This seminar series on Gowers norms and higher-order Fourier analysis is organized by Jihyo Chae at Yonsei University. The group consists of three participants: Ingyu Baek, Jeewon Kim, and Jaehyeon Seo. Jaehyeon Seo also helped with the organization of this seminar. Both the organizer and participants are students of Professor Joonkyung Lee, who also supported this seminar as a faculty sponsor.
The goal of this seminar is to understand the structure-randomness dichotomy in subsets of integers via Fourier analytic techniques. To achieve this, we study the following topics:
Quasirandomness in abelian groups
Higher-order Fourier Analysis
Cauchy-Schwarz complexity of linear forms, and the proof of the Green-Tao Theorem
We aim to learn how the idea of Fourier analysis and the Hardy-Littlewood circle method is generalized to deal with a broad range of problems in additive combinatorics, most of which are related to solution sets of linear configurations and the randomness of subsets of abelian groups.
Summer 2025
Jul 1, Intro: Modern view on the circle method, Capturing randomness in a subset of Z
Jul 11, Quadratic Fourier Analysis, I: Generalized von Neumann Theorems for k=2, 3 in the finite field model [reference]
Jul 24, Quadratic Fourier Analysis, II: Gowers Inverse Theorem for k=3 - Gowers inverse theorems in the finite field model
Scheduled, Quadratic Fourier Analysis, III: Gowers Inverse Theorem - concluding the proof in the finite field setting, Introduction to Z/nZ
References
Quasi-random subsets of Z_n, F.R.K. Chung, R.L. Graham [link]
Montreal Notes on Quadratic Fourier Analysis, Ben Green [link]
Higher-order Fourier Analysis and Applications, Hamed Hatami, Pooya Hatami, Shachar Lovett, [link]
On Higher order Fourier Analysis, Balázs Szegedy, [link]
A new proof of Szemerédi's theorem, Tim Gowers, [link]
Higher order Fourier Analysis, Terrence Tao, [link]