Higher Structures

Beginning with a general description of type III von Neumann algebras, I will proceed to define conformal nets. The latter are a model for conformal field theories, and I will discuss various examples there of (free fermions; free bosons; loop groups).
I will then explain the famous result of Kawahigashi, Longo, Mueger according to which conformal nets with finite mu-index have a representations category that is modular.
Depending on the progress of my own understanding, I might introduce a 3-dimensional graphical calculus for computations in conformal nets.

TBA

Geometric representation theory has a revealed a deep connection between geometry and quantum groups suggesting that quantum groups are shadows of richer algebraic structures called categorified quantum groups. Crane and Frenkel conjectured that these structures could be understood combinatorially and applied to low-dimensional topology. In this lecture series we will categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. We will discuss 2-categories and how they serve as a natural environment for turning algebraic problems into planar diagrammatics that can be manipulated using topological intuition.

We will also survey the applications of categorified quantum groups including a new grading on blocks of the symmetric group and Webster's recent work categorifying Reshetikhin-Turaev invariants of tangles.

The graph complex and the Grothendieck-Teichmüller Lie algebra are probably two of the most mysterious (dg) Lie algebras appearing in mathematics. We will recall their definitions and basic properties. Furthermore, we will discuss their role in Deformation Quantization and see explicit maps relating both objects.