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### Flat and hyperbolic enumerative geometry

**Colloquium**

Speaker: | Anton Zorich, IMJ Paris and MSRI |

Location: | 1147 MSB |

Start time: | Tue, Oct 8 2019, 4:10PM |

Square-tiled surfaces can be interpreted as integer points in the moduli spaces of Abelian and quadratic differentials. This interpretation allows to define the Masur-Veech volume of these moduli spaces in terms of the asymptotic count of square-tiled surfaces.

There are several natural parameters which specify the combinatorial geometry of a flat surface. Together with Vincent Delecroix, Elise Goujard and Peter Zograf we are studying volume contributions of square-tiled surfaces with specified combinatorial geometry. The resulting frequencies of square-tiled surfaces of different types describe geometry of a "random" square-tiled surfaces. These frequencies are closely related to Kontsevich's count of metric ribbon graphs and coincide with Mirzakhani's frequencies of simple closed hyperbolic multicurves.

I will tell what we expect from a "random" square-tiled surface of large genus and from a "random" geodesic multicurve on a surface of large genus. I will include into my talk all necessary background.

There will be a reception in the MSB lobby at 3:30pm before the talk.