THO 101

## From rational billiards to dynamics on moduli spaces

#### Alex Wright from Stanford University

We will begin by giving an elementary introduction to the GL(2,R) action on the Hodge bundle (sometimes called Teichmuller dynamics), after which we will give a survey of some of new developments in this field. This will include restrictions on the structure of orbit closures echoing Ratner's Theorems on homogeneous spaces, the discovery by Moller and Filip that orbit closures can be defined purely in terms of algebraic geometry, and new examples of sub-varieties of the Hodge bundle which provide counterexamples to a conjecture of Mirzakhani. The talk will include joint work with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.