The group SL(2,R) acts naturally on the set of flat surfaces
of higher genus, generalizing the action on the space of flat tori.
This action is naturally related to a number of classical dynamical
systems, like interval exchanges and billiard flows. For example,
asymptotics on a fixed surface are related to the Lyapunov exponents
of the Kontsevich-Zorich (KZ) cocycle.
In this talk, I will discuss some results about the rigidity of the
SL(2,R) dynamics and the KZ cocycle. By work of Eskin, Mirzakhani, and
Mohammadi, orbit closures 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.
The talk will start from the basics, assuming no background in either
Teichmuller dynamics or Hodge theory.