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.

I will introduce the notion of 2-vector bundles, which are a categorified version of vector bundles. This notion is based on the idea that the bicategory of 2-vector spaces is the bicategory of algebras, bimodules, and intertwiners. I will recall the definition of that bicategory, which leads into the definition of 2-vector bundles. As time permits, I will discuss connections to string geometry, and extended TQFT, and classifying results. This is all based on work with Matthias Ludewig and Konrad Waldorf.

We will survey some results of stated $SL_n$ skein algebras and show how to use them to study the ordinary skein algebras of surfaces. We will discuss the integrality of the skein algebra, the injectivity of the cutting homomorphism, and the structure of the skein algebras of the bigon and the triangle. The talk is based on joint work with F. Costantino, J. Korinman, A. Sikora, and T. Yu.