Event Series
Event Type
Seminar
Tuesday, February 4, 2020 4:00 PM
David Jordan (University of Edinburgh)

Around 1990, Prtzytcki and Turaev independently introduced the notion of a skein module of a 3-manifold M.  This is a vector space spanned formally by all possible tangles drawn in M, subject to local "skein" relations allowing one to algorithmically simplify crossings, precisely as in the elementary definition of the Jones polynomial.

Drawing on insights from quantum field theory, Witten conjectured that skein modules of closed 3-manifolds are finite-dimensional whenever the quantization parameter is generic.  Despite their fairly simple definition, skein modules are very difficult to compute in examples, and so very little numerical evidence existed for this conjecture (this is still the case).

In this talk I'll explain a proof of Witten's conjecture which uses ideas from geometric representation theory -- D-modules, local systems, quantum Hamiltonian reduction.  Time permitting I'll explain how the same batch of ideas resolved a question of Bonahon--Wong, concerning the classification of finite-dimensional representations of skein algebras of surfaces.  This is joint work with Gunningham--Safronov, and Ganev--Safronov, respectively.