Schedule, Notes/Handouts & Videos

Classical invariant theory studies the ring of invariants $\Bbbk[x_1, \dots, x_n]^G$ under the action of a group $G$ on a commutative polynomial ring ${\Bbbk}[x_1, \dots, x_n]$. To extend this theory to a noncommutative context, we replace the polynomial ring with an Artin-Schelter regular algebra $A$ (that when commutative is isomorphic to a commutative polynomial ring), and study the invariants $A^G$ under the action of a finite group, or, more generally, a finite dimensional Hopf algebra. We will discuss some open questions on generalizing classical results to this context.

We will discuss some major open problems in the areas of subfactors and unitary tensor categories, together with some related more approachable questions to get started.

The generalization of the Jones polynomial led to constructions of link and 3-manifold invariants as well as of representations of surface mapping class groups.

The talk will focus on the general question of how much of the topology and the Thurston geometric picture of low dimensional manifolds is reflected in these invariants and structures.

I will survey several open problems and conjectures in the area and report on recent progress in this direction.