Curves, Counting, and Correlations

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Topology is a field of mathematics that studies those properties of a space that remain invariant under stretching or bending. A surface of a fixed topological shape can have many different geometries. For example, a two-dimensional shape with no holes is a topological sphere, which could be a round sphere (like the surface of a planet), a polyhedron (a crystal like-shape with facets), or an ellipsoid (an egg-type shape). A shape with one hole (a torus) could look like an inner tube, a tire, or a hula hoop. The collection of all possible geometries is known as a moduli space, and these spaces occur over and over again in fundamental physical questions. This NSF award provides funding for a project to study questions of what typical, or random, surfaces look like, from the point of view of understanding families of curves on these surfaces. The research and outreach activities are mutually reinforcing; the project highlights bridges between different areas of mathematics, and deep connections between research, mentoring, and public engagement programs in mathematics. In addition the project provide research training opportunities for graduate students. Starting with simple to state problems about counting special trajectories on surfaces whose geometry comes from Euclidean polygons and polyhedra, and motivated by theoretical physics (Newtonian mechanics, electron transport, and supersymmetric quantum field theory), this project connects to fundamental questions in many different areas of mathematics, including Teichmuller and homogeneous dynamics; Diophantine approximation and the geometry of numbers; probabilistic models and limit theorems; and the geometry of moduli spaces of meromorphic differentials on surfaces. Building on his prior collaborative work, the PI plans to explore questions on the counting and distributions of curves in varying geometries on higher-genus surfaces and higher-dimensional tori.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.