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.