Geometry and Topology Seminar

The group SL(2,R) acts naturally on the set of flat surfaces (i.e. Riemann surfaces with a holomorphic 1-form) generalizing the action on the space of flat tori. This action is naturally related to a number of classical dynamical systems, like interval exchange transformations and billiard flows. I will discuss applications of Hodge theory to establish rigidity of the SL(2,R) dynamics. By work of Eskin, Mirzakhani, and Mohammadi, orbit closures of the action have a good local structure, in particular are manifolds. I will explain how to obtain further global restrictions, in particular proving that they are algebraic varieties. They are characterized by Hodge-theoretic conditions (real multiplication and torsion of certain points).