# [CANCELLED] Infinite-Dimensional Algebra Seminar: Alexander Goncharov (Yale)

Friday, April 19, 2024 at 3:00pm to 5:00pm

Building 2, Room 135

182 MEMORIAL DR, Cambridge, MA 02139

**Speaker**: Alexander Goncharov (Yale University) [CANCELLED]

**This seminar will be rebooked at a later date.**

** Title: ** Exponential volumes in Geometry and Representation Theory

**Abstract: ** Let S be a topological surface with holes. Let M(S,L) be the moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of the boundary circles. It carries the Weil-Peterson volume form. The volumes of spaces M(S,L) are finite. M.Mirzakhani proved remarkable recursion formulas for them, related to several areas of Mathematics.

However if S is a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that exponential volumes are finite, and satisfies unfolding formulas generalizing Mirzalkhani's recursions.

This part of the talk is based on the joint work with Zhe Sun.

There is a generalization of these moduli spaces for any split simple real Lie group G, with canonical exponential volume forms. When the modular group of the surface P is finite, our exponential volumes are finite for any G. When P are polygons, they provide a commutative algebra of positive Whittaker functions for the group G. The tropical limits of the positive Whittaker are the (zonal) spherical functions for the group G.

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