Perhaps the most (mathematically) well understood 3d TQFT is that of Reshetikhin–Turaev. Famously, the input data for their construction is that of a semisimple modular tensor category (MTC). Attempts at generalizing this construction to the non-semisimple case date back to the 90's with work of Hennings, Lyubashenko and Kerler–Lyubashenko. However, only partial results were achieved. This was until De Renzi et al. defined a 3d TQFT from such non-semisimple modular categories. Importantly, they had to impose an admissibility condition on the cobordism categories they use. My work has been in the direction of defining a once-extended 3d TQFT from this data. However, Bartlett et al. proved that such TQFTs are classified by semisimple modular categories. We will investigate the most natural method of circumventing this. This will lead to the notion of noncompact TQFT. I will then proceed to talk about my work on constructing such a TQFT from the data of a (potentially) non-semisimple MTC, with an emphasis on the key ingredients of this construction. Time permitting, I will also discuss how to extract 3-manifold invariants and a modified trace from such a noncompact TQFT.

Given a periodic link L, we construct a group action on the Khovanov homotopy type defined by Lipshitz and Sarkar. As a result, we prove that the annular Khovanov homology of the quotient link has no larger rank than the Khovanov homology of the periodic link. This is a joint work with Wojciech Politarczyk and Marithania Silvero.

Since the 1980s, it has been well known that there is a close relationship between two-dimensional conformal field theories and three-dimensional topological field theories. This CFT/TFT correspondence provides a tractable example of holography as well as a first example of the symmetry TFT framework.

The Fuchs-Runkel-Schweigert construction is a mathematically precise incarnation of this correspondence and provides a rigorous construction of correlators for rational CFTs using 3D TFTs of Reshetikhin-Turaev type. In this talk, I will review the FRS construction and explain how it can be generalized to non-rational CFTs using the non-semisimple 3D TFTs of De Renzi, Gainutdinov, Geer, Patureau-Mirand, and Runkel.

Linked spaces, originally motivated by applications to TQFTs, simultaneously simplify and generalise stratified spaces. I will briefly introduce the concept and the accompanying exit-path quasi-category construction. To exhibit the nontriviality of the generalisation, I will then consider some fundamental categories (as in "fundamental groupoid") of linked spaces and realise, from a "twist" of the complement of the trefoil knot with a point defect, a two-object category where the hom-set is the modular group PSL(2,Z) and argue that there is no stratified space with this fundamental category.

I will review the notion of a topological (or gapped) domain wall between topological quantum field theories and illustrate an equivalence between domain walls and oplax natural transformations. I will show how this provides a reformulation of Lurie's cobordism hypothesis with singularities.