Speaker: Dr. Ruixiang Zhang (UC Berkeley) 

Time: 16:00-17:00  January 8, 2024 (Monday)

Place: MCM110

Title: Falconer's distance set conjecture and its many faces

Abstract: In 1985, Falconer made the conjecture that whenever $E \subset \mathbb{R}^2$ has dimension $>1$, its distance set $\Delta(E) = \{|x-y|: x, y \in E\}$ will have positive measure. Despite its innocent appearance, this conjecture remains unsolved as of today. We will introduce this conjecture and three interesting results towards it. The methods to prove these results are related to Fourier analysis, classical algebraic geometry/topology and geometric measure theory.

Speaker: Prof. Rafael von Kanel (Tsinghua University）

Time: 10:30-11:30am, January 10, 2024 (Wednesday)

Place: MCM110

Title: Integral points on the Clebsch-Klein surfaces

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch's work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.