Billiards and Surfaces à la Teichmüller and Riemann, Online

organized by Simion Filip, Carlos Matheus, Curtis McMullen and Martin Möller

Third BiSTRO-mini-conference

25 May 2022
Time: 9am (PDT) / 12pm (EDT) / 6pm (CEST)

Leonardo Lerer ( Weizmann Institute)
Title: Bi-algebraic geometry of strata of abelian differentials

Abstract: A stratum of abelian differentials is endowed with an atlas of charts, with linear transition functions, given by mapping a differential to its relative periods. In this talk, we consider the transcendence properties (both arithmetic and functional) of these period coordinates. More precisely, we will discuss the transcendence over \bar{\mathbb{Q}} of the relative periods of abelian differentials, together with a characterization of the "least" transcendental ones and their distribution inside a stratum. On the geometric side, we will discuss the algebraic relations satisfied by the periods of an abelian differential when it varies inside an algebraic subvariety of a stratum. This is joint work with B. Klingler.

Anja Randecker (Heidelberg University)
Title: Topological behaviour of conjugacy classes of big mapping class groups

Abstract: Classical mapping class groups, i.e. for surfaces of finite type, are well-studied objects: they are discrete groups expressing the symmetries of the surface.

When we turn our attention to surfaces of infinite type, the situation changes drastically: In particular, the mapping class groups are now uncountable and we can define an interesting topology on them. This lets us ask many new questions: When considering the conjugacy action of a big mapping class group on itself, can there be comeager orbits? Or at least dense orbits? Or at least somewhere dense orbits?

In this talk, I will give a very short introduction to big mapping class groups, answer the questions above, and give an idea of the tools from model theory that we use in the proofs. This is based on joint work with Jesús Hernández Hernández, Michael Hrušák, Israel Morales, Manuel Sedano, and Ferrán Valdez.

Seung uk Jang (University of Chicago)
Title: Kummer Rigidity for Hyperbolic Hyperkähler Automorphisms

Abstract: Dynamical systems that have volume-class measures of maximal entropy typically have locally homogeneous structures. In complex dynamics, this usually means that the automorphism comes from a torus, as established by Zdunik, Berteloot--Dupont, Cantat--Dupont, Filip--Tosatti, and others. As a successor to this series, we present another result that applies to projective hyperkahler manifolds, a higher-dimensional analogue of K3 surfaces.

We discuss how such a system has a surprisingly simple dynamical structure, and how we can make use of this structure to identify the given automorphism as a "Kummer example" with a (Ricci-flat) flat metric. All the necessary background will be provided.

Each talk will be 30 minutes plus 15 minutes for discussion and questions.