would defeat that

person and move on and and because of

that he could he could defeat all of

them, right? But whereas if they had

fought a bit more intelligently and just

swarmed the guy at once, uh it would

make for much much worse um cinema, but

uh but they would win. Are you usually

uh pen and paper? Are you working uh

with computer and latte? I'm mostly pen

and paper actually. So in my office, I

have four giant blackboards. Um and

sometimes I just have to write

everything I know about the problem on

the four blackboards and then sit my

couch and just sort of see the whole

thing. Is it all symbols like notation

or is there some drawings? Oh, there's a

lot of drawing and a lot of bespoke

doodles that that only make sense to me.

Um I mean and and the beauty of

blackboard is you erase and it's it's

very organic thing. Um I'm beginning to

use more and more computers. Um partly

because AI makes it much easier to do

simple coding things that you know if I

wanted to plot a function before which

is moderately complicated as some

iteration or something you know I'd have

to to remember how to set up a Python

program and and and and and how does a

for loop work and and and debug it and

it would take two hours and so forth and

and now I can do it in 10 15 minutes is

much um yeah I'm using more and more uh

computers to do simple explorations.

Let's talk about AI a little bit if we

could. So um maybe a good entry point is

just talking about computer assisted

proofs in general. Can you describe the

lean formal proof programming language

and how it can help as a proof assistant

and maybe how you started using it and

how uh it has helped you. So um we is a

computer language um much like sort of

standard languages like Python and C and

so forth except that in most languages

the focus is on producing executable

code. Lines of code do things you know

they they flip bits or or they make a

robot move or or they they deliver you

text on the internet or something. Um so

lean is a language that can also do

that. Uh it can also be run as a

standard traditional language but it can

also produce certificates. So a software

like like Python might do a computation

and give you that the answer is seven.

Okay, that does a sum of 3+ 4 is equal

to 7 but uh lean can produce not just

the answer but but a proof that how it

got the the answer of seven as 3+ 4 and

all the steps involved in in so it

creates these more complicated objects

not just statements but statements with

proofs attached to them. um and um every

line of code is just a way of p piecing

together previous statements to to

create new ones. So the idea is not new.

These things are are called proof

assistants and so they provide languages

for which you you can create quite

complicated um intricate mathematical

proofs and um they produce these

certificates that that give a 100% um

guarantee that your arguments are

correct if you trust the compiler of but

they made the compiler really small and

you can there are several different

compilers available for the same for um

can you give people some intuition about

the the difference between writing on

pen and paper versus using lean

programming language How hard is it to

formalize

statement? So lean a lot of

mathematicians were involved in the

design of lean. So it's it's designed so

that individual lines of code resemble

individual lines of mathematical

argument like you might want to

introduce a variable. You want want to

prove a contradiction. You you um there

are various standard things that you can

do and and it's it's written so ideally

it should like a one correspondence. In

practice, it isn't because lean is like

explaining a proof to an extremely

pedantic colleague who will will point

out okay did you really mean this like

what what happens if this is zero? Okay.

Um did you how do you justify this? Um

so lean has a lot of automation in it um

to try to to uh to be less annoying. Um

so for example um every mathematical

object has to come with a type like if I

if I talk about X is X a real number or

um a natural number or or a function or

something um if you write things

informally um it's up in terms of

context you say you know um clearly x is

equal to let x be the sum of y and z and

y and z were already real numbers so x

should also be a real number um so lean

can do a lot of that um but every so

often it it says wait a minute can you

tell me more about what this object is

uh what type of object it is. You see,

you have to think more um at a

philosophical level. Well, not just sort

of computations you're doing, but sort

of what each object actually um is in

some sense. Is he using something like

LLMs to do uh the type inference or like

you mention the real number? It's it's

using much more traditional what's

called good old fashioned AI. Yeah, you

can represent all these things as trees

and there's always algorithm to match

one tree to another tree. So it's

actually doable to figure out if

something is a a real number or a

natural number. Yeah. Every object sort

of comes with a history of where it came

from and you can you can kind of trace.

Oh, I see. Um yeah, so it's it's

designed for reliability. So uh modern

AIs are not used in it's a disjoint

technology. People are beginning to use

AIS on top of lean. So when a

mathematician tries to program um a

proof in lean um often there's a step

okay now I want to use um the

fundamental thing of calculus say okay

to do the next step so the lean

developers have built this this massive

project called methal liib a collection

of tens of thousands of useful facts

about mathematical objects and somewhere

in there is the fundamental theme of

calculus but you need to find it so a

lot the bottleneck now is actually lema

search you know there's a tool that that

you know is in there somewhere and you

need to find it um and so you can there

are various search engines specialized

for math loop that you can do um but

there's now these large language models

that you can say um I need the

fundamental calculus at this point and

it say okay uh um uh for example um when

I code I have GitHub copilot installed

as a plugin to my IDE and it scans my

text and it sees what I need says you

know I might even type here okay now I

need to use the final thing with

calculus okay and then it might suggest

okay try this and like maybe 25% of the

time it works exactly and then another

10 15% of the time it doesn't quite work

but it it's close enough that I can say

oh if I just change it here and here it

it will work and then like half the time

it gives me complete rubbish um so but

people are beginning to use AI a little

bit on top um mostly on the level of

basically fancy autocomplete um but uh

you can type half of one line of a proof

and it will find it will tell you yeah

but a fancy especially fancy with the

sort of capital letter F is uh uh

removes some of the friction

mathematician might feel when they move

from pen and paper to formalizing. Yes.

Yeah. So, right now I estimate that the

effort time and effort taken to

formalize a proof is about 10 times the

amount taken to to write it out. Yeah.

So, it's doable, but uh you don't it's

it's annoying. But doesn't it like kill

the whole vibe of being a mathematician?

Yeah. So, I mean having a pedantic

coworker, right? Yeah. If if that was

the only aspect of it. Okay. But um

Okay. there there are some there's some

case it was actually more pleasant to do

things formally. So there was there was

a theorem I formalized and there was a

certain constant 12 um that that came

out at um in the final statement and so

this 12 had to be carried all through

the proof um and like everything had to

be checked that it goes all the all

these other numbers had to be consistent

with this final number 12 and so we

wrote a paper through this theorem with

this number 12 and then a few weeks

later someone said oh we can actually

improve this 12 to an 11 by reworking

some of these steps and when this

happens with pen and paper um like every

time you change a parameter you have to

check line by line that every single

line of your proof still works and there

can be subtle things that you didn't

quite realize. Some properties on the

number 12 that you didn't even realize

that you were taking advantage of. So a

proof can break down at a subtle place.

Um so we had formalized the proof with

this constant 12 and then when this this

new paper came out uh we said okay let's

so that took like 3 weeks to formalize

and and like 20 people to formalize this

this this original proof. I said oh but

now now let's let's um uh uh let's

update the 12 to 11. And what you can do

with lean is that you just in your

headline theorem you you change a 12 to

11. You run the compiler and like of the

thousands of lines of code you have 90%

of them still work and there's a couple

that are lined in red. Now I can't

justify this these steps but it it

immediately isolates which steps you

need to change but you can skip over

everything which which works just fine.

Um, and if you program things correctly,

um, with sort of good programming

practices, most of your lines will not

be read. Um, and there'll just be a few

places where you, I mean, if if you

don't hard code your constants, but you

sort of, uh, um, um, you use smart

tactics and so forth. Yeah, you can

localize um, the things you need to

change to to a very small um, period of

time. So like within a day or two, we

had updated our proof to this is very

quick process. You um, you make a

change, there are 10 things now that

don't work. for each one you make a

change and now there's five more things

that don't work but but the process

converges much more smoothly than with

pen and paper. So that's for writing are

you able to read it like if somebody

else sends a proof are you able to like

how what's what's the uh versus paper

and yeah so the proofs are longer but

each individual piece is easier to read.

So, um, if you take a math paper and you

jump to page 27 and you look at

paragraph 6 and you have a line of of of

text of math, I often can't read it

immediately because it assumes various

definitions which I have to to go back

and and maybe 10 pages earlier this was

defined and this um the proof is

scattered all over the place and you

basically are forced to read fairly

sequentially. Um, it's it's not like say

a novel where like you know in theory

you could you open up a novel halfway

through and start reading. there's a lot

of context. But when a proven lean, if

you put your cursor on a line of code,

every single object there, you can hover

over it and it would it would say what

it is, where it came from, where stuff

is justified. You can trace things back

much easier than sort of flipping

through a math paper. So, one thing that

lean really enables is actually

collaborating on proofs at a really

atomic scale that you really couldn't do

in the past. So traditionally with pen

and paper um when you want to

collaborate with another mathematician

um either you do it as a blackboard

where you um you can really interact but

if you're doing it sort of by email or

something um basically yeah you have to

segment it say I'm going to I'm going to

finish section three you do section four

but uh you can't really sort of work on

the same thing collaboratively at the

same time but with lean you can be

trying to formalize some portion of the

proof and say I got stuck at line 67

here I need to prove this thing but it

it doesn't quite work here is like the

three lines of code I'm having trouble

with. Um, but because all the context is

there, someone else can say, "Oh, okay.

I recognize what you need to do. You

need to to apply this trick or this tool

and you can do extremely atomic level

conversations. So, because of lean, I

can collaborate, you know, with dozens

of people across the world, most of whom

I don't have never met in person. Um,

and I may not know actually even whether

they're um how reliable they are in in

in their um um in in the process, but

lean gives me a certificate of of of

trust. Um, so I can do I can do

trustless mathematics. So there's so

many interesting questions there's. So

one, you're you're known for being a

great collaborator. So what is the right

way to approach

solving a difficult problem in

mathematics? When you're collaborating,

are you doing a divide and conquer type

of thing or are you brains are you

focusing on a particular part and you're

brainstorming? There's always a

brainstorming process first. Yeah. So

math research projects sort of by their

nature when you start you don't really

know how to do the problem. Um it's not

like an engineering project where

somehow the theory has been established

for decades and it's it's implementation

is the main difficulty. You have to

figure out even what is the right path.

So so this is what I said about about

cheating first you know um it's like um

to go back to the bridge building

analogy you know so first assume you

have infinite budget and and like

unlimited amounts of of of workforce and

so forth. Now can you can you build this

bridge? Okay. Okay. now have infinite

budget but only finite workforce right

now can you do that and so forth um so

uh I mean of course you know no engineer

can actually do this like I say they

have fixed requirements yes there's this

sort of jam sessions always at the

beginning where you try all kinds of

crazy things and you you make all these

assumptions that are unrealistic but you

plan to fix later um and you try to see

if there's even some skeleton of an

approach that might work um and then

hopefully that breaks up the problem

into smaller sub problems which you

don't know how to do but then you uh you

focus on on sub ones and sometimes

different collaborators are better at at

working on on certain things. Um so one

of my themes I'm known for is a theorem

of Ben Green which called the green

tower theorem. Um it's a statement that

the primes contain arithmetic

progressions of any length. So it was a

modification of this theoret

and the way we collaborated was that Ben

had already proven a similar result for

progressions of length three. Um he

showed that sets like the primes contain

lots and lots of progressions of length

three. Um even and even um subsets of

the prime certain subsets do um but his

techniques only worked for um for length

three progressions. They didn't work for

longer progressions. Um but I had these

techniques coming from agotic theory

which is something that I had been

playing with and and uh I knew better

than Ben at the time. Um and so um if I

could justify certain randomness

properties of some set relating to

primes like there there's a certain

technical condition which if I could

have it if if Ben could supply me this

fact I could I could conclude the

theorem but I what I asked was a really

difficult question in number theory

which um he said there's no way we can

prove this can so he said can you prove

your part of the theorem using a weaker

hypothesis that I have a chance to prove

it and he proposed something which he

could prove but it was too weak for me I

can't use this. Um, so there's this

there was this conversation going back

and forth. Um, so different cheats to

Yeah. Yeah. I want to cheat more, he

wants to cheat less. But eventually we

found a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a property which a he

