Dynamics on moduli spaces of hyperbolic surfaces

Two-dimensional geometry has been a cornerstone of mathematics since antiquity. In addition to geometry on the xy-plane, mathematicians today study geometry on surfaces such as the crust of the Earth or the glaze of a donut. Most of these surfaces exhibit hyperbolic geometry, a type of non-Euclidean geometry in which parallel lines spread apart. Because of its ubiquity, two-dimensional hyperbolic geometry is a common meeting point for many branches of mathematics. Recently, the PI strengthened a connection linking hyperbolic geometry with certain transformations of two-dimensional Euclidean geometries, which were motivated by the kinetic motion of particles and the flow of electrons. The goal of this project is to further investigate this connection between non-Euclidean and Euclidean geometries in order to better understand both worlds. The project will also support professionalization, mentorship, and enrichment activities for a broad range of students. The research component of this project has three main prongs of attack. The first is to further develop (and complicate) the link between the earthquake and horocycle flows, dynamical systems on moduli spaces of hyperbolic surfaces and quadratic differentials, respectively. This will allow the PI both to leverage decades of progress on quadratic differentials to study hyperbolic surfaces, and to leverage new geometric features afforded by hyperbolic space to study quadratic differentials. The second prong is to study the structure of optimal Lipschitz maps between hyperbolic surfaces and their related renormalization dynamics. The final prong is to use all of these tools to investigate related counting problems on both flat and hyperbolic surfaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.