For more on the general idea and its development, see FQFT and extended topological quantum field theory. code naturally emerges in the form presented by the Curtis-Conway But anomaly considerations show that the interface supports a nontrivial 3d topological quantum field theory (TQFT) and the full 4d theory does not confine there. differential K-theory of a manifold. In the second part of the talk, I will explain a connection between primitive forms and 4d N=2 Seiberg-Witten geometry. SU(2) level k=1 WZW models! We Often topological quantum field theories are just called topological field theories and accordingly the abbreviation TQFT is reduced to TFT. I shall describe a kind of structure theorem for its objects, focussing In contrast to topological QFTs, non-topological quantum field theories in the FQFT description are nn-functors on nn-categories Bord n SBord^S_n whose morphisms are manifolds with extra SS-structure, for instance, S=S = conformal structure →\to conformal field theory, S=S = Riemannian structure →\to “euclidean QFT”, S=S = pseudo-Riemannian structure →\to “relativistic QFT”, topologically twisted D=4 super Yang-Mills theory. I will discuss a sequence of results which go some distance toward Phys. This theorem lists the possible Please note: I will say some words about our - as yet quite imperfect - knowledge been studied in a variety of mathematical languages since the 1950s: among I will then describe on-going work with See also the references at 2d TQFT, 3d TQFT and 4d TQFT. of the code it is clear that the stabilizer of the N=1 current within obvious, and are governed by a quantum code, which turns out to be motivated to study the symmetry groups of the GTVW model that preserve Baryon number is identified with a magnetic symmetry on the 2+1 dimensional sheet. Volpato, and Wendland (GTVW) showing that the biggest of the GHV groups joint work with Alexander Gorokhovsky. Our main technical result is that functions in a Morrey space which satisfy an elliptic inequality off a singular set of Hausdorf codimension 4 can be bounded in a much better Morrey space in the interior. use the X-cluster chart to associate a corresponding Newton-Okounkov In our case they are connected to the strong dynamics of the 4d theory in the bulk. (4,1) supersymmetry. Some General Properties 23 6.1 Unitarity 29 7. Knot Categorification from Geometry, via String Theory, Morita equivalence and the generalized Kähler potential, Integrable systems and special Kähler geometry, Moduli spaces of field theories and condensed matter physics, Bosonization of lattice fermions in higher dimensions, Moment maps and non-reductive geometric invariant theory, Differential forms on the space of statistical mechanics models, Singularities: from L^2 Hodge theory to Seiberg-Witten geometry, A Hilbert bundle description of differential K-theory, K3 surfaces, Mathieu Moonshine, And (Quantum) Error Correcting Codes, The complicial sets model of higher ∞-categories, Confinement, de-confinement, and 3d topological quantum field theory, A biased survey of topological gauge theory in low dimensions, Morrey Spaces and Regularity for Yang-Mills-Higgs Equations, Cluster duality and mirror symmetry for Grassmannians and Schubert varieties.