Combinatorical Models of Character Varieties

Character varieties are spaces that parameterize isomorphism classes of representations of certain groups or algebras. The theory has many deep and interesting connections with cluster algebras, Teichmuller theory, integrable systems, symplectic and Poisson geometry, and much more. I will give basic definitions and examples, focusing on the case of the fundamental group of a surface with boundary. I will talk about basic combinatorial tools that can be used to model these spaces and perform calculations. Time permitting, I will also discuss Lie algebra structures on these spaces and how they can be modelled combinatorially.