could prove and b I could use um and

then we we could prove our view and um

yeah so there's there's a there all

kinds of dynamics you know I mean it's

every every um collaboration has a has a

has some story no two are the same. And

then on on the flip side of that like

you mentioned with lean programming now

that's almost like a different story

because you can do you can create I

think you've mentioned a kind of a

blueprint

right for a problem and then you can

really do a divide and conquer with lean

where you're working on separate parts

right and they're using the computer

system proof checker essentially to make

sure that everything is correct along

the way. Yeah. So it makes everything

compatible and uh yeah and trustable. Um

yeah so currently only a few

mathematical projects can be cut up in

this way at the current state of the art

most of the lean activity is on

formalizing boos that have already been

proven by humans a math paper basically

is a boop a blueprint in a sense it is

taking a a difficult statement like big

theorem and breaking up into 100 little

lemas um but often not all written with

enough detail that each one can be sort

of directly formalized. A blueprint is

like a really pedantically written

version of a paper where every step is

explained as to as much detail as as as

possible and trying to make each step

kind of self-contained um and or

depending on only a very specific number

of of previous statements that been

proven so that each node of this

blueprint graph that gets generated can

be tackled independently of of the

others and you don't even need to know

how the whole thing works. Um so it's

like a modern supply chain you know like

if you want to create an iPhone or or

some other complicated object um no one

person can can build up um a single

object but you can a specialist who who

just if they're given some widgets from

some other company they can combine them

together to form a slightly bigger

widget. I think that's a really exciting

possibility because you can have if you

can find problems that could be

broken down this way then you can have

you know thousands of contributors right

distributed. So I told you before about

the split between theoretical and

experimental mathematics and right now

most mathematics is theoretical and when

you type it it's experimental. I think

the platform that lean and and other

software tools so um GitHub and things

like that um allow they will allow

experimental mathematics to be to scale

up um to a much greater degree than we

can do now. So right now if you want to

um um do any mathematical exploration 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

errorprone activity it's not practical

to allow other people to collaborate

with you on writing modules for your

code because if one of the modules has a

bug in it the whole thing is unreliable.

Um, so it's these are uh so you get

these bespoke uh spaghetti code that

written by not not professional

programmers but by mathematicians you

know and they're clunky and and and slow

and um and so because of that it's it's

hard to to really massproduce

experimental results um but um yeah but

I think with lean I mean so I'm already

starting some projects where we are not

just experimenting with data but

experimenting with proofs. So I have

this project called the equation

theories project. Basically we generated

about 22 million little problems in

abstract algebra. Maybe should back up

and tell you what what the project is.

Okay. So abstract algebra studies

operations like multiplication and

addition and the abstract properties.

Okay. So multiplication for example is

commutive. X * Y is always Y * X at

least for numbers. Um and it's also

associative. X * Y * Z is the same as X

* Y * Z. Um so um these operations obey

some laws that don't obey others. For

example, x * x is not always equal to x.

So that law is not always true. So given

any any operation, it obeys some laws

and not others. Um, and so we generated

about 4,000 of these possible laws of

algebra that certain operations can

satisfy. And our question is which laws

imply which other ones? Um, so for

example, does commutivity imply

associativity? And the answer is no

because it turns out you can describe an

operation which obeys the commitive law

but doesn't obey the associative law. So

by producing an example you can you can

show that commitivity does not imply

associativity but some other laws do

imply other laws by substitution and so

forth and you can write down some some

algebraic proof. So we look at all the

pairs between these 4,000 laws and this

22 million of these pairs and for each

pair we ask does this law imply this um

law? If so give a give u give a proof.

If not give a counter example. Mhm. Um

so 22 million problems each one of which

you could give to like an undergraduate

algebra student and they had a decent

chance of solving the problem. Although

there are a few of these 22 million

there like 100 or so that are really

quite hard. Okay. But a lot are easy and

the project was just to to work out to

determine the entire graph like like

which ones imply which other ones.

That's an incredible project by the way.

Such a good idea. Such a good test of

the very thing we've been talking about

at a scale that's remarkable. Yeah. So

it would not have been feasible. Yeah, I

mean the state-of-the-art in the

literature was like, you know, 15

equations and sort of how they apply.

That's sort of at the limit of what a

human repentant paper can do. So, so you

need to scale it up. So, you need to

crowdsource, but you also need to trust

all the um I mean no one person can

check 22 million of these proofs. You

needed to be computerized and so it only

became possible with with lean. Um we

were hoping to use a lot of AI as well.

Um so the project is almost complete. Um

so of these 22 million all but two had

been settled. Um wow and uh well

actually and of those two we have a pen

and paper proof of the two uh and we

we're formalizing it. In fact I was this

morning I was working on finishing it.

Um so we're almost done on this um

incredible is yeah fantastic. How many

people were able to get about 50 um

which in mathematics is is considered a

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

crazy. Yeah. So we kind of have a paper

with 50 authors uh and a big appendex of

who contribute to what. Here's an

interesting question. Now to maybe speak

even more generally about it. When you

have this pool of people,

is there a way to uh organize the

contributions by level of expertise of

the people of the contributors? Now

okay, uh I'm asking you a lot of pthead

questions here, but I I'm imagining a

bunch of humans and maybe in the future

some AIS. Can there be like an ELO

rating type of situation where

like a gamification of this? The beauty

of of these lean projects is is that

automatically you get all this data, you

know, so like like everything has to be

uploaded for this GitHub and GitHub

tracks who contributed what. Um so you

could generate statistics from at any at

any later point in time. You can say oh

this person contributed this many this

many lines of code or whatever. I mean

these are very crude metrics. Um I would

I would definitely not want this to

become like you know part of your tenure

review or something. Uh um but um I mean

I think already in in in enterprise

computing right people do use some of

these metrics as part of of the

assessment of of performance of a of an

employee. Um again this is a direction

which is a bit scary for academics to go

down. We we don't like metrics so much

and yet academics use metrics they just

use old ones. Number of papers. Yeah.

Yeah. It's true. It's true that Yeah. I

mean um it feels like this is a metric

while flawed is is going in the more in

the right direction. Right. Yeah. It's

an interesting at least it's a very

interesting metric. Yeah. I think it's

interesting to study. I mean I I think

you can you can do studies of of whether

these are better predictors. Um there's

this problem called good heart's law. If

a statistic is actually used to

incentivize performance, it becomes

gained. Um and then it is no longer a

useful measure. Oh, humans always. Yeah.

Yeah. I know. It's rational. So what

we've done for this project is is

self-report. So um there are actually

standard categories um from the sciences

of what types of contributions people

give. So there's there's concept and

validation and resources and and and and

coding and so forth. Um, so we we we

there's a standard list of troll or so

categories. Um, and we just ask each

contributor to there's a big matrix of

all the of all the authors in all the

categories just to tick the boxes where

they think that they contributed. Um,

and just give a rough idea you know like

oh so you did some coding and and uh and

you provided some compute but you didn't

do any of the pen and paper verification

or whatever. And I think that that works

out traditionally mathematicians just

order alphabetically by surname. So we

don't have this tradition as in the

sciences of you know lead author and

second author and so forth like which

we're proud of you know we make all the

authors equal status but it doesn't

quite scale to this size so a decade ago

I was involved in these things called

polymath projects it was the crowd

sourcing mathematics but without the

lean component so it was limited by you

needed a human moderator to actually

check that all the contributions coming

in were actually valid and and this was

a huge bottleneck actually um but still

we had projects that were you know 10

author

or so. But we had decided at the time um

not to try to decide who did what um but

to have a single pseudonym. So we

created this fictional character called

DHJ Polymath in the spirit of Bwaki.

Baki is is the pseudonym for a famous

group of mathematicians in the 20th

century. But um and so the paper was a

authored under the pseudonym. So none of

us got the author credit. Um this

actually turned out to be not so great

for a couple of reasons. So, so one is

that if you actually wanted to be

considered for tenure or whatever, you

could not use this paper in your uh uh

as your submitted as one of your

publications because it wasn't you

didn't have the formal author credit. Um

um but the other thing that we've

recognized much later is that when

people referred to these projects, they

naturally refer to the most famous

person who was involved in the project.

Oh, so this was Tim Gow's P project.

This was ter project and not mention the

the other 19 or whatever people that

were involved. Yeah. So we're trying

something different this time around

where we have everyone's an author. Um

but we will have an an appendix with

this matrix and we'll see how that

works. I mean uh so both projects are

incredible just the fact that you're

involved in such huge collaborations.

But I think I saw a talk from Kevin

Buzzard about uh the lean programming

language just a few years ago and he was

saying that uh this might be the future

of mathematics. And so it's also

exciting that you're embracing uh one of

the greatest mathematicians in in the

world embracing this

what seems like the paving of the future

of mathematics. Um so I have to ask you

here about

the integration of AI into this whole

process. So deep mind's alpha proof was

trained using reinforcement learning on

both failed and successful formal lean

proofs of IMO problems. So this is sort

of highlevel high school oh very high

level yes very high level high school

level mathematics problems. What do you

think about the system and maybe what is

the gap between this system that is able

to prove the high school level problems

uh versus gradual level uh problems.

Yeah, the difficulty increases

exponentially with the the number of

steps involved in the proof. It's a

commentatorial explosion, right? So the

thing with large language models is is

that they make mistakes. And so if a

proof has got 20 steps and your model

has a 10% failure rate um at each step

um of of going in the wrong direction

like u it's just extremely unlikely to

actually um reach the end. Actually uh

just to take a small tangent here is how

hard is the problem of mapping from

natural language to the formal program?

Oh yeah it's extremely hard actually. Um

natural language you know it's very

fault tolerant. Um like you can make a

few minor grammatical errors and a

speaker in the second language can get

some idea of what you're saying. Um yeah

but but formal language yeah you if you

get one little thing wrong um like the

whole thing is is is nonsense. um even

formal to formal is is is very hard.

There there are different incompatible

um uh proofist languages. Uh there's

lean but also coaul and Isabel and so

forth and actually even converting from

a formal language to formal language um

is is an unsolved basically unsolved

problem. That is fascinating. Okay. So

uh but once you have an informal

language

they're using um their RL train model.

So some something akin to alpha zero

that they used to go to then try to come

up with poos they also have a model I

believe it's a separate model for

geometric problems so what impresses you

about the system and um what do you

think is the gap yeah we talked earlier

about things that are amazing over time

become kind of normalized um so yeah now

somehow it's oh of course geometry is a

silver problem right that's true that's

true I mean it's still beautiful yeah

these are great works it shows what's

possible I mean um it's it um the

approach doesn't scale currently is yeah

3 days of Google's survey server time to

solve one high school math problem. This

is not a scalable uh prospect. Um

especially with the exponential increase

in um as as the complexity um increases.

