Ymchil

Traethawd ymchwil

Categorification of generalised braids

Categorification problems are a mix of algebraic geometry and representation theory. They study representations by functors on derived categories of algebraic varieties. The derived category is the ultimate homological invariant of a variety. Acting on it categorifies acting on more conventional invariants such as cohomology or K-theory.

Unfortunately derived categories were built around the seriously flawed axiomatics of triangulated categories. The theory of DG-enhancements was conceived in early 90s by Bondal and Kapranov to fix these flaws. It was rapidly developed over the last decade in a series of revolutionary results by Keller, Toen, et al. Many previously inaccessible problems are now within our reach.

I am studying these new techniques and applying them towards proving a long-standing conjecture: the category of generalised braids acts on the derived categories of (the cotangent bundles of) full and partial flag varieties. Generalised braids 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.

Goruchwyliaeth