Counting lattice points in moduli spaces of quadratic differentials

Delecroix, Vincent; Goujard, Élise; Zograf, Peter; Zorich, Anton

doi:10.4171/icm2022/56

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Counting lattice points in moduli spaces of quadratic differentials

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Zorich, Anton
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Delecroix, V., Goujard, É., Zograf, P., & Zorich, A. (2023). Counting lattice points in moduli spaces of quadratic differentials. In D. Beliaev, & S. Smirnov (Eds.), ICM international congress of mathematicians 2022 July 6-14. Vol. 3: Sections 1-4 (pp. 2196-2211). Berlin: EMS Press.

Cite as: https://hdl.handle.net/21.11116/0000-000F-77E0-E

Abstract

We show how to count lattice points represented by square-tiled surfaces in the moduli spaces of meromorphic quadratic differentials with simple poles on complex algebraic curves. We demonstrate the versatility of the lattice point count on three different examples, including evaluation of Masur–Veech volumes of the moduli spaces of quadratic differentials, computation of asymptotic frequencies of geodesic multicurves on hyperbolic surfaces, and asymptotic enumeration of meanders with a fixed number of minimal arcs.