Title: The path semigroup of a graph with applications to moduli spaces of geometric structures.

Abstract: I'll introduce some open questions about an elementary object: a commutative, finitely generated semigroup formed by the supports of (unions of) paths in a finite graph. These semigroups are related to affine algebraic schemes called character varieties; also known as moduli of flat principal bundles and moduli of geometric structures. I'll explain how answers to my questions could say something about the symplectic and tropical geometry of these moduli spaces.