There is a deep yet mysterious connection between the hyperbolic and singular flat geometry of Riemann surfaces. Using Thurston and Bonahon's "shear coordinates," Mirzakhani related the earthquake and horocycle flows on Teichmüller space, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. In this talk, I will introduce "shear-shape coordinates" for Teichmüller space, which can be used to extend Mirzakhani's conjugacy. These coordinates also yield information about the global structure of certain subloci in both Teichmüller space and its cotangent bundle of quadratic differentials. This is joint work (in progress) with James Farre. Speaker(s): Aaron Calderon (Yale)