Holomorphic differentials on compact Riemann surface are of central importance in (i) the study of billiards on polygons (with rational angles), and (ii) Teichmuller theory. Many questions about billiards can be approached by studying certain subloci called affine invariant manifolds in strata of holomorphic differentials. Wright's Cylinder Deformation Theorem has proved to be a very useful tool for constraining the structure of affine invariant manifolds.

Meromorphic differentials arise naturally as degenerations of holomorphic differentials. In particular, there is a natural "multi-scale" compactification of strata of holomorphic differentials for which the objects on the boundary correspond to meromorphic differentials. I will discuss work on using this compactification to understand affine invariant manifolds. I'll describe a new proof of the Cylinder Deformation Theorem, which generalizes to the case of meromorphic differentials.

Joint work with Frederik Benirschke and Samuel Grushevsky.

There will be a pretalk by Jenya Sapir (Binghamton).