Counting lattice points in moduli spaces of quadratic differentials (Q6119698)

Quadratic differentials on complex algebraic curves and their moduli spaces is an actively developing area of modern mathematics with close connections to algebraic geometry, dynamics, and mathematical physic. This paper deals with the moduli spaces of meromorphic quadratic differentials with simple poles. These spaces are naturally equipped with linear coordinates (called period coordinates), volume form (called the Masur-Veech volume form), and integer lattice (whose points are represented by square-tiled surfaces). The authors show how to count lattice points in the moduli spaces of meromorphic quadratic differentials and apply this count to three seemingly different problems: the computation of Masur-Veech volumes of these moduli spaces, the distribution of simple closed geodesics on hyperbolic surfaces, and the enumeration of meanders (configurations in the plane that consist of a straight line and a simple closed curve transversely intersecting it considered up to isotopy). For the entire collection see [Zbl 07816357].