We should mention that they got a silver

medal performance the equivalent of I

mean yeah equivalent of a silver so

first of all they took way more time

than was allotted um and they had this

assistance where where the humans

started helped by by formalizing um but

uh also they they're giving us those

full marks for the solution which I

guess is formally verified. So I guess

that that's that's fair. Um yeah um

there there are efforts there was there

will be a proposal at some point to

actually have an an AI math olympiate

where at the same time as the human

contestants get the the actual Olympia

um problems AIS will also be given the

same problems with the same time period

um and the outputs will have to be

graded by the same judges um um and

which means that will have be written in

natural language rather than formal

language. Oh I hope that happens. I hope

that this IMO it happens. I hope I hope

next one it won't happen this IMO the

performance is not good enough in in the

time period and and uh um but there are

smaller competitions um there are

competitions where the the answer is a

is a number rather than a long form

proof um and that's that's um AI are

actually a lot better at um problems

where there's a specific numerical

answer um because it's it's easy to to

to uh to reinforce do reinforcement

learning on it. Yeah, you got the right

answer, you got the wrong answer. It's

it's a very clear signal. But a long

form proof either has to be formal and

then the lean can give it a thumbs up,

thumbs down, or it's informal. Um, but

then you need a human to grade it to

tell uh and if you're trying to do

billions of of reinforcement learning um

you know um um runs, you're not you

can't hire enough humans to uh to grade

those. um it's already hard enough for

for the last language to do

reinforcement learning on on just the

regular text that that people get. But

now if you actually hire people not just

give thumbs up, thumbs down, but

actually check the the output

mathematically. Yeah, that's too

expensive. So if we uh just explore this

possible future,

what what what is the thing that humans

do that's most special in um in

mathematics? So that you could see AI

uh not cracking for a while. So

inventing new theories. So coming up

with new conjectures versus uh proving

the conjectures,

right? Building new abstractions, new

representations, maybe uh an AI turn

style with seeing new connections

between disparate fields. It's a good

question. Um I think the nature of what

mathematicians do over time has changed

a lot. um you know um so a thousand

years ago mathematicians had to compute

the date of Easter uh and there was

really complicated uh calculations you

know but it's all automated been

automated for centuries we don't need

that anymore you know they used to

navigate to do spherical navigation

spherical trigonometry to navigate how

to get from from um the old world to the

new or very complicated calculations

again we've been automated um you know

even a lot of undergraduate mathematics

even before AI um like wolf from alpha

for example It's not a language model,

but it can solve a lot of undergraduate

level math tasks. So on the

computational side, verifying routine

things like having a a problem and um

and say here's a problem in partial

equations. Could you solve it using any

of the 20 standard techniques? Um and

they say yes, I've tried all 20 and here

are the 100 different permutations and

and here's my results. Um and that type

of thing I think it will work very well.

um type of scaling to once you solve one

problem to to make the AI attack 100

adjacent problems. Um the things that

humans do still Yeah. So so where the AI

really struggles right now um is knowing

when it's made a wrong turn. Um that it

can say, "Oh, I'm going to solve this

problem. I'm going to split up this

problem into um into these two cases.

I'm going to try this technique." And um

sometimes if you're lucky and it's a

simple problem, it's the right technique

and you solve the problem and sometimes

it it will get it will have a problem it

would propose an approach which is just

complete nonsense. Um and but like it

looks like a proof. Um so this is one

annoying thing about LM generated

mathematics. So um yeah we we we've had

human generated mathematics as very low

quality um uh like you know submissions

people who don't have the formal

training and so forth. But if a human

proof is bad, you can tell it's bad

pretty quickly. It makes really basic

mistakes. But the AI generated proofs,

they can look superficially flawless. Uh

and that's partly because that's what

the reinforcement learning has actually

trained them to do, right? To to make

things to to produce text that looks

like um what is correct, which for many

applications is good enough. Um uh so

the errors often really subtle and then

when you spot them, they're really

stupid. Um like you know like no human

would have actually made that mistake.

Yeah, it's actually really frustrating

in the programming context because I I

program a lot and yeah, when a human

makes when lowquality code, there's

something called code smell, right? You

can you can tell you can tell

immediately like, okay, there's signs.

But with with a generate code of and

then you're right eventually you find an

obvious dumb thing that just looks like

good code. Yeah. So, um it's very tricky

to and frustrating for some reason to

Yeah. to work. Yeah. So the sense of

smell. Okay, there you go. This is this

is one thing that humans have. Um and

there's a metaphorical mathematical

smell that uh this we it's not clear how

to get the AI to duplicate that

eventually. Um I mean so the way um

Alpha Zero and so forth make progress on

go and and chess and so forth is is in

some sense they have developed a sense

of smell for go and chess positions you

know that that this position is good for

white is good for black. um they can't

initiate why. Um but just having that

that sense of smell lets them

strategize. So if AIs gain that ability

to sort of a sense of viability of

certain proof strategies say so so you

can say I'm going to try to break up

this problem into two small subtasks and

they can say well this looks good two

tasks look like they're simpler tasks

than than your main task and they still

got a good chance of being true. Um so

this is good to try or no you've you

made the problem worse because each of

the two sub problems is actually harder

than your original problem which is

actually what normally happens if you

try a random uh thing to try normally

actually it's very easy to transform a

problem into even harder problem. Mhm.

Very rarely do you problem transport a

simpler problem. Um yeah so if they can

pick up a sense of smell then they could

maybe start competing with human level

mathematicians. So, this is a hard

question, but not competing, but

collaborating. Yeah. If Okay,

hypothetical.

If I gave you an oracle

that was able to do some aspect of what

you do, and you could just collaborate

with it. Yeah. Yeah. What would that

oracle What would you like that oracle

to be able to do? Would you like it to

uh maybe be a verifier? Like check Mhm.

Do the codes like you're Yes. uh

professor to this is the correct this is

a good this is a promising fruitful

direction. Yeah. Yeah. Yeah. Or or would

you like it to

uh generate possible proofs and then you

see which one is the right one? Um or

would you like it to maybe generate

different representation different

totally different ways of seeing this

problem? Yeah, I think all of the above.

Um a lot of it is we don't know how to

use these tools because it's a paradigm

that is not um yeah we have not had in

the past systems that are competent

enough to understand complex

instructions. Mhm. Um that can work at

massive scale but are also unreliable.

Uh like it's it's an interesting uh bit

unreliable in subtle ways while we while

providing sufficiently good output. Um

it's a interesting combination. um you

know I mean you have you have like

graduate students that you work with who

kind of like this but not at scale um

you know and and and we have previous

software tools that um can work at scale

but but very narrow um so we have to

figure out how to how to use um I mean

um so Tim C actually imagine he actually

foresaw like in in 2000 he was

envisioning what mathematics would look

like in in actually two and a half

decades

and that's funny yeah He he wrote in his

in in his article like a a a

hypothetical conversation between a

mathematical assistant of the future um

and himself you know trying to solve a

problem and they would have have a

conversation that sometimes the human

would would propose an idea and the AI

would would evaluate it and sometimes

the AI would propose an idea um and u

and sometimes that computation was

required and a would just go and say

okay I've checked the 100 cases needed

here or um the first you you said this

is true for all n I've checked for n up

to 100 um and it looks good so far or

hang on there's a problem at n equals 46

you so just a free form conversation

where you don't know in advance where

things are going to go but just based on

on I think ideas get proposed on both

sides calculations get proposed on both

sides I've had conversations with AI

where I say okay let's we're going to

collaborate to solve this math problem

and it's a problem that I already know

the solution to so I I try to prompt it

okay so here's the problem I suggest

using this tool and then you'll find

this this lovely argument using a

totally different tool which eventually

goes you know, into the weeds and say,

"No, no, no. If I using this, okay, and

it might start using this and then it'll

go back to the tool that I wanted to to

before." Um, and like you have to keep

railroading it um onto the path you

want. And like I I could eventually

force it to give the proof I wanted. Um,

but it was like hurting cats um like and

the amount of personal effort I had to

take to not just sort of prompt it, but

also check it output because it like a

lot of what it looked like was going to

work. I know there's a problem on online

17 and basically arguing with it. um

like it was more exhausting than doing

it unassisted. So like it but that's the

current state of the art. I wonder if

there's there's a phase shift that

happens to where it's no longer feels

like hurting cats and

maybe it'll surprise us how quickly that

comes. I I believe so. Um so in

formalization I I mentioned before that

it takes 10 times longer to formalize a

proof than to write it by hand with

these modern AI tools is and also just

better tooling um the lean um um

developers are doing a great job adding

more and more features and making it

user friendly. It's going up from 9 to 8

to 7. Okay, no big deal. But one day it

will drop below one. Um and that's a

phase shift because suddenly um it makes

sense when you write a paper to to write

it in lean first or through a

conversation with AI who is generally um

on the fly with you and it becomes

natural for journals to accept you know

maybe they'll offer expedite refereeing

you know if if a paper has already been

formalized in in lean um they'll just

ask the referee to comment on on the

significance of the results and how it

connects to literature and not worry so

much about the correctness.

um because that's been certified. Um

papers are getting longer and longer in

mathematics and actually it's harder and

harder to get good refereeing for um the

really long ones unless they're really

important. It is actually an issue which

and the formalization is coming in at

just the right time for this to be and

the easier and easier to guess because

of the tooling and all the other factors

then you're going to see much more like

math lib will grow potentially

exponentially. It's a it's a it's a

virtuous uh cycle. Okay. I mean one

facet of this type that happened in the

past was the adoption of latte. So so

latte is this type seting language that

all mians use now. So in the past people

use all kinds of word processors and

typewriters and whatever but at some

point latte became easier to use than

all other competitors and that people

just switched you know within a few

years like it was just a dramatic um pay

shift. It's a wild out there question,

but what

what year how far away are we from

a uh AI system being a collaborator

on a proof that wins the Fields medal.

So that level. Okay. Um well, it depends

on the level of collaboration. I mean,

no, like it deserves to be to get the

Fields Medal. like so half and half

already like I I can imagine if it was a

winning paper having some AI systems in

writing it you know uh just you know

like the order complete alone is already

I I use it like it speeds up my my own

writing um um like you know you you can

have a theorem you have a proof and the

proof has three cases and I I write down

the proof of the first case and the

autocomplete just suggests all right now

now here's how the proof of second case

could work and like it was exactly

correct that was great saved me like 5

10 minutes of uh of typing but in that

case The AI system doesn't get the

Fields medal. No. Uh

are we talking 20 years, 50 years, 100

years? What do you think? Okay. So I I

gave a prediction in print. So by 2026,

which is now next year, um there will be

math collaborations, you know, where the

AI, so not Fields Medal winning, but but

like actual research level math like

published ideas that in part generated

by AI. Um maybe not the ideas but at

least uh some of the computations um um

the verifications. Yeah. I mean has that

already happened? Has that already

happened? Yeah. There are there are

problems that were solved uh by a

complicated process conversing with AI

to propose things and the human goes and

tries it and the contract doesn't work

but it might propose a different idea.

Um it it's it's hard to disentangle

exactly. Um there are certainly math

results which could only have been

accomplished because there was a math

method human mathematician and an AI

involved. Um but it's hard to sort of

disentangle credit. Um

I mean these tools they they do not uh

replicate all the skills needed to do

mathematics but they can replicate sort

of some non-trivial percentage of them

you know 30 40%. they can fill in gaps.

Um, you know, so, uh, coding is is is a

is a good example, you know. So, I I um

um it's annoying for me to code in

Python. I'm not I'm not a native um I'm

not a professional um programmer. Um,

but um the with AI that the the friction

cost of of doing it is is is much

reduced. Uh so it it fills in that gap

for me. Um

AI is getting quite good at literature

review. Um I mean there's still a

problem with um hallucinating you know

the references that don't exist. Um but

this I think is a civil war problem if

you train in the right way and so forth

you can you can and um and verify um you

know using the internet um you know um

you should in a few years get to the

point where you you have a a lema that

you need and uh we say has anyone proven

this lema before and it will do

basically a fancy web search AI

assistant and say yeah yeah there are

these six papers where something similar

has happened and I mean it you can ask

it right now and it'll give you six

papers of which maybe one is is

legitimate and relevant. One exists but

is not relevant and four are

hallucinated. Um it has a non-zero

success rate right now, but uh it's

there's so much garbage. Uh so much the

signal to noise ratio is so poor that

it's it's um it's most helpful when you

already somewhat know the literature. Um

and you just need to be prompted to be

reminded of a paper that was already

subconsciously in your memory versus

helping you discover new you were not

even aware of but is the correct

citation. Yeah, that's yeah, that it can

sometimes do. But but when it does, it's

it's buried in in a list of options for

which the other that are bad. Yeah. I

mean, being able to automatically

generate a related work section that is

correct. Yeah. That's actually a

beautiful thing that might be another

phase shift because it assigns credit

correctly. Yeah. It does. It breaks you

out of the silos of Yeah. Yeah. Yeah.

thought, you know. Yeah. No, there's a

big hump to overcome right now. I mean,

it's it's like self-driving cars, you

know. the the safety margin has to be

really high for it to be um uh to be

feasible. So yeah, so there's a last

mile problem um with a lot of AI

applications um that uh you know they

can develop tools that work 20% 80% of

the time but it's still not good enough

um and in fact even worse than good some

ways. I mean another way of asking the

Fields metal question is what year do

you think you'll wake up and be like

real surprised? you read the headline,

the news of something happened that AI

did like you know real breakthrough

something it doesn't you know like feels

metal even hypothesis it could be like

really just

this alpha zero moment with go that kind

of thing right um yeah this this decade

I can I can see it like making a

conjecture

between two unrelated two two things

that people thought was unrelated oh

interesting generating a conjecture

that's a beautiful conjecture Yeah. And

and actually has a real chance of being

correct and and and meaningful and um

because that's actually kind of doable I

suppose but the word of the data is

Yeah. No, that would be truly amazing.

