Mini Course| Institute of Mathematical Sciences
Time: November 12th, 14th, 19th and 21th, 2024, 15:00-17:00
Location: IMS, RS408
Speaker: Yongqiang Zhao, Westlake University
Abstract: How large can a subset of {1,2,...,N} be without a nontrivial three-term arithmetic progression {a, a+d, a+2d} This question, asked by Erdös and Tur´an in 1936, has become one of the central questions in additive combinatorics. The arguments of Roth in 1953, which gave the first quantitative bound, have turned out to be very influential in the developments of the whole subject. In this short course, we will give a friendly introduction to Roth’s theorem and its finite field ramification, the cap set problem. The various topics we plan to discuss are as follows:
(1) Fourier analysis over finite abelian groups;
(2) Roth’s theorem;
(3) The cap set problem and the polynomial method;
(4) The Kelley-Meka bounds over finite fields.
This course is self-contained and should be accessible to advanced undergraduates or beginning graduate students.