Pants decompositions and dynamics on moduli spaces

Every closed hyperbolic surface X (or Riemann surface or smooth algebraic curve over C) can be described by gluing together pairs of pants (three-holed spheres). Each X can be glued out of pants in many different ways, and Mirzakhani showed that the count of these decompositions is closely related to a certain Hamiltonian flow on the moduli space of hyperbolic surfaces. In the field of Teichmüller dynamics, counting problems on flat surfaces can be related to a different dynamical system on a different moduli space, which, by work of Eskin--Mirzakhani--Mohammadi and Filip, is in turn controlled by special algebraic subvarieties. In this talk, I will survey some of these results and describe a bridge between the two worlds that can be used to transfer theorems between flat and hyperbolic geometry.