In spring 2020 I am teaching Math 828 (Topics in Harmonic Analysis). This semester we will talk about connections between Fourier restriction type problems and incidence geometry.
Meeting time and place: T/Th 11:00AM - 12:15PM, 350 Birge Hall
Office Hours: 2:30-3:30pm every Wednesday or by email appointment
Canvas website: https://canvas.wisc.edu/courses/190031. Canvas will only be used to store all videos I recorded for the remote instruction throughout the second half of the semester.
I plan to talk about three topics concerning recent developments of the subject of Fourier restriction type problems. They are all connected to incidence geometry problems.
They include:
(1) Fourier restriction in dimension 3 and the polynomial method,
(2) the Bourgain-Demeter decoupling theorem, and
(3) recent progress on local smoothing in 2+1 dimensions and the Fourier analytic method for incidence problems.
Lecture Notes:
Additional exercises (last updated 4/28/2020)
Lecture 1 Notes Lecture 1 Figures
Lecture 2 Notes Lecture 2 Figures
Lecture 4 Notes Lecture 4 Figures
Lecture 6 Notes Lecture 6 Figures
Lecture 7 Notes Lecture 7 Figures
Lecture 13 Notes Lecture 13 Figures
Lecture 22 Notes Lecture 22 Figures
Lecture 23 Notes Lecture 23 Figures
Lecture 24 Notes Lecture 24 Figures
(To be continued)
Here is the Lecture Notes Template and the Syllabus.
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). We will also be using a lecture notes template very similar to his.
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 is currently teaching 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.