题目：Moduli Spaces of Crowned Surfaces and Their Volumes

主讲：Ivan Telpukhovskiy

时间：12月18日14:00

地点：文理楼B321

主办：数理学院

主讲嘉宾简介：

Ivan Telpukhovskiy，清华大学丘成桐数学中心博士后。2021年博士毕业于多伦多大学（University of Toronto）。主要研究方向几何与拓扑：双曲几何，高维Teichmüller理论，Thurston度量。在《Groups, Geometry, and Dynamics》、《Topology and its Applications》等期刊上发表SCI论文。

报告主要内容：

In 2004, Mirzakhani showed that the Weil-Petersson (WP) volume of the moduli space of hyperbolic structures on a genus g surface with m boundaries, with specified boundary lengths b1,...bm, is a polynomial in Q{＞0}[π2，b2,...,bm2]. A natural generalization of the WP volume form to the moduli spaces of hyperbolic surfaces with boundary punctures (i.e. crowned hyperbolic surfaces) yields “infinite” volumes.

To remedy this, in 2024 Chekhov introduced an action on the moduli spaces of crowned hyperbolic surfaces of fixed neck holonomy, and he refers to the (finite) integrals of the induced measures over moduli spaces as Mirzakhani volumes. Chekhov computed a handful of examples, and we extend his work to full generality, and beyond. In particular, we show that Mirzakhani volume of the moduli space of every crowned hyperbolic surface is naturally expressible as a sum of Gaussian rational multiples of polylogarithms evaluated at ±1 and ±√￣-1. This is joint work with Yi Huang.