Course Description: In this course, we will study elliptic PDE with variable coefficients building up to the minimal surface equation. Then we will study Fourier and harmonic analysis, emphasizing applications of Fourier analysis. We will see some applications in combinatorics/number theory, like the Gauss circle problem, but mostly focus on applications in PDE, like the Calderon-Zygmund inequality for the Laplacian, and the Strichartz inequality for the wave equation and the Schrodinger equation. In the last part of the course, we will study these two dispersive PDE. Along the way, we will work on the craft of proving estimates. For a much more detailed description of the class, you can click on the following link: Course description

Notes and references:

In last year's class, Paul Gallagher, Cole Graham, Kevin Sackel, and Jane Wang typed up lecture notes for the class. A big thank you to them, and also to Spencer Hughes, who gave two guest lectures and typed up notes for them. I hope these notes can be a helpful reference. I'm planning to follow the same plan pretty closely.

Lectures 1 and 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lectures 10 - 12, Lecture 13, Lecture 14, Lecture 15, Lectures 16-17.

Here are notes for the second half of the course, starting after spring break.

Lecture 20, Lecture 21, Lectures 22-24, Lecture 25, Lecture 26, Lecture 27, Lecture 28, Lecture 29, Lecture 30, Lectures 31-34.

If you want to brush up on Fourier analysis, the undergraduate book

Problems:

Problem Set 1 (Due on Wed, Feb 17).

Problem Set 2 (Due on Wed, March 2). Also, here is the latex file of the pset: Problem Set 2 latex file.

Problem Set 3 (Due on Wed, March 16).

Problem Set 4 (Due on Fri, April 15).

Problem Set 5 (Due on Mon, May 2).

Last problem set (Due on Wed, May 11).

Further (optional) problems for study and review