Spectral description of non-commutative character varieties

Seminar: 
Colloquium
Event time: 
Wednesday, September 22, 2021 - 4:15pm
Speaker: 
Alexander Goncharov
Speaker affiliation: 
Yale University
Event description: 

Abstract:

A character variety is the space of representations of the fundamental group of a topological surface to a group, say SL(2, K), GL(m,K), where K is a field, say of real or complex numbers. 

They arise in many areas of Math and Physics,  e.g. in the study of linear differential equations on Riemann surfaces -  as the monodromy data of an equation, in Teichmuller theory &  geometry of hyperbolic 3-folds, Algebraic geometry, mathematics of supersymmetric field theories, etc. 

Character varieties come with a number of features, like Atiya-Hitchin-Goldman Poisson structure. 

Certain finite covers of character varieties carry a collection of coordinate systems in which the Poisson structure 

is especially simple etc. 

I will explain how all this looks in the case when K is an arbitrary non-commutative field, e.g. quaternions. 

I will also explain the story about wild character varieties, aka moduli spaces of Stokes data, which are analogs of the monodromy data for differential equations with irregular singularites.

This is a  joint work with Maxim Kontsevich.