Time: 14h30, Saturday, November 5, 2022

Abstract: Categorification is the process of lifting a mathematical object/theory to a higher categorical level. The resulting objects, which usually possess much richer structures than the original, can then be used to shed light on phenomena that are otherwise invisible at the decategorified level. Geometric categorification takes categorification one step further: it requires that the resulting theory "comes from geometry," which allows one to access powerful tools and intuition from geometry.

In this talk, I will illustrate this philosophy from the perspective of geometric representation theory and categorified knot invariants. More concretely, I will describe the new theory of graded sheaves, which gives a uniform way to construct geometric categorifications of q-deformed objects (such as Hecke algebras) which appear naturally in these two worlds. Classically, such a construction was only available in very special cases, introduced by Beilinson--Ginzburg--Soergel and further developed by Achar, Riche, Rider, and others. This is joint work with Penghui Li.