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 And now,

dear friends, here's Terren

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

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

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 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 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

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

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

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.