The horocycle flow in the moduli space of translation surfaces

The group SL(2,ℝ) acts on the moduli space of translation surfaces. The “magic wand” theorem of Eskin, Mirzakhani, and Mohammadi (~2015) states, in particular, that the closed sets invariant under this action are holomorphic subvarieties. The action of the unipotent upper-triangular subgroup is known as the horocycle flow, by analogy with the horocycle flow on hyperbolic surfaces. One can think of it as a non-homogeneous analogue of unipotent flows, whose dynamics were described by the work of Ratner (~1990). What do the closed invariant sets of the horocycle flow look like? I will discuss some recent work that addresses this question.