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K-State Today

December 13, 2016

Mathematics Colloquium lecture Dec. 13

By Reta McDermott

Timothy Logvinenko, Cardiff University, will present "Generalised Braid Category" as part of the Mathematics Department Colloquium Lecture series at 2:30 p.m. Tuesday, Dec. 13, in 122 Cardwell Hall.

The abstract for the lecture is: Ordinary braid group Br_n is a well-known algebraic structure which encodes configurations of n non-touching strands ("braids") up to continuous transformations ("isotopies"). A classical result of Khovanov and Thomas states that this group acts categorically on the space Fl_n of complete flags in C^n. Logvinenko will begin by reviewing the basics on braid group and flag varieties, and then give a sketch of the geometry involved in the Khovanon-Thomas construction of the categorical action of Br_n on T^* Fl_n.

Logvinenko will then describe a longstanding work-in-progress with Rina Anno: the categorification of generalized 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 generalized braids act categorically on the spaces of full and partial flags in C^n. Categorification will describe our present progress toward it and future expectations.