Skein theory

by David Jordan (Edinburgh)

Thursday 22 May 2025, 14:45 → 15:45 Europe/Berlin

H4 (Geomatikum)

Skeins in 3-manifolds are a mathematical model for Wilson line observables in quantum Chern-Simons theory.  The past five years has seen the opening of many new directions in skein theory, and I will only be able to touch on a small selection in this talk.  I will first give an overview of recent results by many groups computing dimensions of generic skein modules for many closed 3-manifolds.  I will then turn to generalisations:  I'll explain a finiteness theorem for skein categories (which we conjectured with Gunningham and Vazirani, and proved with Detcherry), and a two finiteness theorems for skein modules of 3-manifolds with boundary (both conjectured by Detcherry, one proved by Detcherry-Belletti and another which we proved with Romaidis).  Finally, I'll discuss what appears to be a promising new direction in skein theory developed in joint work with Jennifer Brown: that of skein theory in the presence of defects, such as arise from central 1-form symmetries, Weyl group symmetries, and parabolic induction and restriction.  The latter is closely related to abelianisation and to quantum curves in mathematical physics.

I will assume no prior knowledge of skein theory in the talk, and I will aim to give some of the interesting ideas without proofs or precise details.