Horocycle dynamics on the moduli space of translation surfaces, in the rank one setting

In ongoing joint work with Jon Chaika and Florent Ygouf, we establish a Ratner-type orbit closure theorem for the horocycle flow on rank one loci, in the moduli space of translation surfaces. The question of describing horocycle invariant ergodic measures is still open, and contrary to Ratner’s work, we describe all orbit-closures without a corresponding measure classification theorem. I will give a crash course on the structure of these moduli spaces, emphasizing rel foliations and rel deformations. Then I will describe our approach to the problem, and indicate some of the main difficulties.