LMS South West&South Wales Regional Meeting and Workshop 2025

Titles and abstracts for the Graduate Course and Colloquium:

• Pieter Naaijkens (Cardiff University): Introduction to fusion categories

• Ehud Meir (University of Aberdeen): Generators and relations for representation categories 

Abstract: If G is an algebraic group, the representation category of G, Rep(G), is usually thought of as being built from top to bottom: we take all vector spaces, and we restrict our attention to vector spaces that are equipped with a linear G-action. In this talk, I will discuss a construction from the bottom up of the category Rep(G), which can be thought of as a construction by generators and relations. This enables the construction of new families of symmetric monoidal categories that generalize the interpolation categories of Deligne. I will then explain consequences of this construction in the theory of Hopf algebras and the representation theory of general linear groups over the p-adic numbers. 

• André Henriques (University of Oxford): Bicommutant categories

Titles and abstracts for the Workshop:

• Ana Kontrec (RIMS Kyoto): Kazama-Suzuki duality between certain simple W-algebras

Abstract: One of the most important families of vertex algebras are affine vertex algebras and their associated W-algebras, which are connected to various aspects of geometry and physics. The notion of Kazama-Suzuki dual was first introduced in the context of the duality of the N=2 superconformal algebra and affine Lie algebra $\widehat{\mathfrak sl}(2)$. I will present some old and new Kazama-Suzuki dualities between affine W-algebras and vertex superalgebras. This is joint work with D. Adamovic.

• Robert Laugwitz (University of Nottingham): Induced functors between Drinfeld centers of tensor categories

Abstract: I will discuss how adjoints of tensor functors between tensor categories lift to functors between the respective Drinfeld centers. These functors are either lax or oplax monoidal (depending on whether they are induced by right or left adjoints). Under some additional conditions, the lax and oplax structures define a Frobenius monoidal functor. The general constructions can be demonstrated using examples coming from extensions of Hopf algebra.  This talk is based on joint work with Johannes Flake (Bonn) and Sebastian Posur (Münster).

• Adrià Marín Salvador (University of Oxford): Continuous tensor categories

Abstract: In recent years, there has been an increasing interest in “semisimple” tensor categories with a topological space of simple objects and which allow for “direct integrals” of objects in addition to direct sums. In this talk, I will introduce a new model to discuss these tensor categories using C*-algebras and C*-correspondences. I will present the general theory and provide some basic examples that generalize pointed fusion categories and representation categories of finite groups. Finally, I will discuss generalizations of Tambara-Yamagami fusion categories from finite abelian groups to locally compact abelian groups, and their classification. 

• Nivedita (University of Oxford): Bicommutant Categories from Conformal Nets

Abstract: Two-dimensional chiral conformal field theories (CFTs) admit three distinct mathematical formulations: vertex operator algebras (VOAs), conformal nets, and Segal (functorial) chiral CFTs. With the broader aim to build fully extended Segal chiral CFTs, we start with the input of a conformal net.

In this talk, we focus on presenting three equivalent constructions of the category of solitons, i.e. the category of solitonic representations of the net, which we propose is what theory (chiral CFT) assigns to a point. Solitonic representations of the net are one of the primary class of examples of bicommutant categories (a categorified analogue of a von Neumann algebras). The Drinfel’d centre of solitonic representations is the representation category of the conformal net which has been studied before, particularly in the context of rational CFTs (finite-index nets). Bicommutant categories act on W*-categories analogous to von Neumann algebras acting on Hilbert spaces.

• Lewis Topley (University of Bath): Twisted Yangians, finite W-algebras and symplectic singularities

Abstract: Conic symplectic singularities are an interesting class of algebraic varieties introduced by Beauville, and they include many of the interesting Poisson varieties appearing in representation theory. They admit a rigid theory of Poisson deformations and their quantizations, thanks to results of Namikawa and Losev. The Slodowy slice is an example of a Poisson deformation of a symplectic singularity, and the finite W-algebra is often the universal quantization. This powerful tool allows us to solve the rather long-standing problem of finding presentations of even finite W-algebras in types B and C, using a new truncation of a shifted twisted Yangian. This is a joint work with Kang Lu, Yang-Ning Peng, Lukas Tappeiner and Weiqiang Wang.

• Fiona Torzewska (University of Bristol): Comonoidal structures on 2-categories and tensor products of 2-representations

Abstract: I will explain an attempt to categorify the monoidal structure on the category of modules over a bialgebra to k-linear additive 2-categories. This is joint work in progress with Vanessa Miemietz.

• Thomas Wasserman (University of Oxford): A unitary three-functor formalism for commutative Von Neumann algebras

Abstract: Six-functor formalisms are ubiquitous in mathematics, and I will start this talk by giving a quick introduction to them. A three-functor formalism is, as the name suggests, half of a six-functor formalism. I will discuss what it means for such a three-functor formalism to be unitary, and why commutative Von Neumann algebras (and hence, by the Gelfand-Naimark theorem, measure spaces) admit a unitary three-functor formalism. Based on joint work with André Henriques.