Department of Mathematics Seminar

Research Centre: Department of Mathematics

Title: Generalised braid category

Abstract:

Ordinary braid group Br_n is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that this group acts categorically on the cotangent bundle of the space Fl_n of complete flags in C^n. I will begin by reviewing the basics on braid group and flag varieties, and then give a sketch of the geometry involved in the Khovanov-Thomas construction.

I will then describe an ongoing project with Rina Anno (Kansas): the categorification of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. A decade old conjecture states that generalised braids act categorically on the cotangent bundles of the spaces of full and partial flags in C^n. I will describe our present progress towards it and future expectations.