• Symplectic Geometry and Representation Theory
• Winter School

• Schedule
• Participants
• Poster

Khovanov and Rozansky introduced a knot homology theory which categorifies the HOMFLY polynomial. This homology has a lot of interesting properties, but it is notoriously hard to compute. I will introduce HOMFLY homology and discuss its conjectural relation to algebraic geometry of the Hilbert scheme of points on the plane. For torus links, this conjecture was recently confirmed by Elias, Hogancamp and Mellit. I will also outline a possible strategy of the proof using the recent work of Elias and Hogancamp on categorical diagonalization. All notions will be introduced in lectures, no preliminary knowledge is assumed.

Lecture I: Introduction to HOMFLY homology
Lecture II: Introduction to Hilbert schemes
Lecture III: Categorical diagonalization
Lecture IV: From braids to sheaves on Hilbert scheme

Recorded Lecture I

Recorded Lecture II

Recorded Lecture III

Recorded Lecture IV

TBA

Recorded Lecture I

Recorded Lecture II

Recorded Lecture III

Recorded Lecture IV

The cyclotomic KLR algebras are certain quotients of the quiver Hecke algebras, or Khovanov–Lauda–Rouquier algebras. These algebras are important because they categorify the highest weight representations of the corresponding quantum group. We will start by discussing these algebras in arbitrary type, where surprisingly little is known. We then focus on type A where Brundan and Kleshchev’s graded isomorphism theorem tells us that these algebras are isomorphic to the cyclotomic Hecke algebras of type A, which are a family of deformation algebras that include as special cases the group algebras of the symmetric groups and their Iwahori–Hecke algebras. The main aim of my lectures is to understand the Ariki–Brundan–Kleshchev categorification theorem in terms of the representation of the cyclotomic KLR algebras of type A.

Recorded Lecture I

Recorded Lecture II

Recorded Lecture III

Recorded Lecture IV

The aim of this series of lectures is to introduce the audience to the general area of 2-representation theory of finitary 2-categories. The main emphasis will be made on structural properties of finitary 2-categories and study of classification problems of various classes of 2- representations, in particular, that of simple transitive 2-representations. Both less advanced and more advanced methods will be discussed. The 2-category of projective bimodules over a finite dimensional algebra will be used as a running example. 

Please note: Unfortunately a recording of the first lecture was not possible.

Recorded Lecture II

Recorded Lecture III

Recorded Lecture IV

TBA