Categorification of vertex operators
The basic representation of affine Lie algebras can be constructed via vertex
operators from the Fock space representation of the Heisenberg Lie algebra.
I will review and then explain how to categorify this construction. In particular,
this gives a categorical action of affine Lie algebras on (derived) categories
of coherent sheaves on Hilbert schemes of points on ALE spaces (joint with
Projective geometry = dimensional analysis
We establish a dictionary between these two forms of the study of homogeneous
quantities, illuminating their roles within the tannakian philosophy of algebraic
Soergel bimodules and Kazhdan-Lusztig theory I
The category of Soergel bimodules provides an algebraic categorification
of the Hecke algebra H, closely linked to other geometric and representation-theoretic
categorifications. We give an introduction to this category and to Kazhdan-Lusztig
theory, which is roughly the study of categorifications of H and which bases
of H arise from these categorifications. Then we begin to make Soergel bimodules
explicit using diagrammatics (using work of Khovanov-Elias and of Elias).
In this lecture we focus our attention on the Hecke algebra of the general
dihedral group, where the Temperley-Lieb algebra at roots of unity makes a
surprising appearance. Geordie Williamson will continue with part II.
Buildings and Spiders
Certain graphs, called webs, can be used to produce invariant vectors in
tensor products of standard representations of sl(n). I will explain a geometric
viewpoint on these webs, using the geometric Satake correspondence. From this
perspective, we see why basis webs for sl(3) are non-elliptic. This method
also leads to basis webs for any sl(n). This is joint work with Bruce Fontaine
and Greg Kuperberg.
The odd nilHecke algebra and categorified quantum sl(2)
We will discuss an odd analog of the nilHecke algebra together with its
applications to symmetric functions, cohomology of flag varieties, and categorification
of quantum sl(2).
This is a joint work with Alexander Ellis and Mikhail Khovanov.
Heisenberg categorification and Affine Lie algebras
Given a Dynkin diagram of type ADE, we explain how to categorify the corresponding
lattice Heisenberg algebra and its Fock space representation. We'll also explain
the role of these categorifications in the higher representation theory of
the associated affine Lie algebra. This is a joint work with Sabin Cautis.
The extended Khovanov-Lauda calculus and the colored Rouquier complex
I will explain the extended Khovanov-Lauda (KL) calculus, which can be used
to simplify calculations involving categorified divided powers. I will then
explain how the extended calculus can be used to associate a homology complex,
in the Schur quotient of the categorification of quantum sln (explained by
Vaz in his talk), to a colored braid. The latter construction generalizes
the Rouquier complex for ordinary braids. Finally, I will comment on how conjugation
by the colored Rouquier complex can be lifted to give a categorification of
Lusztig's braid group action on quantum sln.
These results are part of various (ongoing) collaborations; with Stosic and
Vaz, with Khovanov, Lauda and Stosic and with Lauda, Stosic and Vaz.
2-representations of finitary 2-categories
In the talk I will try to describe how one constructs and compares principal
and cell 2-representations of finitary 2-categories. Based on a joint work
with Vanessa Miemietz.
A geometric construction of the quantum Frobenius morphism
We will show how one can lift Lusztig's quantum version of the Frobenius morphism
to the level of perverse sheaves on the moduli space of quiver representations,
and discuss its action on the canonical basis.
Invariants of 4-manifolds from Khovanov homology
I'll explain how to define invariants of 4-manifolds (possibly with boundary,
possibly with a link in the boundary) from Khovanov homology. The construction
relies on a new property of Khovanov homology, roughly 'S^3 functoriality'.
I'll try to indicate where the difficulties may lie establishing this property
for the variations of Khovanov homology based on categorified quantum groups
other than SU(2). Finally, I'll explain how this construction is best viewed
as a partial categorification of the 3+1 dimensional TQFT of which the usual
Witten-Reshetikhin-Turaev theories are the boundary, rather than a direct
categorification of a WRT theory.
Adjoint braid group actions
We show that 2-representations of Kac-Moody algebras admit an action of the
associated braid group (on homotopy categories). This is based on the theory
of perverse equivalences.
This is a joint work with Joe Chuang.
A diagrammatic categorification of the q-Schur algebra
In this talk I will explain how to categorify the q-Schur algebra as a quotient
of Khovanov and Lauda's categorified sl(n). I will also explain the connection
with (singular) Soergel bimodules.
This is a joint work with Mackaay and Stosic.
Categorification of Reshetikhin-Turaev invariants
I'll describe one schema for categorifying the tensor products of representations
of quantum groups and how functors between these categorify the ribbon structure
on representations of quantum groups.
Soergel bimodules and Kazhdan-Lusztig theory II
Following Ben's talk, I will continue the discussion of our presentation of
Soergel bimodules via generators and relations (building on earlier work of
Libedinsky and Elias-Khovanov). I will explain why an important role in the
story is played by actions of Coxeter groups on categories, and how this leads
naturally to certain generalisations of the Zamolodchikov equations related
to the platonic solids. If time permits I will discuss some calculations of
the "p-canonical basis" for Hecke algebras, and try to explain why
this is important in representation theory.
Polynomial functors and categorification
The category P of strict polynomial functors was originally introduced to
study the finite generation of affine group schemes, and it has since been
applied widely in algebraic topology. In this talk we describe actions of
affine special linear Lie algebras and the Heisenberg algebra on P, and relate
these to Schur-Weyl duality.
Back to top