Um the current models struggle a lot. I

mean so um a version of this is um I

mean the physicists have a dream of

getting the AI to discover new new laws

of physics. Um you know the the dream is

you just feed it all this data. Okay.

and and this is here's a new patent that

we didn't see before but it actually

even struggle the current state of the

art even struggles to discover old laws

of physics um from the data uh or if it

does there's a big concern contamination

that that it did it only because like

somewhere in this training data it some

new um you know boils law or whatever

ball you're trying to to to reconstruct

um part of it is that we don't have the

right type of training data for this um

yeah so for laws of physics like we we

don't have like a million different

universes with a million infant laws of

nature. Um

and um like a lot of what we're missing

in math is actually the negative space

of so we have published things of things

that people have been able to prove um

and conjectures that ended up being

verified um or maybe counter examples

produced but um we don't have data on on

things that were proposed and they're

kind of a good thing to try but then

people quickly realized that it was the

wrong conjecture and then they they said

oh but we we should actually change um

our claim to modify it in this way to

actually make it more plausible. Um

there's this there's a trial and error

process which is a real integral part of

human mathematical discovery which we

don't record cuz it's embarrassing. Uh

we make mistakes and and we only like to

publish our wins. Um and uh the AI has

no access to this data to train on. Um I

sometimes joke that basically AI has to

go through um grad school and actually

you know go to grad courses, do the

assignments, go to office hours, make

mistakes, um get advice on how to

correct the mistakes and learn from

that. Let me uh ask you if I may about

uh Gregori Pearlman. Mhm. You mentioned

that you try to be careful in your work

and not let a problem completely consume

you. just you really fall in love with

the problem and really cannot rest until

you solve it. But you also hasted to add

that sometimes this approach actually

can be very successful. An example you

gave is Gregoria Pearlman who proved the

point conjecture and did so by working

alone for 7 years with basically little

contact with the outside world. Can you

explain this one millennial prize

problem that's been solved point

conjecture and maybe speak to the

journey that Gagora Pearlman's been on.

All right. So it's it's a question about

curb spaces. Earth is a good example. So

you can think of a 2D surface in being

round could maybe be a Taurus with a

hole in it or it can have many holes and

there there are many different

topologies up priori that that a surface

could have. um even if you assume that

it's it's bounded and and uh and smooth

and so forth. So we have figured out how

to classify surfaces as a first

approximation everything is determined

by something called the genus how many

holes it has. So a sphere has genus 0 a

donut has genus one and so forth and one

way you can tell these surfaces apart

probably the sphere has which is called

simply connected if you take any closed

loop on the sphere like a big closed

little rope you can contract it to a

point and while staying on the surface

and the sphere has this property but a

taurus doesn't if on a taurus and you

take a rope that goes around say the the

outer diameter taurus there's no way it

can't get through the hole there's no

way to to contract it to a point so it

turns out that the this the sphere is

the only surface with this property of

contractability up to like continuous

deformationations of the sphere. So um

things that I want to call topologically

um equivalent of the sphere. So point

asked the same question in higher

dimensions. Um so this it becomes hard

to visualize because um surface you can

think of as embedded in three dimensions

but a curved free space we don't have

good intuition of 4D space to to to live

and and there are also 3D spaces that

can't even fit into four dimensions. you

need five or six or or higher. But

anyway, uh mathematically you can still

pose this question that if you have a

bounded threedimensional space now which

is also has this simply connected

property that every loop can be

contracted. Can you turn it into a

threedimensional version of a sphere?

And so this is the point conjecture.

Weirdly in higher dimensions four and

five it was actually easier. So uh it

was solved first in higher dimensions.

There's somehow more room to do the

deformation. It's easier to to to move

things around to a sphere. But three was

really hard. So people tried many

approaches. There sort of commentary

approaches where you chop up the the

surface into little triangles or or

tetrahedra and you you just try to argue

based on how the faces interact each

other. Um there were um algebraic

approaches. There's there's various

algebraic objects like things called the

fundamental group that you can attach to

these homology and coology and and and

all these very fancy tools. Um they also

didn't quite work. Um but Richard

Hamilton's proposed a um partial

differential equations approach. So you

take um you take so the problem is that

you so you have this object which is so

secretly is a sphere but it's given to

you in a in a really um in in a weird

way. So like like think of a ball that's

been kind of crumpled up and twisted and

it's not obvious that it's a ball. Um

but um like if you if you have some sort

of surface which is which is a deformed

sphere, you could um u you could for

example think of it as a surface of a

balloon. You could try to inflate it.

You you blow it up. Um and naturally as

you fill it with air um the the wrinkles

will sort of smooth out and it will turn

into um um a nice round sphere. Um

unless of course it was a Taurus or

something in which case it would get

stuck at some point like if you instead

of Taurus it would there'll be a point

in the middle when the inner ring

shrinks to zero you get you get a

singularity and you can't blow up any

further. You can't flow any further. So

he created this flow which is called

Richie flow which is a way of taking an

arbitrary surface or or space and

smoothing it out to make it rounder and

rounder to make it look like a sphere.

And he wanted to show that that either

uh this process would give you a sphere

or it would create a singularity. Um

actually very much like how PDS either

they have global regularity or finite

blow like basically it's almost exactly

the same thing. It's all connected. Um

and so and and he showed that for two

dimensions two dimensional services

surfaces um if you started simply

connected no singularities ever formed

um you never ran into trouble and you

could flow and it would give you a

sphere and it so he he got a new proof

of the two dimensional result but by the

way that's a beautiful explanation of

reach flow and its application in this

context how difficult is the mathematics

here like for the 2D case is it yeah

these are quite sophisticated equations

on par with the Einstein equations

slightly simpler but um Um yeah but but

they were considered hard nonlinear

equations to solve um and there's lots

of special tricks in 2D that that that

helped but in 3D the problem was that uh

this equation was actually super

critical the same problems as Nabia

Stokes as you blow up um maybe the

curvature could get constraint in finer

smaller smaller regions and it um it

looked more and more nonlinear and

things just look worse and worse and

there could be all kinds of

singularities that showed up. um some

singularities um like if there's these

things called neck pinches where where

the uh the surface sort of creates

behaves like like a like a a barbell and

it it pinches at a point. Some some

singularities are simple enough that you

can sort of see what to do next. You

just make a snip and then you can turn

one surface into two and evol them

separately. But there was there was a

the prospect that there's some really

nasty like knotted singularities showed

up that you you couldn't see how to um

resolve in any way that you couldn't do

any surgery to. Um so you need to

classify all the singularities like what

are all the possible ways that things

can go wrong. Um so what Pearlman did

was first of all he he made the problem

he turned the problem a super critical

problem to a critical problem. Um I said

before about how um the invention of the

of of energy the Hamiltonian like really

clarified um Newtonian mechanics. Um uh

so he introduced something which is now

called permanence reduced volume and

permanence entropy. He introduced new

quantities kind of like energy that look

the same at every single scale and

turned the problem into a critical one

where the nonlinearities actually

suddenly looked a lot less scary than

they did before. Um and then he had to

solve he still had to analyze the

singularities of this critical problem.

uh and that itself was a problem similar

to this wake up thing I worked on

actually um so on the on the level of

difficulty of that. So he managed to

classify all the singularities of this

problem and show how to apply surgery to

each of these and through that was able

to to resolve the point Cray conjecture.

um quite like a lot of really ambitious

steps um and like like nothing that a

large language model today for example

could I mean um at best uh I could

imagine model proposing this idea as one

of hundreds of different things to try

um but the other 99 would be complete

dead ends but you'd only find out after

months of work he must have had some

sense that this was the right track to

pursue because you know I it takes years

to get them from A to B so you've done

like you said Actually you see even

strictly mathematically but more broadly

in terms of the process he's done

similarly difficult

things what what can you infer from the

process he was going through because he

was doing it alone what are some low

points in a process like that when you

start to like you've mentioned hardship

like uh AI doesn't know when it's

failing what happens to you you're

sitting in your office when you realize

the thing you did for the last few days

maybe weeks weeks. Yeah. Is a failure.

Well, for me, I switch to different

problem. Uh so, uh as said, I'm I'm a

fox. I'm not a hedgehog. But you

legitimately that is a break that you

can take is is to step away and look at

a different problem. Yeah, you can

modify the problem too. Um I mean um

yeah, you can ask some cheat if if

there's a specific thing that's blocking

you that this um some bad case keeps

showing up that that that for which your

tool doesn't work, you can just assume

by fiat this this bad case doesn't

occur. So you you do some magical

thinking um for the but but but

strategically okay for the point to see

if the rest of the argument goes through

um if there's multiple problems uh with

with with your approach then maybe you

just give up okay but if this is the

only problem that you know but

everything else checks out then it's

still worth fighting um so yeah you have

to do some some sort of forward

reconnaissance sometimes to uh you know

and that is sometimes productive to

assume like okay we'll figure it out oh

yeah yeah eventually um Sometimes

actually it's even productive to make

mistakes. So um one of the I mean um

there was a project which actually u we

won some prizes for actually

four other people. Um we worked on this

PD problem again actually this blow of

regularity type problem. Um and it was

considered very hard. Um Sean Bain who

was another field methodist who worked

on a special case of this but he could

not solve the general case. Um and we

worked on this problem for two months

and we found we thought we solved it. We

we had this this cute argument that if

everything fit and we were excited uh we

were planning celebrationally um to all

get together and have champagne or

something. Um and we started writing it

up. Um and one of one of us, not me

actually, but another co-author said,

"Oh, um in this in this lema here, we um

we have to estimate these 13 terms that

that show up in this expansion." And we

estimate 12 of them, but in our notes, I

can't find the estimation of the 13th.

Can you can someone supply that? And I

said, "Sure, I'll look at this." and

actually yeah we didn't cover we

completely omitted this term and this

term turned out to be worse than the

other 12 terms put together um in fact

we could not estimate this term um and

we tried for a few more months and all

different permutations and there was

always this one thing one term that we

could not control um and so like um this

was very frustrating um but because we

had already invested months and months

of effort into this already um we stuck

at this we we tried increasingly

desperate things and and crazy things um

and after two is we found an approach

which was actually somewhat different by

quite a bit from our initial um strategy

which did actually didn't generate these

problematic terms and and and actually

solve the problem. So we we solve a

problem after 2 years but if we hadn't

had that initial false dawn of nearly

solving a problem we would have given up

by month two or something and and worked

on an easier problem. Um yeah if we had

known it would take two years not sure

we would have started the project. Yeah

sometimes actually having the incorrect

you know it's like Columbus New

incorrect version of measurement of the

size of the earth. He thought he was

going to find a new trade route to India

or at least that was how he sold it in

his perspectus. I mean it could be that

he actually secretly knew but just on

the psychological element.

