Enumerating triangulations of surfaces -- Philip Engel, March 24, 2017

A triangulation of S2 has non-negative curvature if every vertex has at most six triangles adjacent to it. Thurston showed that non-negative curvature triangulations correspond to lattice points in a moduli space of flat cone metrics on S2. In joint work with Peter Smillie, we use an arithmetic technique of Siegel to count such lattice points. The appropriately weighted number of triangulations with 2n triangles is an explicit constant times the ninth divisor power sum of n. If time permits, I will discuss work in progress on the enumeration of triangulations with any set of specified non-zero curvatures. Arithmetic techniques are no longer available but quasi-modularity follows from analyzing the single Hurwitz numbers of an elliptic orbifold, mildly generalizing the techniques of Eskin-Okounkov on covers of a pillowcase.