CE40 - Mathématiques

Moduli spaces of Differentials: Flat surfaces and interactions – MoDiff

Moduli spaces of Differentials: Flat surfaces and Interactions

This project studies 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.

Quantitative problems of dynamics and geometry on moduli spaces of differentials

Dynamics of billiards in rational polygons can be efficiently understood by studying geometrical and dynamical properties of moduli spaces of related differentials (or flat surfaces), using a renormalization process. Quantitative dynamical properties (such as the number of closed orbits, or the diffusion rate in wind-tree models) are similarly obtained by computing geometric and dynamics invariants of these moduli spaces (such as Masur-Veech volumes and Siegel-Veech constants). Although dynamics in billiards was the main motivation for computing these invariants, they happened to be relevant for many other applications in other fields. Our goal is to push the study of these invariants to more general settings, by combining a wide range of techniques and expertise in different fields.

The work program is divided into five tasks. The first one is the large genus asymptotics: here we have very explicit conjectures on the behavior of volumes and Siegel-Veech constants to prove. The second task concerns higher differentials, and in particular the computation of the volumes of the corresponding moduli spaces. The thrid objective is the construction and computation of similar geometric invariants for affine submanifolds. Antoher task is related to combinatorics, more precisely the enumeration of ribbon graphs and meanders for instance, using flat geometry. Finally we want to study boundary of the Teichmüller space and in particular limit sets of Teichmüller disks, and to explore the relations between flat and hyperbolic geometries.

Our main results so far (July 2021) answer some questions regarding the large genus asymptotics as well as higher differentials and affine submanifolds.

The originality of the project lies in its transversality and its focus on computable invariants, that allow to base the intuition on computer experiments, and that makes most of the goals of this project reasonably achievable. The novelty is to associate experts from different fields to start new collaborations and tackle this question as efficiently as possible.

-V. Koziarz et D-M. Nguyen: « Variation of Hodge structure and 
enumerating tilings of surfaces by triangles and squares. » Journal de l'Ecole Polytechnique: Mathématiques 8 (2021),
-V. Delecroix, E. Goujard, P. Zograf, A. Zorich, «Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves «, arXiv:2007.04740.

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.

Project coordination

Elise Goujard (Institut de mathématiques de Bordeaux)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


IMB Institut de mathématiques de Bordeaux

Help of the ANR 69,336 euros
Beginning and duration of the scientific project: November 2019 - 48 Months

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