Seminar Talk

The easiest way to build compact hyperbolic surfaces is by gluing together standard pieces, called pairs of pants, along geodesic boundary circles of matching length. Choosing the gluing data uniformly at random from the torus of possible gluing parameters gives a model for random hyperbolic surfaces. In this talk, I will explain some recent progress on a conjecture of Mirzakhani formalizing the notion that this is a good model for random surfaces. I will describe how this question fits into a broader dynamical and geometrical framework of horocycle and geodesic flows on (spaces of) surfaces and discuss other recent progress on topological and measurable dynamics of horocycle flows.