Haha, all right. The work of Terence Tao. Now, it's a great pleasure and a daunting task to present the work of Terence Tao. Terry has worked on such an incredible variety of problems that I think no

one in the world except Terry himself is really competent to do the presentation. And furthermore, I have 20 minutes and I feel the imperative to snow no one. So I have the following idea as to how to proceed. I'm going to give you a

few selected vignettes and leave out what many may consider to be some of the main contributions. But I want to show you enough to produce all of the following reactions. OK, so math at its highest level can make us think, what amazing technical power. Look on my work, see mighty in despair.

That actually, I think, is the cheapest form of mathematics at its highest level. But then one can think, what a grand synthesis. One can think, this is obvious. How can anyone not have seen this before? And perhaps the rarest of all is mathematics from Mars. Where on earth did this come from?

And my goal in this talk is to produce enough of Terry's work so that we will all feel all of these reactions. Now, by snowing no one, I will perhaps show you a little bit less of what amazing technical power.

But I think you can take my word for it at many points. All right. Here are the three topics that I picked out in order to present Terry's work. These are three that I felt together produced all of the reactions.

The Kakeya needle problem, nonlinear Schrodinger equations and arithmetic progressions of primes. Let's start with the Kakeya needle problem. It's a problem very dear to my own heart. The classic version, which I think goes back to 1917, deals with regions in the plane.

Imagine you have a region in the plane and a needle of length one inside the region. And imagine that you can turn the needle through a full 360 degrees, keeping it entirely inside E at all times. Then how small can we take the area of E? And despite the initial conjectures, the fact is that the area can be taken arbitrarily small.

That was a surprise. All right. Now, the modern version of the problem is to look at a set E in higher dimensions, Rn, and assume that E contains a line segment of length one in any prescribed direction. But of course, once you pick the direction, you will have to make a clever choice of the translate.

Given such a set, which is called a Besicovitch set, what can we say about the fractal dimension of E? Now, there are many definitions of the fractal dimension, and I'm going to concentrate here on the simplest, which is the Minkowski dimension. Simply, you take the set and you cover it by balls of a fixed small radius, delta,

and you ask which power of delta it takes to cover by small balls. And if it takes delta to the minus beta power balls, then roughly then you say that the Minkowski dimension is beta. Okay. All right. Enough about that.

Now, the Kakeya problem is an important problem, and I don't have time to explain why, but you can take my word for it. It's related to Fourier analysis, PDE, combinatorics, and other things. It's a central, interesting problem, but it's very hard. Now, what does a theorem look like regarding the Kakeya problem?

Well, it says that any Besicovitch set in Rn must have Minkowski dimension at least beta of n. Beta of n is obviously a function of n. It's also a function of time, depending on what one knows how to prove. We try for the best beta of n. It's plausible to guess that beta of n should be n.

All right. Now, from the 1970s and 80s, it was known that the best fractal dimension of a Besicovitch set in dimension two was two. That's a result of Davies, and it's related to the early work of Antonio Cordova. Let's see. For higher dimensions, the classic result is due to Drury.

The dimension is at least n plus one over two, and that's related to the work of Chris, Doug Kakeya, and Rubio de la Francia. All right. That's the way things stood once upon a time. But then in the 90s came two breakthroughs.

One started by Bourguin and then continued by Tom Wolf, and another again by Bourguin. The Kakeya problem was related to geometry and then to combinatorics. In terms of geometry, Besicovitch sets of small fractal dimensions, if they exist, have geometric structure which one can analyze.

They contain bouquets and hairbrushes and God knows what. They are sticky. Never mind. The Kakeya problem, as Bourguin realized, is related to work done by Gowers,

related to the earlier work of Balog and Zamoretti, and this is on the way to Gowers' proof of the Zamoretti theorem. And I want to emphasize the fact that the subject grew deep and forbidding. If you want to work on this subject, you had better work very hard and be very smart and be prepared to suffer.

All right. Now, here in particular is the connection with combinatorics. It's in the following fact about abelian groups. Let's see how I'm doing for time. Spent five minutes? Yes. Okay. So imagine that we have two subsets of an abelian group, and we suppose that both of them

Oh, I have a pointer. Look at that. I won't use the pointer. Let's see. But notice that both of these sets, A and B, have at most n points.

