Tuesday, October 6, 2020
10/06/2020 - 4:00pm
There is a deep yet mysterious connection between the hyperbolic and singular flat geometry of Riemann surfaces. Using Thurston and Bonahon’s “shear coordinates” for maximal laminations, Mirzakhani related the earthquake and horocycle flows on moduli space, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. In this talk, I will describe joint work with James Farre in which we construct “horospherical” coordinates for Teichmüller space adapted to any measured lamination which generalize both Fenchel–Nielsen and shear coordinates. These coordinates simultaneously parametrize both flat and hyperbolic structures, and consequently allow us to extend Mirzakhani’s conjugacy and gain insight into the ergodic theory of the earthquake flow. If time permits, I will also mention some work in progress that relates this viewpoint to Masur–Veech volumes, the equidistribution of Weil–Petersson horoballs, and intersection numbers on moduli space.