Do you have like emotional or

like self-doubt that just overwhelms you

moments like that? You know, because

this stuff it feels like math is is so

engrossing

that like it can break you when you like

invest so much yourself in the problem

and then it turns out wrong. You could

start to

similar way chess has broken some

people. Yeah. Um I I think different

mathematicians have different levels of

emotional investment in what they do. I

mean I think for some people it's just a

job. you know you you have a problem and

if it doesn't work out you you you go on

the next one. Um yeah so the fact that

you can always move on to another

problem um it reduces the emotional

connection. I mean

there are cases you know so there are

certain problems that are what I call

back diseases where where where just

latch on to that one problem and they

spend years and years thinking about

nothing but that one problem and um you

know maybe their career suffers and so

forth but okay this big win this will

you know once I once I finish this

problem I will make up for all the years

of of of lost opportunity but that's

that's I mean occasionally occasionally

it works But I I um I really don't

recommend it for people without the the

right fortitude. Yeah. So I I've never

been super invested in any one problem.

Um one thing that helps is that we don't

need to call our problems in advance. Uh

um well uh when we do grant proposals we

s say we we will study this set of

problems. But even then we don't promise

definitely by 5 years I will supply a

proof of all these things. you know, or

um you promise to make some progress or

discover some interesting phenomena. Uh

and maybe you don't solve the problem,

but you find some related problem that

you you can say something new about and

that's that's a much more feasible task.

But I'm sure for you there's problems

like this. You have you have

um made so much progress towards the

hardest problems in the history of

mathematics. So is there is there a

problem that just haunts you? It sits

there in the dark corners, you know,

twin prime conjecture, reman hypothesis,

global conjecture. Twin prime that

sounds again. So, I mean, the problem is

like a reman hypothesis, those are so

far out of reach. Why do you think so?

Yeah. there's no even viable strate like

even if I activate all my all the cheats

that I know of in this problem like it

there's just still no way to get me to

be um like it's um I think it needs a

breakthrough in another area of

mathematics to happen first and for

someone to recognize that it that would

be a useful thing to transport into this

problem. So we we should maybe step back

for a little bit and just talk about

prime numbers. Okay. So they're often

referred to as the atoms of mathematics.

Can you just speak to the structure that

these uh atoms the natural numbers have

two basic operations attached to them?

Addition and multiplication. Um so if

you want to generate the natural

numbers, you can do one of two things.

You can just start with one and add one

to itself over and over again and that

generates you the natural numbers. So

additively they're very easy to generate

1 2 3 4 5. Or you can take the prime if

you want to generate multiplicatively

you can take all the prime numbers 2 3

57 and multiply them all together. um

and together that gives you all the the

natural numbers except maybe for one. So

there these two separate ways of

thinking about the natural numbers from

an additive point of view and point of

view. Um and separately they're not so

bad. Um so like any question about that

only was addition is relatively easy to

solve and any question that only was

multiplication is easy to solve. Um but

what has been frustrating is that you

combine the two together. Um and

suddenly you get this extremely rich I

mean we know that there are statements

in number theory that are actually as

undecidable. There are certain polomials

in some number of variables. You know is

there a solution in the natural numbers

and the answer depends on on an

undecidable statement um like like

whether um the aims of of mathematics

are consistent or not. Um

but um yeah but even this the simplest

problems that combine something

multiplicative such as the primes with

something additive such as shifting by

two uh separately we understand both of

them well but if you ask when you shift

the prime by two do you can you get a

how often can you get another prime we

it's been amazingly hard to relate the

two and we should say that the twin

prime conjecture is just that it posits

that there are infinitely many pairs of

prime numbers that differ by do. Yes.

Now the interesting thing is that you

have been very successful at pushing

forward the field in answering these

complicated questions uh of this variety

like you mentioned the green tile

theorem. It proves that prime numbers

contain arithmetic progressions of any

length, right? Which is mind-blowing

that you can prove something like that,

right? Yeah. So, what we've realized

because of this this this type of of

research is that there's different

patterns have different levels of uh

indestructibility. Um so, so what makes

the twin prime problem hard is that if

you take all the primes in the world,

you know, 3, 5, 7, 11, so forth, there

are some twins in there. 11 and 13 is a

twin prime pair of twin primes and so

forth. But you could easily if you

wanted to um redact the primes to get

rid of to get rid of the um these twins

like the twins they show up and they're

infinitely many of them but they're

actually reasonably sparse. Um not there

there's not I mean initially there's

quite a few but once you got to the

millions the trillions they become rarer

and rarer and you could actually just

you know if if someone was given access

to the database of primes you just edit

out a a few primes here and there they

could make the trim pan conjure false by

just removing like 01% of the primes. or

something um just well well chosen to to

um to do this. And so you could present

a censored database of the primes which

passes all of the statistical tests of

the primes. You know that it it obeys

things like the paralle theorem and and

other texts about the primes but doesn't

contain any true primes anymore. Um and

this is a real obstacle for the twin

prime conjecture. It means that any

proof strategy to actually find twin

primes in the ecto primes must fail when

applied to these slightly edited primes.

And so it must be some very um subtle

delicate feature of the primes that you

can't just get from like like aggregate

statistical analysis. Okay. So that's

all yeah on the other hand progressions

has turned out to be much more robust.

um like you can take the primes and you

can eliminate 99% of the primes actually

you know and you can take take any 99%

you want and uh it turns out and another

thing we prove is that you still get

arithmetic progressions um arithmetic

progressions are much you know they're

like cockroaches of arbitrary length yes

that's crazy I mean so so this for for

people who don't know arithmetic

progressions is a sequence of numbers

that differ by some fixed amount yeah

but it's again like it's infinite monkey

type phenomenon for any fixed length of

your set. You don't get arbitrary as

progressions. You only get quite short

progressions. But you're saying twin

prime is not an infinite monkey

phenomena. I mean, it's a very subtle

monkey. It's still an infinite monkey

phenomen. Yeah. If the primes were

really genuinely random, if the primes

were generated by monkeys, um then yes,

in fact, the infinite monkey theorem

would Oh, but you're saying that twin

prime is it doesn't you can't use the

same tools like the it doesn't appear

random almost. Well, we don't know.

Yeah, we we we we believe the prior

behave like a random set. And so the

reason why we care about the trim

conjecture is is a test case for whether

we can genuinely confidently say with

with 0% chance of error that the primes

behave like a random set. Okay. Random.

Yeah. Random versions of the primes we

know contain twins. Um at least with

with 100% probability or probably

tending to 100% as you go out further

and further. Um yeah. So the primes we

believe that they're random. Um the

reason why arithmetic progressions are

indestructible is that regardless of

whether you looks random or looks um

structured like periodic in both cases

um arithmetic regressions appear but for

different reasons. Um and this is

basically all the ways in which the

there are many proofs of of these sort

of arithmetic region epithems and

they're all proven by some sort of

dichotomy where your set is either

structured or random and in both cases

you can say something and then you put

the two together. Um but in twin primes

if if the primes are random then you're

happy you win. But if your primes are

structured they could be structured in

in a specific way that eliminates the

twin the twins. Uh and we can't rule out

that one conspiracy and yet you were

able to make a as I understand progress

on the Kupal version. Right. Yeah. So um

the one funny thing about conspiracies

is that any one conspiracy theory is

really hard to disprove that you know if

if you believe the water is won by

lizards you say here's some evidence

that that it it's not run by lizards

well that that evidence was planted by

the lizards. Yeah. Right. You may have

encountered this kind of phenomen. Yeah.

So like like um a pure like there's

there's almost no way to um definitively

rule out a con and the same is true in

mathematics that a con is to solely

devote devoted to learning twin primes

you know like it would you have to also

infiltrate other areas of mathematics to

sort of but but like it could be made

consistent at least as far as we know

but there's a weird phenomenon that you

can make one um one conspiracy rule out

other conspiracies so you know if the if

the world is is run by lizardist it

can't also be run by Right.

Right. So one unreasonable thing is is

is is hard to dispute but but more than

one there are there are tools. Um so

yeah so for example we we know there's

infinitely many primes that are um no

two which are um so there infinite pair

of primes which differ by at most um 246

actually is is a is the current. So

there's like a bound yes on the right.

So like there's twin primes this thing

called cousin primes that differ by by

four. Um there's called sexy primes that

differ by six. Uh, what are sexy primes?

Primes that differ by six. The name is

much less the concept is much less

exciting than the name suggests. Got it.

Um, so you can make a conspiracy rule

out one of these, but like once you have

like 50 of them, it turns out that you

can't rule out all of them at once. It

just it requires too much energy somehow

in this conspiracy space. How do you do

the bound part? How do you how do you

develop a bound for the difference

between the prize that okay so um that

there's an infinite number of so it's

ultimately based on what's called the

pigeon hole principle um so the pigeon

hole principle uh it's a statement that

if you have a number of pigeons and they

all have to go into into pigeon holes

and you have more pigeons than pigeon

holes then one of the pigeon holes has

to have at least two pigeons in so there

has to be two pigeons that that are

close together. So for instance if you

have 100 numbers and they all range from

one to a thousand um two of them have to

be at most 10 apart. Mhm. because you

can divide up the numbers one to 100

into 100 pigeon holes. Let's let's say

you have if you have 101 numbers 100 one

numbers then two of them have to be

distance less than 10 apart because two

of them have to belong to the same

pigeon hole. So it's a basic um basic

feature of uh a basic principle in

mathematics. Um so it doesn't quite work

with the primes directly because the

primes get sparer and sparser as you go

out that fewer and fewer numbers are

prime. But it turns out that there's a

way to assign weights to the to to

numbers like um so there are numbers

that are kind of almost prime but

they're not they they don't have no

factors at all other than themselves in

one but they have very few factors. Um

and it turns out that we understand

almost primes a lot better than primes.

Um and so for example it was known for a

long time that there were twin almost

primes. This has been worked out. So

almost primes are something we can't

understand. So you can actually restrict

attention to a a suitable set of almost

primes and uh whereas the primes are

very sparse overall relative to the

almost primes actually are much less

sparse. They make um you can set up a

set of almost primes where the primes

have density like say 1%. Um and that

gives you a shot at proving by applying

some sort of original principle that

that those pairs of primes are just only

100 100 apart. But in order to with the

twin pan conjecture you need to get the

density of primes inside the also size

up to up to a first of 50%. Um once you

get up to 50% you will get twin primes.

But uh unfortunately there are barriers.

Um we know that that no matter what kind

of good set of almost primes you pick

the density primes can never get above

50%. It's called the parody barrier. Um

and I would love to find yes. So one of

my long-term dreams is to find a way to

breach that barrier because it would

open up not only the trip conjecture the

go back conjecture and many other

problems in number theory are currently

blocked because our current techniques

would require improve going beyond this

theoretical um parody barriers. It's

like it's like pulling past the speed of

light. Yeah. So we just say a twin prime

conjecture one of the biggest problems

in the history of mathematics go by

conjecture also um they feel like

nextdoor neighbors. Uh has there been

days when you felt you saw the path? Oh

yeah. Um um yeah uh sometimes you try

something and it it works super well. Um

you you again again the sense of methac

smell uh we talked about earlier uh you

learn from experience when things are

going too well

because there are certain difficulties

that you sort of have to encounter. Um

um I think the way a colleague might put

it is that um you know like if if you

are on the streets in New York and you

put in a blindfold and you put in a car

and and um after some hours um you the

blindfold's off and you're in Beijing.

