Baby Pizza ZMP Seminar

Introduction to Quantum Groups (Christopher Raymond)

I have the pleasure of introducing quantum groups, which (in)famously are not quantum and not groups. My goal for the talk will be to describe a little bit of the physics that motivates the definition of what we now refer to as a "quantum group", then give some version of a definition, and finish off with some motivation for the talks that are to come. I'll be sacrificing some precision and detail in order to make things as understandable as possible for those with either physics or math backgrounds.

Notes from the talk.

Modular Tensor Categories associated to Quantum Groups (Felipe Ruiz)

Modular Tensor Categories are relevant objects in mathematical physics partly because they encode algebraic data of  3d TQFT's and 2d RCFT´s. The goal of this talk is to show how certain quantum groups produce modular tensor categories and state the Kazhdan-Lusztig correspondence relating them to categories of representations of affine vertex operator algebras.

Notes from the talk.

Yangians at three levels of complexity (Lukas Hardi)

This talk surveys three interconnected appearances of Yangians in mathematical physics. We begin with the algebraic Bethe Ansatz for the Heisenberg spin chain, emphasizing the RTT presentation of the Yangian and its use in solving the model. We then pass to the construction of the rational spin Ruijsenaars-Schneider model of N particles as a representation of the affine Yangian by way of Schur--Weyl duality. Finally, we take N to infinity, which recovers the Fock space representation of the affine Yangian, and realize it via a geometric action on the equivariant cohomology of instanton moduli spaces. Time permitting, we discuss how this implies the AGT correspondence.

From quantum groups to quantum topology (Bangxin Wang)

We will see how quantum groups give rise to topological invariants—from the Jones polynomial to the Hennings invariant—and identify the underlying representation-theoretic input in each case.

The reflection equation (Tyler Franke)

Similarly to how we first met the Yang-Baxter equation, the reflection equation appears as a consistency condition for factorised  scattering on the boundary of an integrable system. The associated algebraic structure has many facets: It  ``duals" of quantum groups, has a coend construction, and is prominent in the study of braided module categories. I will review these points, and mention connections to quantum topology and integrable systems.