Math 542 (Harmonic Analysis 2)

Spring 2021

Basic information

  • Instructor: Pablo Shmerkin.

  • Times: The published times are Tuesday-Thursday 9:30-11:00, but these may be reviewed at the start of the semester to take into account possible clashes with other courses, students in other time zones, etc. Feel free to get in touch if the official times do not work for you.

  • Mode of delivery: Online, via zoom and discussion boards. Details will be posted on canvas.

  • 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 some topics of current research interest concerning the geometry of sets and measures on Euclidean space, in particular fractal sets and measures. I will aim to present a variety of ideas and tricks that are used troughout harmonic analysis, such as 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

I will post exercises throughout the semester (tentatively, 4 sets of exercises). You can choose to either submit at least 75% of the exercises in each set, or give a presentation on a topic chosen together with me at the end of the semester. The presentation will consist of a written paper (5+ pages) and a zoom presentation of the main ideas. It is recommended that you try the exercises even if you choose the presentation. Both the exercises and the presentation can be worked on with other students (and of course questions are welcome) but the final writeup/delivery of the presentation must be done individually.

Course topics (subject to minor revisions)

  1. Notions of regularity for singular measures: Frostman exponent, notions of dimension, Fourier decay.

  2. Hausdorff and Fourier dimensions. Frostman's Lemma. Salem sets.

  3. Projection theorems: Marstrand's projection theorem and some finer versions.

  4. Distance sets. Falconer's problem. Some elementary bounds. The Mattila-Liu integral.

  5. The Kakeya problem. Existence of zero measure Kakeya-Besicovitch sets. Dimension and maximal Kakeya conjectures, and their solution in the plane.

  6. Multilinear Kakeya. Guth's proof via Loomis-Whitney and induction on scales.

  7. Depending on time and interest, I may add some discussion of the following topics: (i) restriction theory (connection with Kakeya, Stein-Tomas bounds for curved surfaces and fractal measures, multilinear restriction), (ii) Bernoulli convolutions (Fourier transform, overview of results on dimension and smoothness).

Bibliography

There is no required textbook. Much of the material in the course can be found in the book Fourier Analysis and Hausdorff Dimension by P. Mattila. Other books that can be useful for parts of the course include:

  • Geometry of sets and measures in Euclidean space, by P. Mattila.

  • Fractals in probability and analysis, by C. Bishop and Y. Peres.

  • Fourier restriction, decoupling and applications, by C. Demeter.

All these books are available for free online to current UBC students. For the Mattila-Liu integral, we will follows B. Liu's paper An L^2 identity and pinned distance problem, available here. For multilinear Kakeya, we will follow L. Guth's article A short proof of the multilinear Kakeya inequality, available here.

I will provide notes for some of the topics.