Algebra, combinatorics, and geometry in skein theory

Skein theory uses quantum groups to associate algebras to topological surfaces and modules to 3-manifolds with boundary. In this talk we discuss these algebras/modules and some relationships to both algebraic geometry (character varieties of surfaces) and symplectic geometry (the Fukaya category of the surface). Much of this discussion is motivated by the example where the surface is a torus and the 3-manifold is a solid torus; in this case, skein theory produces operators on symmetric functions with interesting combinatorial properties. This will be a survey-style talk; in particular, no background in skein theory or quantum groups will be assumed.