Geometry and Topology Seminar: "Flat and hyperbolic geometry of surfaces"

Abstract: The uniformization theorem tells us that the deformation space of constant curvature metrics on a surface also describes the moduli space of its complex structures.   Mirzakhani proved that two flows, namely Thurston’s earthquake flow from hyperbolic geometry and Teichmüller horocycle flow on the principal stratum of quadratic differentials from complex analysis, are measurably isomorphic. We will spend some time reviewing some constructions coming from hyperbolic geometry and some coming from complex analysis to explain Mirzakhani’s correspondence.  Then, we will introduce a new coordinate system for Teichmüller space that allows us to extend Mirzakhani’s conjugacy to other strata of quadratic differentials, answering a question of Alex Wright.  This is joint work (in progress) with Aaron Calderon.

Host: Michael Landry