Horocycle flow on the moduli space of translation surfaces

I will discuss the dynamics of the horocycle flow on a stratum of translation surfaces (which is an invariant subvariety of the bundle Omega M_g of holomorphic one forms over the moduli space of genus g Riemann surfaces). This flow can be defined as the action of upper triangular matrices with eigenvalue 1, acting linearly on flat charts. Work of Ratner on unipotent flows on homogeneous spaces leads to the question of whether the orbit-closures and invariant measures for this action can be meaningfully classified. I will quickly survey both positive and negative results in this direction. The talk will be based on joint work with Bainbridge, Chaika, Smillie, and Ygouf (in various combinations).