Math 542 (Harmonic Analysis 2)

Spring 2023

Basic information

  • Instructor: Pablo Shmerkin.

  • Times: The published times are MWF 12:00-12:50, but these may be reviewed at the start of the semester to take into account possible clashes with other courses, etc. Feel free to get in touch if the official times do not work for you.

  • Office hours: via zoom, times will be published at the beginning of the semester.

  • Prerequisites: MATH 420/507 and 404/541, or equivalent background in real and harmonic analysis (please contact me in case of any doubts)

Course description

Harmonic Analysis is a core area of modern mathematics with connections to pretty much all mathematical fields, such as PDEs, combinatorics and number theory. In this course we will focus on two (closely connected) problems at the heart of modern analysis: Kakeya and restriction. I will aim to present a variety of ideas and tricks that are used throughout harmonic analysis, such as (non)stationary phase, dyadic pigeonholing and decomposition, induction on scales, the role of curvature and multilinearity, etc. I will try to provide a glimpse of current research as well.

Grading policy

For students taking the course for credit, the grade will be based on submitting homework exercises. There will be 4 sets of exercises, tentatively due at the end of weeks 3, 6, 9 and 12. Each set of exercises will be weighted equally. Correct, clearly written solutions to 80% of the problems in a given set counts for a perfect mark. Students are allowed and encouraged to work together on the HW, but solutions must be written down individually.

Tentative topics (subject to revision)

  1. Some basic concepts: Stationary phase. Dual balls and rectangles. Hausdorff and Minkowski dimension.

  2. The restriction problem. Stein-Tomas argument. The planar case.

  3. The Kakeya problem: maximal and dimension formulations, and their connections. Restriction implies Kakeya.

  4. Some partial results towards the Kakeya conjecture. Kakeya sets in finite fields.

  5. The Loomis-Whitney inequality and Guth's proof of the multilinear Kakeya theorem.

  6. Introduction to decoupling.

Bibliography

There is no required textbook. Much of the material in the course can be found in the books:

 

  • Fourier Analysis and Hausdorff Dimension by P. Mattila . Available online through UBC library.

  • Fourier restriction, decoupling and applications, by C. Demeter. Available online through UBC library.

  • Lectures on Harmonic Analysis, by. T. Wolff. Available here.

For multilinear Kakeya, we will follow L. Guth's article A short proof of the multilinear Kakeya inequality, available here.