Horocycle-invariant measures on the moduli space of translation surfaces

A translation surface is a closed surface obtained by gluing edges of a polygon by translations. The group GL_2(R) acts on the collection translation surfaces of a fixed genus g. Eskin and Mirzakhani classified probability measures that are invariant under SL_2(R) and, more generally, under the upper triangular subgroup. In the talk we will discuss a new extension that describes probability measures invariant under the horocyclic flow, conjectured by Forni. We also present an application to billiards with rational angles.