Série d'exposés sur les
"Outils algébriques en théorie conforme des champs"

Résumé : We give an overview about applications and appearances of Grothendieck-Verdier duality, a weakening of rigidity, in conformal field theory, quantum topology, and representation theory. This is an exciting field, with many developments in recent years and we will give (a necessarily incomplete) overview about these trying to highlight some open questions. A focus will be on connections between algebra and low-dimensional topology.

Résumé: Originally arising from two-dimensional conformal field theory, vertex algebras have also found wide-ranging applications across mathematics (eg. the Monster group and the Moonshine conjecture, modular forms and combinatorial identities, and the geometric Langlands programme just to mention a few). These two lectures aims to discuss the basic concepts and some important challenges of the theory of vertex algebras. We will first introduce our main objects of interest and illustrate with the concrete and fundamental examples arising from Lie theory. The second lecture will focus on the representation theory of vertex algebras and the deep connection with modular tensor categories.

Résumé : Grothendieck-Verdier categories are monoidal categories with a duality structure generalising rigid duality. Examples include categories of bimodules, modules over Hopf algebroids, and modules over vertex operator algebras. Unlike rigid categories, Grothendieck-Verdier categories admit non-invertible associativity constraints. These can be studied using a surface-diagrammatic calculus, extending Joyal and Street’s string-diagrammatic calculus into a third dimension. I will illustrate this calculus in the context of Frobenius algebras and higher Frobenius–Schur indicators. To make the geometry tangible, I will share 3D-printed surface diagrams created with homotopy.io. If time permits, I will also present coherence theorems for Grothendieck-Verdier categories. These combinatorial results, from ongoing joint work with Christian Reiher and Christoph Schweigert, simplify calculations in the surface-diagrammatic calculus.

Résumé: Originally arising from two-dimensional conformal field theory, vertex algebras have also found wide-ranging applications across mathematics (eg. the Monster group and the Moonshine conjecture, modular forms and combinatorial identities, and the geometric Langlands programme just to mention a few). These two lectures aims to discuss the basic concepts and some important challenges of the theory of vertex algebras. We will first introduce our main objects of interest and illustrate with the concrete and fundamental examples arising from Lie theory. The second lecture will focus on the representation theory of vertex algebras and the deep connection with modular tensor categories.

Résumé : In the first half of the talk, I will explain how string-net models, and skein theory more broadly, can be organized as an operadic construction. In the second half of the talk, I will explain how to use string-net models to construct a consistent system of correlators in a rational conformal field theory with defects in all codimensions, and how this construction can be understood functorially using double categories.