Therefore, if you add them together, if you take all possible sums, you may expect to get as many as n squared terms. But perhaps A and B have some special structure so that there may be far fewer. Well, we imagine we have a set of pairs, G, a set of pairs in A cross B,

and if we look at the sums of A and B where the pairs A and B belong to G, this is also at most n, so that there are far, far fewer sums than you might expect. If this is true, then the theorem says that there are also fewer differences than there might be.

Again, there might be as many as n squared differences because A and B have n points, but in fact, the number is bounded by a power of n strictly less than two. You gain 1 thirteenth. So the point is to do better than the trivial n squared. This combinatorial fact immediately leads to information about Besicovitch sets

because you simply slice the Besicovitch set by two typical parallel hyperplanes, and A and B are roughly the intersection of the Besicovitch set with each of the two hyperplanes, and G is the set of all the lines in the Besicovitch set. All right, so that was deep, hard, and forbidding.

And now in 1999, we have in place of the deep theorem, we have a little lemma by Netz, Katz, and Terence Tao. We have the same assumptions as the deep theorem, but instead of gaining a power of n to the 1 thirteenth over the trivial estimate, we now gain a power of n to the 1 sixth.

And relating that and a slightly fancier version of the little lemma, one gets an inequality for the fractal dimension of a Besicovitch set, which at the time was the best known. All right, that was the sharpest result known.

The point that I want to make is that the little lemma is strictly better than the deep theorem, and yet the proof avoids somehow all difficulties. You could read this proof in an hour.

It consists of, I would say, of two pieces, and each piece has the flavor of a nice solution to an Olympiad problem. Where on earth did that come from? It couldn't be simpler. And yet after you've read this, it's not at all clear how the hell they thought of it. All right, well, progress continues.

And in particular, I want to mention the Tour de Force by Netz, Katz, Isabella Laba, and Tao. And the subject is still deep and forbidding, and if you want to work in it, you have to be prepared to suffer intensely, and unfortunately it seems that a complete solution is far away. But nevertheless, we have an example of mathematics from Mars,

and so although I don't know how Terry personally feels about this particular result among his many, it's among my favorites. All right, now let's look at the next subject, PDE. I want to talk about interaction Morowitz estimates. Interaction Morowitz estimates come from the I-team, I presumably for interaction.

Here's the I-team, okay? Now, this is one ingredient in a program to understand certain non-linear PDEs

and understand very deep stuff. I'm going to suppress the very, very difficult technical problems and conceptual problems, and I'm simply going to focus in on exactly the interaction Morowitz estimate itself and what that might hope to do for you.

Okay, so let's start with the non-linear Schrodinger equation. There it is. P is a parameter. Plus or minus is an important parameter, but once you pick the sign, plus or minus, and the power P, then we have a particular PDE with an initial condition, U is a complex valued function of space,

so X is in R3, T is in time. Now, the behavior of the equation is very crucially influenced by the sign. The minus sign causes it to be focusing, and at least for certain P, one can expect that a singularity will form, and the problem is then to understand, for instance, what a singularity looks like.

On the other hand, if there was a plus sign, then the equation is defocusing, and then that means that in any bounded region of space,

U grows small. It somehow diffuses out to infinity, and therefore a power of U greater than the first power will be negligibly small, and so we expect that the non-linear term will become less and less important, and U will resemble the solution of a free Schrodinger equation, and so that is scattering.

Okay, so that's what one expects. Now, how can you actually prove that any of this actually happens? It's all very nice to say that you expect this or that, but how does it actually happen? Before we get to that, let me just point out two obvious conserved quantities, the L2 norm, which is the mass,

and this combination of the gradient of U squared and an integral of a power of U, that's the energy. Notice that when P is 5, that's precisely critical in that the Sobolev inequality controls the potential energy in terms of the kinetic energy. The fact that the equation is defocusing means that the energy has a plus sign here.

We can't see here. The kinetic and potential terms are connected by a plus sign rather than a minus sign, so they're both under control thanks to the energy. Okay, right, so here's how we start to prove that in the defocusing case, that the solution spreads out.

This is the Morovitz estimate, a beautiful, simple, wonderful idea of Kathleen Morovitz, originally introduced by her for a different equation, but in the context of this equation, let's take defocusing, nonlinear Schrodinger, and let's take P equals 3 to start with. And here you can see up on the screen

what the Morovitz estimate says. And the point is that the right-hand side is controlled by the mass and the energy, whereas the left-hand side will blow up if, let's say, U continues to be about as big as 1 throughout the unit ball in space for all time. Okay, so the Morovitz estimate, as it stands,

rules out the idea that U should stay about as big as 1 for X in the unit ball for all time. That's good. That starts to tell us that the equation defocuses. But unfortunately, it doesn't rule out the idea that the solution should remain about as big as 1 in a unit ball centered at a moving point.

