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 Rep(Uq(sl2)) \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 Rep(Uq(sl2)) \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