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Math 542 (Harmonic Analysis 2)

Spring 2023

Basic information

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  • Instructor: Pablo Shmerkin.

  • Time and location: MWF 12:00-12:50 in MATH 204. Please get in touch if you're interested in the course and the times do not work for you.

  • Office hours: times will be published on Canvas 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, see also point 1 in the list of topics below)

Course description

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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

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For students taking the course for credit, the grade will be based on submitting and presenting homework exercises. There will be 4 sets of exercises, all 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. Each student will be asked to present solutions to two HW problems in the course of the semester (with at least 2 days of advance warning). This is a requirement but does not affect the grade.

Tentative topics (subject to revision, updated December 25)

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  1. Quick review of required background: change of variables, L^p duality, the Schwartz class, definition and basic properties of the Fourier transform, Riesz-Thorin interpolation theorem.

  2. Dual rectangles. Rapid decay and local constancy. Fourier decay of surface measures.

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

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

  5. Some partial results towards the Kakeya conjecture. Kakeya sets in finite fields (if time allows).

  6. The Loomis-Whitney inequality and Guth's proof of the multilinear Kakeya theorem (if time allows).

Bibliography

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There is no required textbook. Except for the last topic (multilinear Kakeya), the material in the course can be found in at least one of the following excellent 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.

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

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