This equation has infinite propagation speed, and therefore you don't really know where the moving point might be, and so that's the weakness in the Morovitz estimate. Now, let's see. I have used up 10 minutes. All right. I'm going to accelerate. So if we take the proof of the Morovitz estimate,

then it's based on looking at the right quantity. It's this M0, and you check that it's bounded by the right-hand side of the Morovitz estimate. That's sort of trivial. And then you do a computation in which you plug in the nonlinear Schrodinger equation, and you discover the left-hand side of the Morovitz estimate

before it's integrated in time, plus some positive stuff. And so therefore, the Morovitz estimate follows trivially from that. Now, this is all very nice, except that the quantity M0, well, the quantity M0 gives a special role to the origin.

And maybe the origin isn't where U lives. Maybe U lives far away from the problem. And so, I'm sorry, U lives far away from the origin, and so maybe M0 of T is irrelevant. And so the solution is just so simple and so natural that this is an example, I claim, of why didn't people think of it before?

It's completely obvious. Well, it's completely obvious once you read it. So we move the quantity M0 to where U lives. So if you look at the middle line on the screen, you have in place of M0, you have My, with an arbitrary point Y playing the role of the origin, and that's the analog of the M0,

with the origin translated to Y. And then the quantity that you start with in trying to derive a Morovitz estimate is a weighted average of those M of Y, where the weight is U squared. So wherever U is big, wherever U lives, that's where you put the quantity M. What could be more sensible and natural than that?

And so you obtain the interaction Morovitz estimate, which you see there before you. The left-hand side now no longer picks out the origin as a special point, and the right-hand side is still controlled by the mass and energy. And that's all it takes to rule out U being

of the order of one in a moving ball. Now, to pass from this to results on PDE is very hard, but nevertheless, this is at the heart of the results of the I-team on PDEs. So, let's see. All right, what does one do with this estimate?

Very quickly, for P equals three, look at the interaction Morovitz estimate, and notice that you're taking the Laplacian to the one-fourth power. That means that you need one-half a derivative, whereas the energy involves a whole derivative. So, therefore, you don't need finite energy in order to get control over the equation.

It's enough to assume that you're in a Sobolev space short of the energy space for P equals three. Now, that's a big, big, long, hard story. That's the first species of mathematics,

of first-rate mathematics. My God, it's hard. I'm suppressing here all the ideas except the interaction Morovitz estimates, but in principle, that's an application. But I want to come now to quintic defocusing NLSP equals five. Remember, that's the critical case. So, that's a particularly important and hard equation

because it is critical for the energy. Short-time solutions are easily seen to exist for finite energy. Global solutions exist for small energy. I remind you that the energy is what's written there. And the challenge is to prove global existence for large, finite energy. So, now this is a tremendously difficult thing.

I'm going to pick out only one small, I'm sorry, one small but very important aspect, one small crucial aspect of the challenge, which is this. So, Bourguin, in a great tour de force, succeeded in improving the well-posedness of this equation

in the radial case. And that is big, hard, and forbidding. I promise you. But the general case is yet much harder. And one way you can see that is that in the radial case, singularities can form only at the origin, whereas in the general case, singularities can form anywhere. So, the I-team proved,

well, let's not go through exactly the statement of the theorem. Time, oh dear, am I out of time? A few more minutes? Okay, thanks. All right, so here, for finite energy initial data,

the quintic defocusing NLS has global solutions provided that you're in the Sobolev space where the energy is finite, and furthermore, there exist solutions of the free Schrodinger equation, which mimic, which reflect very well the behavior as T goes to plus or minus infinity. Fine. All right. And so we use the interaction,

they use the interaction more of its estimates with cutoffs that make life much harder. Many ideas from Bourguin, in particular something called induction on the energy, and a whole lot of ideas that I can't describe here. All right. Let's say something about arithmetic progressions of primes very quickly. Let's see.

First of all, so it's perhaps a little silly that you're getting a sound bite from me. We'll hear from the man himself tomorrow, but here goes the little sound bite attempt very quickly. Given K, let's take K equals 173, you can find an arithmetic progression, in fact, arbitrarily many arithmetic progressions

of length 173 among the primes. And in fact, if you look at the primes up to N for sufficiently large N, there are roughly as many such arithmetic progressions as there ought to be. So here's the theorem. Okay. Now, how on earth do you prove that? Well, to begin with, one starts with the great theorem of Samoretti,

which says that if you have, let's say, the numbers from one to N, and N is sufficiently enormous, and you pick out a subset of, let's say, one one-millionth of the numbers from one to N, then they will contain a K term arithmetic progression, and in fact, about as many as you think there ought to be.

