# 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
- Penrose and Yamada polynomials
- Chromatic polynomials, the flow category, and the Temperlely-Lieb category
- The virtual flow category and graphs in surfaces

- Topological Field theories
- Modular categories
- Reshetikhin-Turaev theory

- Potential other topics
- Fusion categories and conformal field theory
- Tensor categories in statistical mechanics