# Mathematics Colloquia and Seminars

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### Volumes of moduli spaces of hyperbolic surfaces

**Geometry/Topology**

Speaker: | Paul Norbury, University of Melbourne |

Location: | 2112 MSB |

Start time: | Wed, Feb 1 2006, 1:10PM |

The moduli space of genus g curves with n marked points can be reformulated as the moduli space of genus g hyperbolic surfaces with n cusps. This viewpoint brings a symplectic structure to the moduli space, so in particular volume makes sense. Mirzakhani calculated the volume of a more general moduli space - the moduli space of genus g hyperbolic surfaces with n geodesic boundary components of specified lengths - and showed that it is a polynomial in the boundary lengths. Mirzakhani showed the coefficients in these polynomials are related to the intersection numbers on the moduli space and used this to reprove the Witten-Kontsevich theorem. I will explain this work and some further consequences that the hyperbolic geometry has on intersection numbers.

Lunch with the speaker at 11:30, please contact Joel Hass if interested.