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Speaker: James Leng (UCLA)

Title:  Szemerédi’s theorem, primes, and nilsequences

Abstract: Let $r_k(N)$ be the largest subset of $[N] = {1, dots, N}$ with no k-term arithmetic progression. Szemerédi’s theorem states that $r_k(N) = o_k(N)$. We will go over the proof that achieves the best known upper bounds for $r_k(N)$ for general $k$. We will discuss how the mathematics behind the proof relates to counting primes along linear forms and the distribution of orbits on $G/Gamma$ with $G$ nilpotent and $Gamma$ discrete and cocompact. This is (partly) based on joint work with Ashwin Sah and Mehtaab Sawhney.