In this talk, I will present a recent construction that enables one to interpolate symmetric monoidal categories by interpolating algebraic structures and their automorphism groups. I will explain how one can recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), the constructions of Knop for wreath products with S_t and GL_t(O_r), where O_r is a finite quotient of a discrete valuation ring, and also the TQFT categories recently constructed from a rational function by Khovanov, Ostrik, and Kononov.

Hopf monads are generalizations of Hopf algebras that bring to bear the rich theory of monads from category theory. Even though this generalization overarches many Hopf-like structures, quasi-Hopf algebras seem to be out of its reach. This talk reports on advances in the study of slack Hopf monads, including how these generalize quasi-Hopf algebras. (Joint work with A. Bruguieres and M. Haim).

In this talk we present classical examples of exact indecomposable module categories over a finite non-semisimple rigid tensor category, as well as recently constructed new examples.

Quantum groups and Nichols algebras appear as symmetries in conformal field theories, as so-called non-local screening operators. I will start by giving a motivation of conformal field theory and explain how this leads from the analysis side to a modular tensor category, quite often one that is well known from algebra. I then explain my recent work that non-local screening operators generate Nichols algebras in this category, and also some recent work with T. Creutzig and M. Rupert that conclude a braided category equivalence between a certain physical model and the representation category of the smallest quantum group \(u_q(\mathfrak{sl}_2),\;q^4=1\).

The \(A_{q,t}\) algebra was introduced by Carlsson and Mellit in their proof of the celebrated shuffle theorem, which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This algebra arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Carlsson and Mellit constructed an action via plethystic operators on the space of symmetric functions which was then realized geometrically on parabolic flag Hilbert schemes by them and Gorsky. The original algebraic construction was then extended to an infinite family of actions by Mellit and shown to contain the generators of elliptic Hall algebra. However, despite the various formulations of \(A_{q,t}\), performing computations within it is complicated and non-intuitive.

In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of \(A_{q,t}\) (at t=-1) and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of \(A_{q,t}\) and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations in this new framework. This purely diagrammatic formulation allows us to lift the operators as certain functors, thus providing a categorification of the \(A_{q,t}\) action on the derived trace of the Soergel category.

I will discuss recent work with T. Gannon in which we provide a ribbon tensor equivalence between the representation category of small quantum SL(2) at q=exp(pi*i/p) and the representation category of the triplet VOA at a corresponding central charge 1-6(p-1)^2/p. Such an equivalence was conjectured in work of Gainutdinov, Semikhatov, Tipunin, and Feigin from 2006.

Quadratic categories are fusion categories with a unique non-trivial orbit from the tensor product action of the group of invertible objects. Familiar examples are the near-groups (with one non-invertible object) and the Haagerup-Izumi cate- gories (with one non-invertible object for each invertible object). Frobenius-Schur indicators are an important invariant of fusion categories generalized from the theory of finite group representations. These indicators may be computed for objects in a fusion category C using the modular data of the Drinfel’d center Z(C) of the fusion category, which is itself a modular tensor category. Recently, Izumi and Grossman provided new (conjectured infinite) families of modular data that include the modular data of Drinfel’d centers for the known quadratic fusion categories. We use this information to compute the FS indicators; moreover, we consider the relationship between the FS indicators of objects in a fusion category C and FS indicators of objects in that category’s Drinfel’d center Z(C).

The categorical version of a module over a ring is the notion of a module category over a fusion category. We will first discuss what it means for a module category to be represented by an algebra and introduce the notion of Morita equivalence of algebras in fusion categories. Sonia Natale and Victor Ostrik described algebras representing the Morita equivalence classes in pointed fusion categories. We will explain how this result can be generalized to group-theoretical fusion categories, based on joint work with Monique Müller, Julia Plavnik, Ana Ros Camacho, Angela Tabiri, and Chelsea Walton.

It was conjectured by Andruskiewitsch, Angiono and Heckenberger that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system; they did in fact prove this for braidings of affine type or when the rank is two. We shall review some tools developed with the intention of proving this conjecture positively, in a work of Angiono and the author, and exhibit the proof for the rank 3 case, as well as for braidings of Cartan type.