Topological recursion and consequences (continued)

Abstract

We have completed the demonstration of Mc Shane's formulas, generalized by Mirzakhani. These are remarkable geometric identities, valid on any hyperbolic surface with an edge, relating the lengths of the edges of the "hyperbolic pants" contained in the surface, and the length of an edge component.

We then show how Mirzakhani used these identities to obtain "topological recursion" formulas for volumes of moduli spaces.

We then begin to study the asymptotics of these "volume functions" in large genera.

References

M. Mirzakhani,Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Inventiones mathematicae 2007

M. Mirzakhani, Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus, J. Differential Geom. 2013.

M. Mirzakhani, P. Zograf, Towards large genus asymptotics of intersection numbers on moduli spaces of curves, Geometric and Functional Analysis 2015

Corrected exercises and notes taken during the presentations are available on Thibaut Lemoine's page

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Speaker(s)

Nalini Anantharaman

Professor at the Collège de France

Events

Topological recursion and consequences (continued)

Abstract

Speaker(s)

Nalini Anantharaman

Events

Models of regular graphs and random hyperbolic surfaces

Models of regular graphs and random hyperbolic surfaces (continued)

Mirzakhani integration formulas

Topological recursion and consequences

Topological recursion and consequences (continued)

Asymptotic geometry of random hyperbolic surfaces

"Volume" Functions for Any Curves

Friedman-Ramanujan Functions

Spectral Hole of Random Surfaces

See also