Perverse Schobers and the McKay Correspondence

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.

The student working on this project would study these new techniques and apply 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.

This remained open for a decade because most of its building blocks weren't discovered yet: spherical, P- and Grassmanian functors and the equivalences they induce. Of these, only spherical functors are now completely worked out: via DG-techniques by Anno and Logvinenko. P-functors we only have limited examples of, while Grassmanian functors are an uncharted territory.

The student working on the project would join an international team of collaborators spanning UK, US, Japan and Denmark. He will work either on further developing the currently in-demand theories of spherical and P-functors, or on computing our first examples of Grassmanian functors, or on computing the generalised braid relations in the conjectured categorical action. If successful, his work would contribute to the ambitious goal of turning this conjectured action into reality.