Horocycle flows in moduli spaces of abelian differentials

There is a fruitful analogy between Lie group actions and the dynamics of SL(2,R) on the moduli space of abelian differentials. Along the lines of this dictionary, the horocycle flow corresponds to a unipotent flow on a homogeneous space, for which Ratner’s theory is available. I will report on recent progress regarding the dynamics of the horocycle flow in the moduli space of abelian differentials.