Les journées de topologie quantique ont pour but de réunir, dans un format proche de celui d’un séminaire d’équipe, des chercheuses/chercheurs qui s’intéressent à la topologie quantique dans un sens assez large. Elles auront lieu en alternance entre Paris et Dijon. La première édition a eu lieu le lundi 21 novembre 2022 à Paris.
Based on the theory of quantum dilogarithms over locally compact Abelian groups, I will talk about a particular example of a quantum dilogarithm associated with a local field FFF which leads to a generalized 3d TQFT based on the combinatorial input of ordered Δ\DeltaΔ-complexes. The associated invariants of 3-manifolds are expected to be specific counting invariants of representations of π1\pi_1π1 into the group PSL2FPSL_2FPSL2F. This is an ongoing project in collaboration with Stavros Garoufalidis.
I will start by reviewing deformation quantisation of algebras, and explain how we in a similar spirit can define deformation quantisation of categories. The motivation is to understand how deformation quantisation interacts with categorical factorization homology, or more explicitly: how deformation quantisation interacts with “gluing” local observables to obtain global observables. One important and well-known example of factorization homology is given by skein categories, which I will briefly introduce. We generalise the theory of skein categories to fit into the deformation quantisation-setting, and use it as a running example. This is based on joint work (in progress) with Corina Keller, Lukas Müller and Jan Pulmann.
Skein modules are invariants of 3-manifolds which were introduced by Józef H. Przytycki (and independently by Vladimir Tuarev) in 1987 as generalisations of the Jones, HOMFLYPT, and Kauffman bracket polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Over time, skein modules have evolved into one of the most important objects in knot theory and quantum topology, having strong ties with many fields of mathematics such as algebraic geometry, hyperbolic geometry, and the Witten-Reshetikhin-Turaev 3-manifold invariants, to name a few. One avenue in the study of skein modules is determining whether they reflect the geometry or topology of the manifold, for example, whether the module detects the presence of incompressible or non-separating surfaces in the manifold. Interestingly enough this presence manifests itself in the form of torsion in the skein module. In this talk we will discuss various skein modules which detect the presence of non-separating surfaces. We will focus on the framing skein module and show that it detects the presence of non-separating 2-spheres in a 3-manifold by way of torsion.