Abstract: The moduli space of translation surfaces is a space parametrizing certain flat metrics
on compact orientable surfaces. The dynamics of the action of G=SL(2,R) and its subgroup U of
upper triangular unipotent matrices on this space has been intensively investigated in recent years,
and the aim has been to prove results analogous to Ratner's celebrated theorems classifying invariant
measures and orbit-closures on homogeneous spaces of Lie groups. For the G-action, this was
achieved by McMullen in 2005 in genus 2, and in full generality by very recent breakthrough results
of Eskin-Mirzakhani-Mohammadi. For the U-action, the problem is still open. I will explain the questions,
motivating them by seemingly unrelated elementary geometrical questions, and survey the recent
breakthroughs. I will also present results of joint work with Samuel Lelievre and John Smillie.
The talk is intended for a general audience and all relevant notions will be defined.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006