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Character varieties are moduli spaces of representations of fundamental groups. They play an important role in physics, as basic examples of gauge theories. In this talk, I'll explain how character varieties of surfaces and 3-manifolds can be "quantized", leading to the construction of a fully extended 4-dimensional topological field theory, which has applications to many important constructions in representation theory of quantum groups.

Computations for surfaces shed some new light on many common "quantum" algebras, like reflection equation algebras and double affine Hecke algebras. The foundations needed to define the quantum character theory for 3-manifolds require us to revisit classical works of Schur and Frobenius.

bring lunch!