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Séminaire DynamiX

14h00 – Anton Zorich (Institut de Mathématiques de Jussieu)

"Flat and hyperbolic enumerative geometry"

Abstract: In the first part of the talk I will mostly try to remind how Maryam Mirzakhani has counted simple closed geodesics on hyperbolic surfaces. Her count of Weil-Peterson volumes and her proof of Witten's conjecture would be only mentioned. In the second part I will show how ideas of Mirzakhani work in problems of flat enumerative geometry. This part is based on the ongoing project with V. Delecroix, E. Goujard and P. Zograf.

Namely, I will prove equidistribution of square-tiled surfaces of a fixed combinatorial type. I will combine this result with computation of Masur-Veech volumes of the moduli space of quadratic differentials in genus zero (obtained earlier with Athreya and Eskin) to count asymptotical number of meanders of fixed combinatorial type. Our formulae are particularly efficient for classical meanders in genus zero.

I will conclude by constructing a bridge between flat and hyperbolic worlds. On the one hand, I will give a formula for the Masur--Veech volume of the moduli space of quadratic differentials in terms of psi-classes (in the spirit of Mirzakhani's formula for Weil--Peterson volume of the moduli space of pointed curves). On the other hand, I will show that Mirzakhani's frequencies of simple closed hyperbolic geodesics of different combinatorial types coincide with the frequencies of the corresponding square-tiled surfaces.

 

Fichiers de l'exposé : Flat and hyperbolic enumerative geometry

"Meanders" et "Hyperbolic geometry of surfaces"