So that's Samoretti's theorem. Here's the variant of Samoretti's theorem for functions, immediately equivalent to the version for sets. Notice the form of the theorem. I don't have time to say more, but we have a function. It's between zero and one. It's defined on the numbers from one through N for huge N,

and yet its average is bounded below by, let's say, one one-millionth. And then we can control from below an average of a product of translates of the function, as you see. Okay? So that's Samoretti's theorem for functions. Now, there are three, I would say, completely different proofs of Samoretti's theorem.

The original proof by Samoretti, which on the soundbite level I would call combinatorics, the proof by Furstenberg, which on the soundbite level I would call ergodic theory, and the proof by Gowers, which on the soundbite level I would call non-linear Fourier analysis. Now, I say completely different to Green and Tao.

These theorems are not so different, and these proofs are not so different at all, but for the rest of us, that comes as interesting news. All right, so the work of Green and Tao synthesizes all the previous work on Samoretti's theorem in the sense that it quotes from Samoretti's theorem itself, and it uses ideas from the proofs of Furstenberg and Gowers.

Am I now officially out of time? Okay, thank you. All right, all right. So the idea on the soundbite level is that we look at a function which is essentially log of x if x is a prime and zero otherwise.

Now, by the prime number theorem, that has average comparable to one. I multiplied by a half so that the average would be a half, okay? And so if we could apply the Samoretti theorem, then it would say that the average of translates of this function is bounded below by some constant, and that would give the number of arithmetic progressions. But oops, Samoretti's theorem doesn't apply

because this function f is not bounded. That's the whole point. The primes don't have positive density, otherwise the theorem would be a lot easier. But Green and Tao prove an extension of Samoretti's theorem in which the hypothesis that f is bounded is replaced by the idea that f is less than or equal

to some function new of x with some special properties. This is called a pseudo-random measure by Green and Tao. And so here are their defining conditions for a pseudo-random measure. The average of new is one, and there are upper bounds on certain averages of products of never mind what.

Okay, and so here's the Green-Tao version of Samoretti's theorem, which you will not even have time to read, but it is just like Samoretti's theorem except that instead of f being bounded, it's less than or equal to a pseudo-random measure. First discuss the proofs. Haha, no time to discuss the proofs. We take the function and we split it up into two pieces,

an easy piece and a hard piece. We then look at the average of the products. That's a sum of a whole lot of terms in which every time you see an f, you replace it either by the easy part or by the hard part. If you see any easy part, then thanks to Gower's ideas

that go back to Gower's proof of Samoretti's theorem, you can control that, and somehow that's not a problem. This leaves us with a single main term, namely the case in which you take the product of all the hard terms. And in that case what you do is that you partition the numbers from one to n into subsets and replace the hard function

by its average over each of the sets of the partition. You prove that that doesn't make any difference, a significant difference in the final answer, and yet when you've done this averaging, the averaged out function, the smeared out function, satisfies the hypotheses of the classic Samoretti theorem. All right, proving those properties

of the hard part of the function follow from the ideas that go back to first in Berg's proof of the Samoretti theorem and now the Green-Tao-Samoretti theorem follow it once by simply applying the classic Samoretti theorem. And so that's all there is to it. Okay, now how does this apply to the primes? All you have to do is find a new of x which bounds the function that you want

and yet is a pseudo-random measure and such a new comes from the work of Goldston Yildirim for using not so hard analytic number theory. Let me emphasize that not so hard is very different from not so good. It's wonderful and very clever, just not so hard.

It's curious that the analytic number theory that you have to use is not so hard. All right, yes, fine, yes. So that's all I have time for. In fact, I didn't, yes, I went over my time. I apologize. There is much, much more first-rate work. There's a program to study wave maps,

bio-harmonic maps. There's a solution of the saturation conjecture from representation theory joint with Knudsen. This is algebra. It's very, very unusual for an analyst to solve a conjecture in algebra. Okay, and finally, what next? There's somehow, I mean, well, that says it all.

What next? How can, how can, what can possibly follow what he's already done, but we look forward to it with great anticipation. Thank you very much.