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Modern Developments in Fourier Analysis

During 2019 -- 2020 semester 2, I am running an advanced graduate course on Fourier analysis.

Time and location: Lectures begin on Tuesday 7th January 2020.

09.15 - 10.15 Tuesdays, 5.02 ICMS Seminar Room (Bayes Centre).
09.15 - 10.15 Fridays, 5.46 Webinar room (Bayes Centre).

Lecture notes:

Lecture 1	Lecture 5	Lecture 9	Lecture 13	Lecture 17	Lecture 21	Lecture 25
Lecture 2	Lecture 6	Lecture 10	Lecture 14	Lecture 18	Lecture 22
Lecture 3	Lecture 7	Lecture 11	Lecture 15	Lecture 19	Lecture 23
Lecture 4	Lecture 8	Lecture 12	Lecture 16	Lecture 20	Lecture 24

References: Here are some references for additional material on topics covered in the course:

Lectures 1 - 3	Fefferman's paper, Tao's lecture notes (see Lecture notes 3)
Lecture 4	Wolff's survey, Tao's lecture notes (see Lecture notes 5)
Lectures 5-6	Cordoba's Bochner-Riesz argument, Tao's lecture notes (see Lecture notes 3), Gafakos (results on distributions).
Lecture 7	Tao's paper (Kakeya vs Nikodym), Kroc's masters' thesis (Kakeya vs Nikodym).
Lectures 8-9	Bougain-Guth paper, Tao's survey (bilinear methods).
Lecture 10	Sogge's book: Chapter 6 covers fix-time estimates. Seeger-Sogge-Stein.
Lectures 11-12	Square functions vs local smoothing: Mockenhaupt-Seeger-Sogge, Tao-Vargas. Cordoba's square function.
Lectures 13-18	Guth-Wang-Zhang, Lee-Vargas.
Lectures 19	Vaughan's book, Pierce's survey.
Lectures 20-	Bourgain-Demeter-Guth, Guo-Li-Yung-Zorin-Kranich.

Overview: Recently, there have been a number of remarkable developments in euclidean harmonic analysis, related to the Fourier restriction conjecture. Broadly speaking, one is interested in studying functions

$f$ whose Fourier transform is supported in a neighbourhood of a submanifold of

$\mathbb{R}^n$ , such as a paraboloid or a cone or a sphere. Such situations arise naturally in PDE, as well as in harmonic analysis and analytic number theory.

One of the goals of this course is to understand the so-called decoupling inequalities of Wolff and Bourgain-Demeter. The idea is that whilst

$f$ may be difficult to analyse, it can be broken up as a sum of pieces

$f_{\theta}$ which are much easier to understand (in particular, the pieces

$f_{\theta}$ are localised in frequency to small regions where the submanifold is essentially flat). The key question is then to understand how the various

$f_{\theta}$ interact with one another. In decoupling theory this is achieved via a norm inequality of the form

$\begin{equation*} \|f\|_{L^p(\mathbb{R}^n)} \lessapprox \Big(\sum_{\theta} \|f_{\theta}\|_{L^p(\mathbb{R}^n)}^2 \Big)^{1/2}. \end{equation*}$ The key feature of the above inequality is that an

$\ell^2$ expression appears on the right-hand side, rather than the trivial

$\ell^1$ expression given by the triangle inequality; this crucially takes into account complex destructive interference patterns between the different

$f_{\theta}$ .
Decoupling theory has had a profound impact on a wide range of (ostensibly) distinct areas of mathematical analysis. A large portion of the course will investigate applications.

Possible topics include:

Fourier analysis philosophy and uncertainty principle heuristics.
Multilinear harmonic analysis: the Bennett--Carbery--Tao theorem via induction-on-scale.
The Bourgain--Guth method for estimating oscillatory integral operators.
Proof of the $\ell^2$ -decoupling theorem of Bourgain--Demeter.
Relation to incidence geometry.
Applications of decoupling to PDE: Strichartz estimates on the torus, spectral theory, local smoothing for the wave equation.
Applications of decoupling to harmonic analysis: Bochner--Riesz means, Fourier restriction, $L^p$ -Sobolev and maximal bounds for generalised Radon transforms.
Applications of decoupling to analytic number theory: diophantine equations, the proof of the Vinogradov mean value theorem, Weyl sum bounds, the Lindelöf hypothesis.
Variable coefficient extensions and analysis on manifolds.

A detailed syllabus can be found here.

A list of project titles can be found here.

Contact

E-mail: jonathan.hickman [at] ed [dot] ac [dot] uk

Address:
School of Mathematics,
James Clerk Maxwell Building,
The King's Buildings,
Peter Guthrie Tait Road,
Edinburgh,
EH9 3FD.

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