Abstract:
The Jones polynomial is a powerful invariant of knots in $S^3$, whose discovery ushered in the modern era of topological quantum field theory.  Despite its deep origins in operator algebra, the Jones polynomial can be computed by a pleasingly simple and combinatorial "skeining" algorithm, comprehensible to a high school student.  Around 1990, Turaev and Przytycki independently extended this algorithm to any oriented manifold $M$ in place of $S^3$, producing in this way a vector space -- the "skein module" -- housing invariants of knots embedded in $M$, but its basic properties remain mysterious.  Based on manipulations in quantum field theory, Witten conjectured around 2014 that skein modules of closed 3-manifolds should be finite-dimensional.  The conjecture came as a surprise to mathematicians, because there was almost no numerical evidence:  precious few classes of skein modules had been computed in the intervening 25 years. 
     In this colloqium I will survey the ingredients from geometric representation theory, quantum field theory, and complex Floer theory which led to the resolution of Witten's conjecture -- in out joint work with Sam Gunningham and Pavel Safronov -- and I will highlight examples where these dimensions have been computed.