Um you know I mean that was too easy

somehow like like there was no ocean

being crossed. Even if you don't know

exactly what how what what was done

you're suspecting that that something

wasn't right. But is that still in the

back of your head to do you return to

these to the prime do you return to the

prime numbers every once in a while to

see yeah when I have nothing better to

do which is less and less 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 to

work on my sort of real research

projects and I also don't want to do my

administrative stuff I don't want to do

some errands for my family um I can play

with these these things um for fun uh

and usually you get nowhere Yeah, you

have you have to learn to just say okay

fine once again nothing happened I I

will move on. Um yeah very occasionally

one of these problems I actually solved

or sometimes as you say you think you

solved it and then you're euphoric for

maybe 15 minutes and then you think I

should check this because this is too

easy too good to be true and it usually

is. What's your gut say about when these

problems would be uh solved when prime

and go back? Prime I think we'll keep

getting keep getting more partial

results. Um

it does need at least one this parody

barrier is is the biggest remaining

obstacle. Um there are simpler versions

of the conjecture where we are getting

really close. Um so I think we will in

10 years we will have many more much

closer results. May not have the whole

thing. Um yeah so trens is somewhat

close reman hypothesis I have no I mean

it has to happen by accident I think so

the reman hypothesis is a kind of more

general conjecture about the

distribution of prime numbers right yeah

it's it's states are sort of viewed

multiplicatively like for questions only

involving multiplication no addition the

primes really do behave as randomly as

as you could hope so there's a

phenomenon in probability called square

root cancellation that um you know like

if you want to poll say America upon on

on some issue. Um, and you you ask one

or two voters and you may have sampled a

bad sample and then you get you get a

really imprecise um measurement of of

the full average, but if you sample more

and more people, the accuracy gets

better and better and it actually

improves like the square root of the

number of people you you sample. So

yeah, if you sample a thousand people,

you can get like a 2 3% margin of error.

So in the same sense if you measure the

primes in a certain multiplicative sense

there's a certain type of statistic you

can measure and it's called the reman's

data function and it fluctuates up and

down but in some sense um as you keep

averaging more and more if you sample

and 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 reman hypothesis is a very

elegant way that captures this but um as

with many others in mathematics we have

very few tools to show that something

really genuinely behaves like really

random And this is actually not just a

little bit random but it's it's asking

that it behaves as random as it actually

random set this this 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. Um the proof has to come out

of left field. Um

yeah but uh what that is yeah no one has

any serious proposal. Um yeah and and

there's there's various ways to sort of

as I said you can modify the primes a

little bit and you can destroy the human

hypothesis. Um so like it has to be very

delicate. You can't apply something that

has huge margins of error. It has to

just barely work. Um and like um there's

like all these pits pitfalls that you

have like dodge very adeptly. The prime

numbers are just fascinating. Yeah.

Yeah. What what to you is um most

mysterious about the prime numbers.

So that's a good question. So like

conjecturally we have a good model of

them. I mean like as I said I mean they

have certain patterns like the primes

are usually odd for instance but apart

from this of obvious patterns they

behave very randomly and just assuming

that they behave so there's something

called the crema random model of the

primes that that after a certain point

primes just behave like a random set. Um

and there's various slight modifications

this model but this has been a very good

model. It matches the numeric. It tells

us what to predict. Like I can tell you

with complete certainty the truth is

true. Uh the random model gives

overwhelming odds it is true. I just

can't prove it. Most of our mathematics

is optimized for solving things with

patterns in them. Um and the primes have

this anti-attern um as do almost

everything really. But we can't prove

that. Yeah. I guess it's not mysterious

that the prize be kind of random because

there no reason for them to be um uh to

have any kind of secret pattern but what

is mysterious is what is the mechanism

that really forces the randomness to

happen. Uh and this is just absent.

Another incredibly surprisingly

difficult problem is the colots's

conjecture. Oh yes. simple to state,

beautiful to visualize in its simplicity

and yet extremely

uh difficult to solve and yet you have

been able to make progress. Uh Paular

said about the coloss conjecture that

mathematics may not be ready for such

problems. Others have stated that it is

an extraordinarily difficult problem

completely out of reach this is in 2010

out of reach of present- day mathematics

and yet you have made some progress. Why

is it so difficult to make? Can you

actually even explain what it is? Oh,

yeah. So, it's it's it's a problem that

you can explain. Um yeah, it um it helps

with some um visual aids, but yeah, so

you take any natural number like say 13.

And you apply the the following

procedure to it. So, if it's even, you

divide it by two and if it's odd, you

multiply by three and add one. So, even

numbers get smaller, odd numbers get

bigger. So, 13 will become 40 because 13

* 3 is 39. Add one, you get 40. So, it's

a simple process for odd numbers and

even numbers. They're both very easy

operations. And then you put it

together. It's still reasonably simple.

Um, but then you ask what happens when

you iterate it. You take the output that

you just got and feed it back in. So, 13

becomes 40. 40 is now even divide by 2

is 20. 20 is still even divide by 10 2

10 5 and then 5 * 3 + 1 is 16. And then

8 4 2 1. So, uh, and then from 1 it goes

1 4 2 1 421. It cycles forever. So this

sequence I just described um yeah 13 40

20 10 so these are also called hailstone

sequences because there's an

oversimplified model of of hailstone

formation yeah which is not actually

quite correct but it's so somehow taught

to high school students as a first

approximation is that um like a a little

nugget of ice gets gets an ice crystal

forms in a cloud and it it goes up and

down because of the wind and sometimes

when it's cold it get acquires a bit

more mass and maybe it melts a little

bit and this process of going up down

creates this s of partially melted ice

which event hell stone and eventually it

falls out the earth. So the conjecture

is that no matter how high you start up

like you take a number which is in the

millions or billions you go this process

that that goes up if you're odd and down

if you're even eventually um goes down

to to earth all the time no matter where

you start with this very simple

algorithm you end up at one and you

might climb for a while right yeah so

yeah if you plot it um these sequences

they look like brownie in motion um they

look like the stock market you know they

just go up and down in a in a seemingly

random pattern and in Usually that's

what happens that that if you plug in a

random number, you can actually prove at

least initially that it would look like

um random walk. Um and that's actually a

random walk with a downward drift. Um

it's like if you're always gambling on

on roulette at at the casino with odds

slightly weighted against you. So

sometimes you you win, sometimes you

lose, but over in the long run you lose

a bit more than you win. Um and so

normally your wallet will hit will go to

zero um if you just keep playing over

and over again. So statistically it

makes sense. Yes. So, so the result that

I I proved roughly speaking such that

that statistically like 99% of all

inputs would would drift down to maybe

not all the way to one, but to be much

much smaller than what you started. So,

it's it's like if I told you that if you

go to a casino, most of the time you end

up if you keep playing for long enough,

you end up with a smaller amount in your

wallet than when you started. That's

kind of like the what the result that I

proved. So why is that result like can

you continue down that thread

to prove the full conjecture? Well, the

problem is that um my I I used arguments

from probability theory um and there's

always this exceptional event. So you

know, so in probability we have this

this law of large numbers um which tells

you things like if you play a casino

with a um a game at a casino with a

losing um expectation over time you are

guaranteed or almost surely with

probably probability as close to 100% as

you wish you're guaranteed to lose

money. But there's always this

exceptional outlier. Like it is

mathematically possible that even in

when the game is is the odds are not in

your favor, you could just keep winning

slightly more often than you lose. Very

much like how in Navia Stokes there

could be, you know, um most of the time

um your waves can disperse. There could

be just one outlier choice of initial

conditions that would lead you to blow

up. And there could be one outlier

choice of um um special number that they

stick in that shoots off infinity while

all other numbers crash to earth uh

crash to one. Um in fact um there's some

mathematicians um who Alex Kovvich for

instance who've proposed that um that

actually um these collat uh iterations

are like the similar automator um

actually if you look at what they happen

on in binary they do actually look a

little bit like like these game of life

type patterns. Um and in an analogy to

how the game of life can create these

these massive like self-plicating

objects and so forth possibly you could

create some sort of heavier than air

flying machine a number which is

actually encoding this machine which is

just whose job it is is to encode is to

create a version of itself which which

is larger heavier than air machine

encoded in a number that flies forever.

Yeah. So Conway in fact worked on worked

on this problem as well. Oh wow. So

Conway so similar in fact that was one

of inspirations for the Nebby Stokes

project that Conway studied

generalizations of the collapse problem

where instead of multiplying by three

and adding one or dividing by two you

have a more complicated branch but but

instead of having two cases maybe you

have 17 cases and then you go up and

down and he showed that once your

iteration gets complicated enough you

can actually encode touring machines and

you can actually make these problems

undecidable and and do things like this.

In fact, he invented a programming

language for uh these kind of fractional

linear transformations. He called a

factrat as a play on forrat. Uh and he

showed that that you could um you can

program it was too incomplete. You could

you could you could uh um you could make

a program that if if your number you

insert in was encoded as a prime, it

would sync to zero. It would go down

otherwise it would go up uh and things

like that. Um so the general class of

problems is is really uh as complicated

as all of mathematics. some of the

mystery of the cellular automa that we

talked about uh having a fra

mathematical framework to say anything

about cellular automa maybe the same

kind of framework is required yeah

injecture yeah if you want to do it not

statistically but you really want 100%

of all inputs to to fall to earth yeah

so what might be feasible is is

statistically 99% you know go to one but

like everything yeah that looks hard

what would you say is out of these

within reach famous problems is the

hardest problem we have today. Is there

a reman hypothesis? We want is up there.

Um POS MP is a good one because like uh

that's that's a meta problem like if you

solve that in the um in the positive

sense that you can find a PMP algorithm

that potentially this solves a lot of

other problems as well and we should

mention some of the conjectures we've

been talking about. You know a lot of

stuff is built on top of them. Now

there's ripple effects. P equ= 1 P has

more ripple effects than basically any

other right if the reman hypothesis is

disproven um that would be a big mental

shock to the number theorist uh but it

would have follow on effects for um

cryptography

um because a lot of cryptography uses

number theory um it uses number theory

constructions involving primes and so

forth and um it relies very much on the

intuition that number theories are built

over many many years of what operations

involving prime behave randomly and what

ones don't. Um, and in particular, our

encryption

um methods are designed to turn text

with information on it into text which

is indistinguishable from um from random

noise. So um and hence we believe to be

almost impossible to crack um at least

mathematically. Um but uh if something

has core to our belief as human

hypothesis is is wrong it means that

there are there are actual patterns of

the primes that we not aware of and if

there's one there's probably going to be

more. Um and suddenly a lot of our

crypto systems are in doubt. Yeah.

But then how do you then say stuff about

the the primes? Yeah. That you're going

towards the collect conjecture again. Um

because if I I you you want it to be

random, right? You want it to be

randomly. Yeah. So more broadly, I'm

just looking for more tools, more ways

to show that that that things are

random. How do you prove a conspiracy

doesn't happen, right? Is there any

chance to you that P equals NP? Is there

some Can you imagine a possible

universe? It is possible. I mean there's

there's various uh scenarios. I mean

there there's one where it is

technically possible but in practice is

never actually implementable. The

evidence is sort of slightly pushing in

favor of no that we probably is not

equal to NP. I mean it seems like it's

one of those cases similar similar to

reman hypothesis that I think the

evidence is le leaning pretty heavily on

the no. Certainly more on the no than on

on the yes. The funny thing about

picompy is that we have also a lot more

obstructions than we do for almost any

other problem. Um so while there's

evidence we also have a lot of results

ruling out many many types of approaches

to the problem. Uh this is the one thing

that the computer scientists have

actually been very good at. It's

actually saying that that certain

approaches cannot work. No go theorems.

It could be undecidable. We don't Yeah,

we don't know. There's a funny story I

read that when you won the Fields Medal,

somebody from the internet wrote you

and asked uh you know what are you going

to do now that you've won this

prestigious award? and and then you just

quickly very humbly said that, you know,

this a shiny metal is not going to solve

any of the problems I'm currently

working on. So, I'm just I'm going to

keep I'm going to keep working on them.

It's just first of all, it's funny to me

that you would answer an email in that

context, and second of all, it um it

just shows your humility. But anyway, uh

maybe you could speak to the Fields

Medal, but it's another way for me to

ask uh

about Gregoria Pearlman. 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

he's somewhat of an outlier. Um even

among mathematicians who tend to uh to

have uh somewhat idealistic views. I've

never met him. I think I'd be interested

to meet him one day, but I I never had

the chance. I know people who met him,

but he's always had strong views about

certain things. Um, you know, I mean,

it's it's not like he was completely

isolated from the math community. I

mean, he would he would give talks and

write papers and so forth. Um, but at

some point he just decided not to engage

with the rest of the community. He was

he was disillusioned or something. Um, I

don't know. Um, and he decided to to uh

uh to peace out uh and you know, collect

mushrooms in St. Petersburg or

something. And then that's that's fine.

you know and you can you can do that. Um

I mean that's another sort of flip side.

