MoDiff

Since the foundational work of Masur and Veech in 1982 the last thirty years have been extremely fruitful for the theory of translation surfaces. Various techniques issued from combinatorics, geometry, and ergodic theory have been used to tackle extremely delicate questions. The goal of the project MoDiff is to push further some of the techniques introduced by the coordinator together with a consortium with broad spectrum of expertise. In particular we want to study new problems of dynamics and geometry on moduli spaces of differentials (also viewed as flat surfaces), by exploring their interactions with combinatorics, enumerative geometry, complex geometry, and hyperbolic geometry. The core of the project is computation of volumes and geometric invariants of these moduli spaces, such as the Masur-Veech volumes and Siegel-Veech constants. This would have direct consequences in the dynamics of rational billiards. The project extends to related problems such as the geometry of the boundary of moduli spaces, asymptotic combinatorics, interaction between the flat and hyperbolic metrics on surfaces.