In Summer 2020, I made these videos for the Nearly Carbon Neutral Geometric Topology Conference, speaking about work with James Farre in which we extend Mirzakhani’s conjugacy between the earthquake and horocycle flows.
Part 1 deals with coordinates for hyperbolic structures adapted to arbitrary measured laminations, generalizing Fenchel-Nielsen coordinates and the shear coordinates of Bonahon and Thurston.
In part 2, I define both the earthquake and horocycle flows, and explain how to use the orthogeodesic foliation map of part 1 to extend Mirzakhani’s conjugacy between them.
You can also check out a talk I gave about an application of this work to random hyperbolic surfaces and Mirzakhani’s “twist torus conjecture” at the Pacific Dynamics Seminar in May 2021 (see also the recording of James’s talk).
Aaron Calderon 