I mean we are not a lot of our problems

that we solve you know they some of them

do have practical application and that's

that's great but uh like if you stop

thinking about a problem you know so

he's he hasn't published since in in

this field but that's fine there's many

many other people who've done so as

well. Um yeah so I guess one thing I

didn't realize initially with the fields

medal is that it it sort of makes you

part of the establishment. Um you know

so you know most mathematicians you

there's uh just career mathematicians

you know you just focus on publishing

the next paper maybe getting one to

promote one one rank you know and and

starting a few projects maybe taking

some students or something. Yeah. But

then suddenly people want your opinion

on things and uh you have to think a

little bit about you know things that

you might just so foolishly say because

you know no one's going to listen to

you. Uh it's more important now. Is it

constraining to you? Are you able to

still have fun and be a rebel and try

crazy stuff and well play with ideas? I

have a lot less free time than I had

previously. Um I mean mostly by choice.

I mean I I I obviously I have the option

to sort of uh decline. So I decline a

lot of things. I I could decline even

more. Um or I could acquire a reputation

for being so unreliable that people

don't even ask anymore. Uh this is I

love the different algorithms here. This

is great. This is it's always an option.

Um but you know um there are things that

are like

I mean so I mean I I I don't spend as

much time as I do as a postto you know

just just working on one problem at a

time or um fooling around. I still do

that a little bit but yeah as you

advance in your career somehow the more

soft skills so math somehow frontloads

all the technical skills to the early

stages of your career. So um yeah, so

it's as a post office publisher or

parish you're you're incent you're

incentivized to basically focus on on

proving very technical themsel

um as well as proof the theorems. Um but

then as as you get more senior you have

to start you know mentoring and and and

and giving interviews uh and uh and

trying to shape um direction of the

field both research wise and and you

know uh sometimes you have to uh u you

know do various administrative things

and it's kind of the right social

contract because you you need to to work

in the trenches to see what can help

mathematicians. the other side of the

establishment sort of the the really

positive thing is that um you get to be

a light that's an inspiration to a lot

of young mathematicians or young people

that are just interested in mathematics.

It's like it's just how the human mind

works. This is where I would probably uh

say that I like the fields metal

that it does inspire a lot of young

people somehow. I don't this just how

human brains work. Yeah. At the same

time, I also want to give sort of

respect to somebody like Gregoria

Pearlman who

is critical of awards in his mind. Those

are his principles and any human that's

able for their principles to like do the

thing that most humans would not be able

to do. It's beautiful to see. Some

recognition is is necessarily important.

Uh but yeah, it's it's also important to

not let these things take over your

life. um and like only be concerned

about uh getting the next big award or

whatever. Um I mean yeah so again you

see these people try to only solve like

a really big math problems and not work

on on on things that are less uh sexy if

you wish but but but actually still

interesting and instructive as you say

like the way the human mind works it's

um we understand things better when

they're attached to humans um and also

uh if they're attached to a small number

of humans like this this way our human

mind is is wired we can comprehend and

the relationships between you know 10 or

20 people you know but once you get

beyond like 100 people like there

there's a there's a limit I think

there's a name for it um beyond which uh

it just becomes the other um and so we

have you have to simplify the pole

master you know 99.9% of humanity

becomes the other um and uh often these

models are are incorrect and this causes

all kinds of problems but um so yeah so

to humanize a subject you know if you

identify a small number of people and

say you know these

representative people of the subject

role models for example um that has some

role um but it can also be um uh yeah

too much of it can be harmful because

it's

I'll be the first to say that my own

career path is not that of a typical

mathematician um I the very accelerated

education I skipped a lot of classes um

I think I was had very fortunate

mentoring opportunities um and I think I

was at the right place at the right time

just because someone does doesn't have

my um trajectory, you it doesn't mean

that they can't be good mathematicians.

I mean they be ma good mathematician in

a very different style. Uh and we need

people of a different style. Um and you

know even if and sometimes too much

focus is given on the on the person who

does the last step to complete um a

project in mathematics or elsewhere

that's that's really taken you know

centuries or decades with lots and lots

of building lots of previous work. Um,

but that's a a story that's difficult to

tell um if you're not an expert because,

you know, it's easier to just say one

person did this one thing. You know, it

makes for a much simpler history. I

think on the whole it um is a hugely

positive thing to to talk about Steve

Jobs as a representative of Apple when I

personally know and of course everybody

knows the incredible design, the

incredible engineering teams, just the

individual humans on those teams.

They're not a team. They're individual

humans on a team. And there's a lot of

brilliance there. But it's just a nice

shorthand like a very like pi. Yeah.

Steve Jobs. Yeah. Yeah. As as a starting

point, you know, as a first

approximation that's how you and then

read some biographies and then look into

much deeper. First approximation. Yeah.

That's right. Uh so you mentioned you

were a Princeton to um Andrew Wilds at

that time. He's a professor there. It's

a funny moment how history is just all

interconnected. And at that time he

announced that he proved the form last

theorem. What did you think maybe

looking back now with more context about

that moment in math history? Yes. So I

was a graduate student at the time. I

mean I I vaguely remember you know there

was press attention and uh um we all had

the same um we had pigeon holes in the

same mail room you know. So we all

picked our mail and like suddenly Andrew

W's mailbox exploded to be overflowing.

That's a good that's a good metric.

Yeah. um you know so yeah we we all

talked about it at at tea and so forth I

mean we we didn't understand most of us

didn't understand the proof um we

understand sort of high level details um

fact there's an ongoing project to

formalize it in lean right Kevin puzzly

yeah can can we take that small tangent

is it is it how difficult does that cuz

as as I understand the for last the

proof for uh for last theorem has like

super complicated objects yeah really

difficult to formalize now yeah I guess

yeah you're right the objects that they

use um you can define them. Uh so

they've been defined in lean. Okay. So

so just defining what they are can be

done. Uh that's really not trivial but

it's been done. But there's a lot of

really basic facts about um these

objects that have taken decades to prove

and that they're in all these different

math papers and so lots of these have to

be formalized as well. Um Kevin's uh

Kevin Buzzard's goal actually he has a

five-year grraft to formalize fossil

theorem and his aim is that he doesn't

think he will be able to get all the way

down to the basic axioms but he wants to

formalize it to the point where the only

things that he needs to rely on as black

boxes are things that were known by 1980

to um to number theorist at the time. Um

and then some other person some other

work would have to done to to to get

from there. Um so it's it's a different

area of mathematics than um the type of

mathematics I'm used to. Um um in

analysis, which is kind of my area, um

the objects we study are kind of much

closer to the ground. We study I study

things like prime numbers and and

functions and things that are within

scope of a high school um math education

to at least uh define. Um yeah, but then

there's this very advanced algebraic

side of number theory where people have

been building structures upon structures

for quite a while. Um and it's it's a

very sturdy structure. It's it's been

it's been very um at the base at least

is extremely well developed in the

textbooks and so forth. But um um it

does get to the point where um if you if

you haven't taken these years of study

and you want to ask about what what is

going on at um like level six of of this

tower, you have to spend quite a bit of

time before they can even get to the

point where you can see you see

something you recognize. What uh

inspires you about his journey that we

similar as we talked about seven years

mostly working in secret? Yeah. Uh that

is a romantic uh Yeah. So it kind of

fits with sort of the the romantic image

I think people have of mathematicians to

the extent they think of them at all as

these kind of eccentric uh you know

wizards or something. Um so that

certainly kind of uh uh accentuated that

perspective you know I mean it's it is a

great achievement his style of solving

problems is so different from my own um

but which but which is great. I mean we

we need people speak to it like what uh

in in terms of like the you like the

collaborative I like moving on from a

problem if it's giving too much

everybody. Um got it. But you need the

people who have the tenacity and the

fearlessness. Um you I've collaborated

with with people like that where where I

want to give up uh cuz the first

approach that we tried didn't work and

the second one didn't approach but

they're convinced and they have the

third fourth and the fifth approach

works. Um and I have to eat my words.

Okay. I didn't think this was going to

work, but yes, you were right all along.

And we should say for people who don't

know, not only are you known for the

brilliance of your work, but the

incredible productivity, just the number

of papers, which are all of very high

quality. So there's something to be said

about being able to jump from topic to

topic. Yeah, it works for me. Yeah, I

mean there also people who are very

productive and they focus very deeply on

Yeah. I think everyone has to find their

own workflow. Um like one thing which is

a shame in mathematics is that we have

mathematics there's sort of a one size

fits all approach to teach teaching

mathematics um and you know so we have a

certain curriculum and so forth I mean

you know maybe like if you do math

competitions or something you get a

slightly different experience but um I

think many people um they don't find

their their native math language uh

until very late or usually too late so

they they stop doing mathematics and

they have a bad experience with a

teacher who's trying to teach them one

way to do mathematics. They don't like

it. Um my theory is that um humans don't

come evolution has not given us a math

center of a brain directly. We have a

vision center and a language center and

some other centers um which have

evolution has honed but we it doesn't we

don't have innate sense of mathematics.

Um but our other centers are

sophisticated enough that different

people we we we can repurpose other

areas of our brain to do mathematics. So

some people have figured out how to use

the visual center to do mathematics and

so they think very visually when they do

mathematics. Some people have repurposed

their their language center and they

think very symbolically. Um, you know,

um, some people like if they are very

competitive and they they like gaming,

there's a type there's this part of your

brain that's very good at at at uh at

solving puzzles and games and and and

that can be repurposed. But like when I

talked about the mathematicians, you

know, they don't quite think they I can

tell that they're using some different

styles of of thinking than I am. I mean,

not not disjoint, but they they may

prefer visual. Like I I don't actually

prefer visual so much. I need lots of

visual aids myself. Um, you know,

mathematics provides a common language.

So, we can still talk to each other even

if we are thinking in in different ways.

But you can tell there's a different

set of subsystems being used in the

thinking process like they take

different paths. They're very quick at

things that I struggle with and vice

versa. Um, and yet they still get to the

same goal. Um, that's beautiful. And

yeah, but I mean the way we educate

unless you have like a personalized

tutor or something. I mean education

sort of just by natural scale has to be

mass-produced you know you have to teach

to 30 kids and you know if they have 30

different styles you can't you can't

teach 30 different ways. On that topic

what advice would you give to students

uh young students who are struggling

with math and but are interested in it

and would like to get better. Is there

something in this Yeah. um in this

complicated educational context, what

what would you Yeah, it's a tricky

problem. One nice thing is that there

are now lots of sources for mathematical

enrichment outside the classroom. Um so

in in in my day there already there are

math competitions. Um and you know there

also like popular math books in the

library. Um yeah but but now you have

you know YouTube uh there there are

forums just devoted to solving you know

math puzzles and um and math shows up in

other places you know like um for

example there there are hobbyists who

play poker for fun uh and um they they

you know they for very specific reasons

are interested in very specific

probability questions um and and they

actually know there's a community of

amateur proists in in in poker um in

chess, in baseball. I mean, there's

there's there's uh yeah um there's math

all over the place. Um and I'm I'm I'm

hoping actually with with these new sort

of tools for lean and so forth that

actually we can incorporate the broader

public into math research projects um

like this is almost is doesn't happen at

all currently. So in the sciences

there's some scope for citizen science

like astronomers uh they amateurs who

discover comets and there's biologists

there people who could identify

butterflies and so forth. Um and in

mathematics

whereum amateur mathematicians can like

discover new primes and so forth but but

previously because we have to verify

every single contribution um like most

mathematical research projects it would

not help to have input from the general

public. In fact, it would it would just

be be timeconuming because just error

checking and everything. Um but you know

one thing about these formalization

projects is that they are bringing

together more bringing in more people.

