In homogeneous spaces, Ratner theorems say that orbit closures for the action of a unipotent-generated subgroup are nice (actually homogeneous). Moduli spaces are somewhat generalizations of homogeneous spaces; in this setting, are sets invariant under unipotent-generated groups nice too?

A theorem by A. Eskin and M. Mirzakhani states that SL_2(R) orbit closures are actually affine, pointing towards a "yes." I will present an ongoing project of J. Smillie and B. Weiss about some not-so-nice horocycle-invariant subset in moduli space; here not-so-nice means: manifold with boundary and infinitely-generated fundamental group.

This talk is intended for a general graduate audience; no special knowledge is required.