ABSTRACT OF "SOME CONJECTURES IN HARMONIC ANALYSIS"
PADOVA, MARCH/APRIL 2023
ANTHONY CARBERY
In these lectures, which are aimed at Masters level or beginning PhD students in mathematical analysis, we shall discuss some of the important conjectures which have underpinned and motivated much of the most exciting work in harmonic analysis over the last 50 years or so.
Rather than aiming for a detailed survey of the latest progress on these conjectures, we shall introduce and motivate them, try to place them in a conceptual context, and most of all, try to convince the audience that they are really interesting to study.
We will focus each of the four lectures on one of these conjectures. The four conjectures we discuss are however, intimately related, and we will see a lot of interactions between them.
The headings under which we shall discuss the four conjectures are as follows:
• The Kakeya conjecture: In its simplest form, the Kakeya conjecture asserts that any compact set in euclidean space R^n which contains a unit line segment in each direction (such as a solid ball) must in fact have full dimension n. This is known in dimension n = 2, but is not known in higher dimensions.
• The Fourier restriction conjecture: The simplest form of this conjecture asserts that for every function f ∈ L^p(R^n), where 1 ≤ p < 2n/(n + 1), its Fourier transform Ff can be meaningfully restricted to the unit sphere S^(n−1). At first glance this question doesn’t even make sense once p > 1, and our first task will be to convince you that it does at least make sense after delving a little deeper.
• The Mizohata–Takeuchi conjecture: This is a sort of dual problem to the Fourier restriction problem, and it seeks to understand in some detail the geometric structure of the level sets arising from a superposition of plane waves with common frequency, but varying directions of propagation. It is especially attractive because it is a purely L^2-based conjecture, meaning that it should be susceptible to Hilbert space methods. But things are not so straightforward...
• Multilinear conjectures: Here we draw together several of the strands previously discussed, and in particular reconsider the Kakeya and restriction conjectures from a multilinear perspective. While this is definitely a simpler class of problems, there are nevertheless still various surprises and interesting open questions.
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, U.K.
Email address: A.Carbery@ed.ac.uk