In fall 2024 I am teaching Math 278 (Topics in Analysis). This semester we will talk about Fourier restriction type problems.
Meeting time and place: T/Th 3:30PM - 5:00PM, 41 Evans Hall
Office Hours: 11:00a-12:00p T, 1:00-3:00p Th
Our three problem sets will be posted here.
Problem Set 2 [Extended due 11/19/24]
I plan to talk about recent developments of the subject of Fourier restriction type problems and emphasize their connections to incidence geometry problems. I expect myself to cover topics similar to a past course (Math 828) I taught at UW-Madison.
Tentative topics include:
(1) Fourier restriction in dimension 3 and the polynomial method,
(2) the Bourgain-Demeter decoupling theorem,
(3) Local smoothing for the wave equation in 2+1 dimensions.
Some useful references:
Stein's book Harmonic Analysis has classical theory in oscillatory integrals in Chapters 8 and 9. A proof of the Stein-Tomas theorem can be found in Chapter 9.
Terence Tao's survey paper Recent progress on the Restriction conjecture has a good introduction to important benchmarks (local restriction, the idea of the wave packet decomposition, parabolic rescaling), some history of the Fourier restriction problem and some connections with PDE.
For Fourier restriction we will focus on the dimension 3 result in the paper "A restriction estimate using polynomial partitioning" by Guth. Among other things it has a very nice and precise statement of the wave packet decomposition (Proposition 2.6).
Larry Guth has a webpage on his course on decoupling that contains very high quality and detailed lecture notes about decoupling for the paraboloid (and the moment curve that we do not plan to cover).
Our proof of the multilinear Kakeya theorem follows the proof of Guth in his paper "A short proof of the multilinear Kakeya inequality".
This is a short note on Guth's webpage showing an "essentially correct" way to obtain multilinear restriction from multilinear Kakeya.
This course by Guth is a good reference on polynomial method (mainly) in incidence geometry.
A recent book Fourier Restriction, Decoupling, and Applications by Ciprian Demeter touches a lot of topics and techniques we have covered/will cover throughout the semester. It is a great reference for our first two topics and part of the third.
Terence Tao taught a course "Classical Fourier Analysis" on Restriction theory/Strichartz estimates/Decoupling estimates/Time-frequency analysis/Paraproducts/Carleson's theorem. This course is related to some of our topics.