# GRASP

This semester, GRASP will focus on diagram categories. Diagram categories are spherical tensor categories whose morphisms are formal linear combinations of pictures called “diagrams”. (A spherical tensor category is a monoidal category with good duals whose morphism spaces are modules over a commutative ring.) The prototypical example is the Temperley-Lieb category.

Diagram categories are central objects in quantum topology, with connections to:

• representation theory of groups and algebras (and their quantum versions)
• subfactors and operator algebras
• quantum invariants of knots and manifolds
• topological (quantum) field theories
• combinatorial topology
• conformal field theory and statistical mechanics

## Schedule

August 29
Introduction to diagram categories via the Temperley-Lieb category and $\operatorname{Rep}(\mathcal{U}_q(\mathfrak{sl}_2))$
Calvin McPhail-Snyder Notes
September 5
Introduction to the Jones polynomial and the Temperley-Lieb category, Part I
Kyle Miller Notes
September 12
Introduction to the Jones polynomial and the Temperley-Lieb category, Part II
Kyle Miller Notes
September 19
Reshetikhin-Turaev invariants, Part I
Aaron Brookner Notes
September 26
Reshetikhin-Turaev invariants, Part II
Aaron Brookner
October 3
Chromatic polynomials and the Temperley-Lieb category
Ben Wormleighton
October 10
Lie algebras and the four-color theorem
Calvin McPhail-Snyder
October 17
Representations of the general linear supergroup
Alexander Sherman Notes
October 24
Guest lecture General relativity on manifolds with corners in the BV-BFV formalism
Giovanni Canepa (University of Zurich)
October 31
No talk
November 7
How to get 3-manifold invariants from link invariants
Calvin McPhail-Snyder
November 14
Representations of the general linear supergroup, part II
Alexander Sherman Notes

## Topics

Tentative list of talks (some bullets may correspond to more than one talk!)

• Basic examples
• Introduction: The Temperley-Lieb category and $\operatorname{Rep}(\mathcal{U}_q(\mathfrak{sl}_2))$
• The Jones polynomial via the Temperley-Lieb category, invariants of tangles via ribbon categories
• Other bracket polynomials, connections to other quantum groups
• Kuperberg spiders
• Deligne’s partition category
• Combinatorial applications
• Lie algebras and the four-color theorem