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Sam Gunningham, University of Montana

Abstract: Skein modules are linear spaces spanned by knots and links in a given 3-manifold, modulo certain skein relations. They were defined about 30 years ago independently by Przytycki and Turaev and have been extensively studied in the subsequent years. In this talk I will propose a new role for skein modules: as (a component of) the state space of a certain 4-dimensional topological quantum field theory, which according to the work of Kapustin and Witten, encodes the mathematical features of the geometric Langlands program. This realization leads to some surprising conjectures (which can be directly verified in some key cases), relating two different flavors of skein modules on a given closed 3-manifold. This is joint work with David Ben-Zvi, David Jordan, and Pavel Safronov.

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