Seminars & Colloquia Calendar
Flat and hyperbolic geometry of surfaces
James Farre (Yale)
Location: Hill 705
Date & time: Tuesday, 11 February 2020 at 3:50PM - 4:40PM
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. We will review some constructions coming from hyperbolic geometry and some coming from complex analysis. Using Bonahon and Thurston’s shear coordinates for Teichmüller space, Mirzakhani proved that two flows, namely earthquake flow from hyperbolic geometry and Teichmüller horocycle flow from complex analysis, are measurably isomorphic. Mirzakhani’s correspondence does not factor through uniformization and it is defined only on a measurably generic set. We will introduce a new coordinate system for Teichmüller space that allows us to extend Mirzakhani’s conjugacy everywhere in a natural way relating the hyperbolic geometry of a surface to certain singular flat metrics induced by quadratic differentials. This is joint work (in progress) with Aaron Calderon.