Langlands duality for skein modules

Skein modules of 3-manifolds are quantum versions of character varieties of 3-manifolds. In this talk I will review some (conjectural) properties of skein modules for generic quantum parameters, such as finite-dimensionality, relationship to critical cohomology and an unexpected new phenomenon: Langlands duality. In the case of G = SL_2 Langlands duality becomes a result that can be checked. This is a report on work (partially in progress) joint with David Jordan and Sam Gunningham.