Some research in analysis

Terence Tao

Abstract: I describe some analysis in my research area which could be a starting point for a graduate thesis.

Possible areas of research:

• The Kakeya problem (geometric combinatorics, arithmetic combinatorics, geometric measure theory, harmonic analysis)

• L^{p} bounds for oscillatory integrals (harmonic analysis, linear PDE; closely related to the Kakeya problem)

• Korteweg de Vries, non-linear Schrödinger, wave maps, and other nonlinear dispersive equations (nonlinear PDE, harmonic analysis, possible use of numerics)

• Multilinear operators, Schrodinger eigenfunctions (harmonic analysis, spectral theory; Thiele works more heavily in this area than I do)

• …

• I’ll only talk about the first two topics today.

The Kakeya problem

• The Kakeya problem is not a single problem, but is really a class of problems, all trying to address the following issue: to what extent can lines in different directions overlap each other?

• The original question of Kakeya was the Kakeya needle problem posed in 1917: Given a needle of unit length and zero thickness, how much area is needed in the plane in order to turn the needle completely around? Besicovitch gave the surprising answer in 1928 that one can turn this needle using arbitrarily small area. (On the other hand, if the needle has non-zero thickness 0 < δ ≪ 1, the area needed is ∼ 1/log(1/δ), which goes to zero only very slowly as δ → 0).

• There is a corresponding problem in three dimensions: given a tube of unit length and thickness δ, how much volume is needed in order that the tube can be oriented in each direction in space? If δ = 0, one can use arbitrarily small area. When δ > 0, the answer is unknown. In 1995 Wolff showed that one needs ≳ δ^{1/2} volume; in 1999 Keich showed that one can use ∼ 1/ log(1/δ). The conjecture is that one

needs ≳ δ^{ε} volume for any fixed 0 < ε ≪ 1.

• There are more fancy variants of this conjecture involving Hausdorff dimension, or a maximal function (similar to the Hardy-Littlewood maximal function), which I won’t detail here. These fancier variants have application to oscillatory integrals (more on this later).

• There is a “toy” version of this problem which is cleaner to state, and takes place in a finite field F. Suppose that E is a subset of the n-dimensional finite field geometry F^{n} which contains a line in each direction. (A line in F^{n} is a finite collection of points, |F | to be exact). What is the least possible cardinality of E? In two dimensions n = 2 this is known exactly (basically, |E| can be as low as \frac{1}{2}|F |^{2} but not much lower). In three and higher dimensions it is conjectured that |E| has to essentially be as large as all of |F |^{n}, but this is not known. Numerous partial results (lower bounds in |E|, mostly) are known.

Methods used:

• Extremal graph theory. One can use geometric facts, such as “every two lines intersect in at most one point”, to limit the number of times lines can overlap. This requires some basic combinatorics (pigeonhole principle, Cauchy-Schwartz, etc.). Also the geometry required can get a little fancy, verging on algebraic geometry (example: in F^{3}, the number of lines which intersect three given lines in general position is O(|F |)).

• Arithmetic combinatorics. Lines are very similar to arithmetic progressions, for instance if x, y belong to a line then so does (x + y)/2. However, the difference x − y is parallel to the direction of the line. Since our lines all point in different directions, we thus are faced with a situation where E − E is large but E + E is small. This leads to arithmetic combinatorics - the combinatorics of sums, differences, and arithmetic progressions of finite sets. These are of course very simple concepts, but the problems here are quite challenging - the Fields medallists Bourgain and Gowers have worked in this area recently, for instance.

• Geometric measure theory. Once you leave the finite field setting and go into Euclidean space, some other technical issues arise regarding Hausdorff dimension, covering lemmas, etc. - in short, a lot of 245AB gets used here.

• Many, many open problems, some more difficult than others. Many of the tools used are new and not fully developed, and we need people to extend and refine them to more general contexts. One nice thing about this problem is that one can focus on one aspect at a time; for instance, one can deal exclusively with the combinatorial aspects, or with the measure theory aspects, etc.

Oscillatory integrals

• I’m interested in understanding oscillatory linear operators which have a form such as

T_{λ}f(x) = ∫ e^{iλϕ(x,y)}a(x, y)f(y) dy

where a is some fairly smooth amplitude function, ϕ is a real smooth phase function, and λ ≫ 1 is a large parameter. A typical example can come from appropriate modification of the Fourier transform

f̂(ξ) = ∫ e^{2πix·ξ}f(ξ) dξ.

• More examples come from solving PDE. For instance the Schrödinger equation

iu_{t}(t, x)+Δu(t, x) = 0 for all (t, x) ∈ R×R^{n}; u(0, x) = f(x)

has the explicit solution

u(t, x) = \frac{1}{(4πt)^{n/2}} ∫ e^{i|x−y|^{2}/4t}f(y) dy.

• The basic question arises: to what extent does the size of f control the size of T f? For the Fourier transform, this is answered by the Plancherel and Hausdorff-Young inequalities. For the Schrödinger equation, the question is (partially) answered by the Strichartz inequalities, which play a major role in the theory of non-linear Schrödinger equations.

• For more general T, the question is to obtain the best possbile estimates of the form

∥T_{λ}f∥_{L^{q}} ≤ Cλ^{−α}∥f∥_{L^{p}}.

• These estimates have many applications to harmonic analysis, PDE, spectral theory, and may possibly have application to number theory. When p = 2 or q = 2 the theory is fairly well understood, but the general case is not. Model cases include some long-standing conjectures in harmonic analysis, notably the restriction conjecture, the Bochner-Riesz conjecture, and the local smoothing conjecture.

• Interestingly, this problem is closely related to the Kakeya problem. This comes from a “wave packet decomposition” of f, which decomposes f as a superposition of oscillating waves supported on tubes. (From a Schrödinger point of view, we are trying to isolate the particle trajectories inside a quantum wave function). To control T_{λ}f one then needs to control the overlap of these tubes, which leads to Kakeya issues.

• However, knowing the Kakeya problem is not enough; one also needs to understand how various oscillatory functions superimpose on each other. This requires harmonic analysis techniques (notably, use of the Fourier transform, uncertainty principle, and orthogonality). A little bit of symplectic geometry also comes in, in order to handle both space and frequency simultaneously.

• I’m mostly interested in clarifying the interconnection between the oscillatory integral problems and the Kakeya-type problems, but there are many, many interesting and open problems in oscillatory integrable problems which can be looked at.

Skills

• Level of mathematical background is not so important to me; one nice thing about analysis is that one can already get into some interesting unsolved problems without needing too much formalism. But being able to pick up ideas quickly is a plus. So is maintaining a broad interest in analysis, and mathematics as a whole; one can get quite far by narrowly focusing on one particular sub-field, but often the breakthrough comes from keeping an ear open on developments in nearby fields (or sometimes fields not so nearby). Please pester me with crazy ideas as to how to solve a problem!

• The nature of these problems can change over time. A problem can be too difficult, and we move to a simpler model problem; or it can be too easy, and we look for generalizations. Also it may turn out that a technique we thought would be effective is not, and so we have to try something rather different. Expect some early disappointments and some unexpected successes; the problems we’ll look at will not just be routine applications of known techniques (at least, not completely).

• I can travel for periods of up to a month, though usually I’m reachable by e-mail. So a certain amount of self-sustainability is desirable.