So I'm sure there are high school

students who've already contributed to

some of these these formalizing projects

who contributed into math liib. Um you

know you don't need to be a PhD holder

to just work on one atomic thing.

There's something about the

formalization here that also at as a

very first step opens it up to the

programming community too. The people

who are already comfortable with

programming. It seems like programming

is somehow maybe just the feeling but it

feels more accessible to folks than

math. Math is seen as this like extreme

especially modern mathematics seen as

this extremely difficult to enter area

and programming is not. So that could be

just an entry point. you can execute

code and you can get results. You know,

you can print a hello world pretty

quickly. Um, you know, like if uh if

programming was taught as almost

entirely theoretical subject where you

just taught the the computer science,

the theory of functions and and and

routines and so forth and and outside of

some some very specialized homework

assignments, you're not actually program

like on the weekend for fun. Yeah. Or

Yeah. They would be as considered as

hard as math. Mhm. Um Yeah. Yeah. So, as

I said, you know, there are communities

of non- mathematicians where they're

deploying math for some very specific

purpose, you know, like like optimizing

their poker game and and for them then

math becomes fun for them. Uh what

advice would you give in general to

young people how to pick a career, how

to find themselves like that's a tough

tough tough question. Yeah. So um

there's a lot less certainty now in the

world you know I mean I there was this

period after the war where uh at least

in the west you know if you came from a

good demographic you uh you know like

you there was a very stable path to to a

good career you go to college you get an

education you pick one profession and

you stick to it becoming much more a

thing of the past so I think you just

have to be adaptable and flexible I

think people have to get skills that are

transferable you know like like learning

one specific programming language or one

specific subject of mathematics or

something. It's it's it's that itself is

not a super transferable skill but sort

of knowing how to um reason with with

abstract concepts or how to problem

solve when things go wrong. So these are

things which I think we will still need

even as our tools get get better and you

know you you would be working with AI

sport and so forth. But actually you're

an interesting case study. I mean you're

like a

one of the great living mathematicians

right and then you had a way of doing

things and then all of a sudden you

start learning I mean first of all you

kept learning new fields but you learn

lean that's not that's a non-trivial

thing to learn like that's a that's a

for a lot of people that's an extremely

uncomfortable leap to take right yeah

mathematicians um first of all I've

always been interested in new ways to do

mathematics I I I feel like a lot of the

ways we do things right now are

inefficient. Um I I I I spend me my

colleagues, we spend a lot of time doing

very routine computations or doing

things that other mathematicians would

instantly know how to do and we don't

know how to do them. Uh and why can't we

search and get a quick response and so

that's why I've always been interested

in exploring new workflows.

About four or five years ago, I was on a

committee where we had to ask for ideas

for interesting workshops to run at a

math institute. And at the time, Peter

Schulzer had just formalized one of his

his um new theorems. And um there are

some other developments in computer

assisted proof that look quite

interesting. And I said, "Oh, we should

we should uh um we should run a workshop

on this. This be a good idea." Um and

then I was a bit too enthusiastic about

this idea. So I I got volunte.

Um, so I did with a bunch of other

people, Kevin Bisard and Jordan

Ellenburg and and a bunch of other

people. Um, and it was it was a a nice

success. We brought together a bunch of

mathematicians and computer scientists

and other people and and we got up to

speed and state um and it was really

interesting um developments that that

most mathematicians didn't know was

going on. Um that lots of nice proofs of

concept, you know, just sort of hints of

of what was going to happen. this was

just before chat GBD but there was even

then there was one talk about language

models and the potential um capability

of those in the future. So that got me

excited about the subject. So I started

giving talks um about this is something

we should more of us should start

looking at um now that I' arranged to

run this conference and then chat GPT

came out and like suddenly AI was

everywhere and so uh I got interviewed a

lot um about about this topic um and in

particular um the interaction between AI

and formal proof assistance and I said

yeah they should be combined this this

is this is um this perfect synergy to

happen here and at some point I realized

that I have to actually do not just talk

the talk but walk the book you know like

you know I don't work in machine

learning I and I don't work in proof

formalization and there's a limit to how

much I can just rely on authority and

saying you know I I'm a I'm a warn

mathematician just trust me you know

when I say that this is going to change

athletics and I'm not doing it any when

I don't do any of it myself so I felt

like I had to actually uh uh justify it

yeah a lot of what I get into actually I

don't quite see in advance as how much

time I'm going to spend on it and it's

only after I'm sort of waste deep in in

in in in a project that I I I realized

by that point I'm committed. Well,

that's deeply admirable that you're

willing to go into the fray be in some

small way a beginner, right? Or have

some of the sort of challenges that a

beginner would, right?

new concepts, new ways of thinking also,

you know, sucking at a thing that others

I think I think in that talk you could

be a fields med metal winning

mathematician and undergrad knows

something better than you. Yeah. Um I

think mathematics inherently I mean

mathematics is so huge these days that

nobody knows all of modern mathematics.

Um and inevitably we make mistakes and

um you know uh you can't cover up your

mistakes with just sort of bravado and

and uh I mean because people will ask

for your proofs and if you don't have

the proofs you don't have the proofs. Um

I don't love math. Yeah. So it does keep

us honest. I mean not not I mean you can

still it's not a perfect uh panacea but

I think uh we do have more of a culture

of admitting error than because we're

forced to all the time. Big ridiculous

question. I'm sorry for it once again.

Who is the greatest mathematician of all

time? Maybe one who's no longer with us.

Uh who are the candidates? Zyler, Gaus,

Newton, Raman, Hilbert. So, first of

all, as as mentioned before, like

there's there's some time dependent

on the day. Yeah. Like like if if you if

you if you plot cumulatively over time,

for example, Uklid like like sort of

like is is one of the leading

contenders. Um and then maybe some

unnamed anonymous mathematicians before

that um you know whoever came up with

the concept of of numbers you know you

know um do mathematicians today still

feel the impact of Hilbert just oh yeah

directly of everything that's happened

in the 20th century yeah Hilbert spaces

we have lots of things that are named

after him of course just the arrangement

of mathematics and just the introduction

of certain concepts I mean 23 problems

have been extremely influential

there's some strange power to the

declaring ing which problems are hard to

solve. The statement of the open

problems. Yeah. I mean this is bystander

effect in everywhere. Like if no one

says you should do X, everyone just

moves around waiting for somebody else

to to uh to do something and and like

nothing gets done. Um so and and like it

like it's one one thing that actually uh

you have to teach undergraduates in

mathematics is that you should always

try something. So um you see a lot of

paralysis um in an undergraduate trying

a math problem if they recognize that

there's a certain technique that that

can be applied they will try it but

there are problems for which they see

none of their standard techniques

obviously applies and the common

reaction is then just paralysis I don't

know what to do or um or I think there's

a quote from the Simpsons I've tried

nothing and I'm all out of ideas um so

you know like the next step then is to

try anything like no matter how stupid

um and in fact almost as stupid of the

better um which you know and one a

technique which is almost guaranteed to

fail but the way it fails is going to be

instructive um like it fails because you

you you're not at all taking into

account this hypothesis oh this

hypothesis must be useful that's a clue

I I think you also suggested somewhere

this this fascinating approach which

really stuck with me I started using it

and really works I think you said it's

called structured procrastination no yes

it's when you really don't want to do a

thing. Do you imagine a thing you don't

want to do more? Yes. That's worse than

that. And then in that way, you

procrastinate by not doing the thing

that's worse. Yeah. Yeah. It's a nice

It's a nice hack. It actually works.

Yeah. Yeah. This um I mean with anything

like you know I mean like you um

psychology is really important like you

you talk to athletes like marathon

runners and so forth and and they talk

about what's the most important thing is

it their training regimen or the diet

and so forth. Actually so much of it is

actually psychology. Um you know just

tricking yourself to to think that the

problem is feasible um so that you can

you're motivated to do it. Is there

something our human mind will never be

able to comprehend?

Well I sort of as a mathematician I mean

you

there must be some suffer that you can't

understand. That was the first thing

that came to mind. So that but even

broadly is there are we li is there

something about our mind that's we're

going to be limited even with the help

of mathematics well okay I mean like how

much augmentation are you willing like

like for example if if I didn't even

have pen and paper um like if I had no

technology whatsoever okay so I'm not

allowed blackboard pen and paper right

you're already much more limited than

you would be incredibly limited even

language the English language is a

technology

It's a It's one that's been very

internalized. So, you're right. There

really the the the formulation of the

problem is incorrect because there

really is no longer a just a solo human.

We're already augmented in extremely

complicated intricate ways, right? Yeah.

Yeah. We're already like a collective

intelligence. Yes. Yeah. Yes. So,

humanity plural has much more

intelligence in principle on it good

days than than the individual humans put

together. It can also have less. Okay.

But uh um yeah, so yeah, mathemat

mathematical community plural is is is

incredibly super intelligent uh entity

um that uh no single human mathematician

can can come closer to to replicating.

You see it a little bit on these like

question analysis sites. Um so this math

overflow which is the math version of

stack overflow and like sometimes you

get like this very quick responses to

very difficult questions from the

community. Um, and it's it's it's a

pleasure to watch actually as a as an

expert. I'm a fan spectator of that uh

of that site, just seeing the brilliance

of the different people, the um the

depth of knowledge that people have and

the the willingness to engage in the in

the rigor and the nuance of the

particular question. It's pretty cool to

watch. It's fun. It's almost like just

fun to watch. Uh what gives you hope

about this whole thing we have going on,

human civilization? I think uh yeah. Um

the younger generation is always like

like really creative and enthusiastic

and and inventive. Um it's a pleasure

working with with with uh with uh with

young students. Um

you know the uh the progress of science

tells us that the problems that used to

be really difficult can become extremely

you know can become like trivial to

solve. you know, I mean, like it was

like navigation, you know, just just

knowing where you were on the planet was

this horrendous problem. People died um

you know, or or lost fortunes because

they couldn't navigate, you know, and we

have devices in our pockets that do this

automatically for us, I guess, a

completely solved problem, you know. So

things that are seem unfeasible for us

now could be maybe just sort of homework

exercises for

Yeah. But one of the things I find

really sad about the finitness of life

is that I won't get to see all the cool

things we create as a civilization. You

know that cuz in the next 100 years, 200

years, just imagine showing showing up

in 200 years. Yeah. Well, already plenty

has happened, you know, like if if you

could go back in time and and talk to

your teenage self or something, you know

what I mean? Yeah. and just the internet

and and our AI. I mean again they

they've been in they're beginning to be

internalized and say yeah of course an

AI can understand our voice and and give

reasonable you know slightly incorrect

answers to to any question but yeah this

was mind-blowing even 2 years ago and in

the moment it's hilarious to watch on

the internet and so on the the drama uh

people take everything for granted very

quickly and then they we humans seem to

entertain ourselves with drama out of

anything that's created somebody needs

to take one opinion another person needs

to take an opposite opinion, argue with

each other about it. But when you look

at the arc of things, I mean just even

in progress of robotics. Yeah. Just to

take a step back and be like, "Wow, this

is beautiful that we humans are able to

create this." Yeah. When the

infrastructure and the culture is is

healthy, you know, the community of

humans can be so much more intelligent

and mature and and and rational than the

individuals within it. Well, one place I

can always count on rationality is the

comment section of your blog, which I'm

a big fan of. There's a lot of really

smart people there. And thank you, of

course, for uh for putting those ideas

out on the blog, and it's I can't tell

you how

uh honored I am that you would spend

your time with me today. I was looking

forward this for a long time, Terry. I'm

a huge fan. Um you inspire me. You

inspire millions of people. Thank you so

much for talking. Oh, thank you. It was

a pleasure.

Thanks for listening to this

conversation with Terrence Tao. To

support this podcast, please check out

our sponsors in the description or at

lexfreedman.com/sponsors.

And now, let me leave you with some

words from Galileo Galile.

Mathematics is a language with which God

has written the universe.

Thank you for listening and hope to see

you next time.