I will begin with a review of how a hyperbolic surface/Riemann surface/algebraic curve can degenerate. By adding degenerate surfaces to the moduli space, we get the Deligne-Mumford compactification, an object which has been studied intensively from many different viewpoints.

I will then discuss analogous compactification questions for strata of translation surfaces (a translation surface can be thought of as a Riemann surface together with a holomorphic 1-form). Strata admit different compactifications depending on how much information about the shape of degenerating surfaces is remembered in the limit. I will give an overview of these, and discuss how they interact with Deligne-Mumford. There will be many pictures! Speaker(s): Ben Dozier (Stony Brook)