Masur-Veech volumes of odd strata of quadratic differentials

Translation / half-translation surfaces are surfaces equipped with a singular flat metric with a condition on holonomy. They are equivalent to Abelian / quadratic differentials on Riemann surfaces. Their moduli spaces (strata) are extensively studied from the perspectives of flat geometry, dynamical systems and algebraic geometry. Computing the volumes of strata (Masur-Veech volumes) is important for understanding the geometry of individual or random flat surfaces (such as counting geodesics / saddle connections for example).

I will present a new recursive formula for the volumes of certain strata in the quadratic case, which is a joint work with Eduard Duryev and Elise Goujard. It relies on reformulating the volume computation as a certain combinatorial enumeration problem. The main technical point is a refinement of a result of Kontsevich on the enumeration of metric ribbon graphs. Using our formula, we hope to prove conjectures about the large genus asymptotics of volumes and about the properties of random square-tiled surfaces.