Skein modules of 3-manifolds and their finite dimensionality

by Giulio Belletti (UCLouvain)

Thursday 2 Oct 2025, 10:00 → 11:00 Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

In the first part of the talk (approximately 50 minutes), I will give a broad overview of skein modules of 3-manifolds, which are modules that somehow “encode” 1-dimensional submanifolds in a way similar to homology. They are rich algebraic objects with interesting connections to physics, representation theory, knot theory (via the Jones polynomial), and non-commutative algebra. I will present some examples and explain the kinds of questions that are natural to ask about these objects.

In the second part of the talk (approximately 30 minutes), I will move on to the main result of the talk: the finite dimensionality of the skein modules of closed manifolds. This striking result was first proved by Gunningham–Jordan–Safronov (answering a conjecture of Witten) using very sophisticated algebraic techniques, including DQ-modules and the Kashiwara–Schapira theorem. I will sketch a new proof, joint with Renaud Detcherry, of finite dimensionality, which is essentially obtained directly from the definitions. I will also explain some interesting consequences that follow naturally from our method.