Variation of Hodge structure and enumerating triangulations and quadrangulations of surfaces

Abstract: Since the work of Eskin-Okounkov (in 2001), it has been known that in any stratum of translation surfaces the number of square-tiled surafces constructed from at most n squares grows like $c\pi^{2g}n^d$, where $d$ is the (complex) dimension of the stratum, $g$ is the genus of the surfaces, and $c$ is a rational number. Similar phenomenon also occurs in strata of quadratic differentials. Counting square-tiled surfaces in a given stratum is more or less the same as counting quadrangulations of a topological surface, with some prescribed conditions on the singularities and the holonomy of the associated flat metric. More recently, Engel showed that the asymptotics of the numbers of quadrangulations and triangulations, satisfying some prescribed conditions at the singularities, with at most $n$ tiles are of the form $\alpha n^d$, where $\alpha$ is a constant in $Q[\pi]$ or $Q[\sqrt{3}\pi]$. In this talk, we will explain how the asymptotics above can be related to the computation of the volume of some moduli spaces, and how one can show that in some situations the constant $\alpha$ belongs actually to either $Q\cdot\pi^d$, or $Q\cdot(\sqrt{3}\pi)^d$ by using tools from complex algebraic geometry. This is joint work with Vincent Koziarz.

differential geometry

Audience: researchers in the topic

Pangolin seminar

Series comments: TIME HAS CHANGED: 15:30 Paris 10:30AM Rio de Janeiro

Description: Differential geometry seminar

Zoom link posted on the webpage 15 minutes before each lecture: https://sites.google.com/view/pangolin-seminar/home