Topics in Analysis: Decoupling

Instructor Info: Larry Guth, 2-278,

Class times: Tu Th 11 - 12:30, 2-139

Office hours: Tu 3-4.

No class on the week of Sep. 25-29.

Course description: Decoupling theory is a recent development in Fourier analysis with applications in partial differential equations and analytic number theory. It studies the ``interference patterns'' that occur when we add up functions whose Fourier transforms are supported in different regions. The geometry of the regions in Fourier space influences how much constructive interference can happen in physical space. Here is a detailed outline: Course description.

Prerequisites: Background in analysis and especially Fourier analysis at the level of 18.155-156.

Lecture notes:

Lecture 1
Lecture 2 Lecture 2 figures.
Lecture 3 Lecture 3 figures.
Lecture 4
Lecture 5
Lecture 6
Lecture 7
In Lecture 8, we discussed problem set 2. We decided not to scribe the lecture, but you can see the solutions to problem set 2 below, which contains similar content.
Lecture 9
Lecture 10 Lecture 10 figures.
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20

Here is a template for lecture notes that you can use when you write them up: Lecture notes template

A study guide for the decoupling theorem by J. Bourgain and C. Demeter.
A short proof of the multilinear Kakeya inequality by L. Guth. Related to Lecture 4.
On the multilinear restriction and Kakeya conjectures by J. Bennett, T. Carbery, and T. Tao.

Problem Sets:

Problem Set 1
  (Due on Thursday, Sep. 21.) Problem Set 1 Solutions
Problem Set 2
  (Due on Thursday, Oct. 5.) Problem Set 2 Solutions
Problem Set 3
  (Due on Thursday, Nov. 2.) Problem Set 3 Solutions
Problem Set 4
  (Due Thur. Dec. 7.)