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:
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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