Title: Returns in flat surface dynamics
Abstract: The dynamics on the space of flat surfaces are related to both billiard dynamics and algebraic geometry. In the talk, I will introduce the motivation for the study of flat surface dynamics, discuss some of its striking properties, and focus on the following problem: How many ways can a surface return to itself under dynamics?
More explicitly, one can get a flat surface by gluing parallel edges of a polygon using translations. One can act on a surface using a 2×2 matrix. We discuss the following result: for every flat surface S, either (1) the group of matrices taking S to itself is either a lattice in SL_2(R) in which case it contains ~ R^2 matrices with operator norm ≤ R for every R, or (2) this group contains less then R^{2-ε} matrices with operator norm ≤ R for every R, where ε depends only on the genus of S.
Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=98820aac-5290...