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

- The following is a conversation with Terence Tao,

widely considered to be one

of the greatest mathematicians in history,

often referred to as the Mozart of math.

He won the Fields Medal

and the Breakthrough Prize in mathematics,

and has contributed groundbreaking work

to a truly astonishing range of fields

in mathematics and physics.

This was a huge honor for me for many reasons,

including the humility and kindness that Terry showed

to me throughout all our interactions.

It means the world.

This is the Lex Fridman podcast.

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

What was the first

really difficult research level math problem

that you encountered?

One that gives you pause, maybe?

- Well, I mean,

in your undergraduate education you learn

about the really hard,

impossible problems like the Riemann Hypothesis,

the Twin-Primes Conjecture.

You can make problems arbitrarily difficult.

That's not really a problem.

In fact, there's even problems that we know

to be unsolvable.

What's really interesting are the problems just

on the boundary between what we can do relatively easily

and what are hopeless.

But what are problems

where existing techniques can do like 90% of the job,

and then you just need that remaining 10%?

I think as a PhD student,

the Kakeya problem certainly caught my eye

and it just got solved, actually.

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

in my early research.

Historically, it came from a little puzzle

by the Japanese mathematician Soichi Kakeya in 1918 or so.

So the puzzle is that you have a needle on the plane,

or think like driving on a road or something,

and you wanted to execute a U-turn.

You want to turn the needle around,

but you want to do it in as little space as possible.

So you want to use this little area

in order to turn it around,

but the needle is infinitely maneuverable.

So you can imagine just spinning it around as a unit needle.

You can spin it around its center

and I think that gives you a disk of area,

I think pi over four.

Or you can do a three-point U-turn,

which is what we teach people

in their driving schools to do.

And that actually takes area pi over eight.

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

And so for a while,

people thought that was the most efficient way

to turn things around.

But Besicovitch showed that

in fact you could actually turn the needle

around using as little area as you wanted.

So 0.001.

There was some really fancy multi back and forth

U-turn thing

that you could do that you could turn 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.

And yeah, so we understand everything in two dimensions.

So the next question is, what happens in three dimensions?

So suppose like the Hubble Space Telescope

is a tube in space

and you want to observe every single star in the universe,

so you want to rotate the telescope

to reach every single direction.

And here's the unrealistic part.

Suppose that space is at a premium, which it totally is not.

You want to occupy

as little volume as possible in order to rotate your needle

around in order to see every single star in the sky.

How small a volume do you need to do that?

And so you can modify Besicovitch's construction.

And so if your telescope has zero thickness,

then you can use as little volume as you need.

That's a simple modification

of the two dimensional construction.

But the question is that

if your telescope is not zero thickness, but just very,

very thin, some thickness delta,

what is the minimum volume needed

to be able to see every single direction

as a function of delta?

So as delta gets smaller, as your needle gets thinner,

the volume should go down.

But how fast does it go down?

And the conjecture was that it goes down very, very slowly,

like logarithmically, roughly speaking.

And that was proved after a lot of work.

So this seems like a puzzle.

Why is it interesting?

So it turns out to be surprisingly connected to a lot

of problems in partial differential equations,

in number theory, in geometry, combinatorics.

For example, in wave propagation,

you splash some water around,

you create water waves

and they travel in various directions.

But waves exhibit both particle and wave-type behavior.

So you can have what's called a wave packet,

which is like a very localized wave that is localized

in space and moving at a certain direction in time.

And so if you plot it in both space and time,

it occupies a region which looks like a tube.

And so what can happen is that you can have a wave

which initially is very dispersed,

but it all focuses at a single point later in time.

Like you can imagine dropping a pebble into a pond

and ripples spread out.

But then if you time reverse that scenario

and the equations of wave motion are time reversible.

You can imagine ripples that are converging

to a single point, and then a big splash occurs,

maybe even a singularity.

And so it's possible to do that.

And geometrically what's going on is that there's always

sort of light rays.

So like if this wave represents light, for example,

you can imagine this wave

as the superposition of photons all traveling

at the speed of light.

They all travel on these light rays

and they're all focusing at this one point.

So you can have a very dispersed wave focus

into a very concentrated wave at one point in space

and time, but then it defocuses again and it separates.

But potentially, if the conjecture had a negative solution,

so what that meant is that there's a very efficient way

to pack tubes pointing in different directions into a very,

very narrow region of very narrow volume,

then you would also be able to create waves that start out,

there'll be some arrangement of waves

that start out very, very dispersed,

but they would concentrate not just at a single point,

but there'll be a large,

there'll be a lot of concentrations in space and time.

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.

And so in mathematical physics,

we care a lot about whether certain equations

in wave equations are stable or not,

whether they can create these singularities.

There's a famous unsolved problem

called the Navier-Stokes regularity problem.

So the Navier-Stokes equations that govern a fluid flow

or incompressible fluids like water.

The question asks,

if you start with a smooth velocity field of water,

can it ever concentrate

so much that the velocity becomes infinite at some point?

That's called a singularity.

We don't see that in real life.

If you splash around water in the bathtub,

it won't explode on you,

or have water leaving at the speed of light.

But potentially it is possible.

And in fact, in recent years,

the consensus has drifted towards the belief that in fact,

for certain very special initial configurations of, say,

water, that singularities can form,

but people have not

yet been able to actually establish this.

The Clay Foundation

has these seven Millennium Prize Problems,

has a million dollar prize

for solving one of these problems.

This is one of them.

Of these seven, only one of them has been solved.

The Poincare conjecture by Perelman.

So the Kakeya conjecture is not directly, directly related

to the Navier-Stokes problem,

but understanding it would help us understand some aspects

of things like wave concentration,

which would indirectly probably help us understand

the Navier-Stokes problem better.

- Can you speak to the Navier-Stokes?

So the existence and smoothness, like you said,

Millennium Prize problem.

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

In 2016, you published a paper, "Finite Time Blowup

for an Averaged Three-Dimensional Navier-Stokes Equation."

- [Terence] Right.