QUAntum groups, Categorification, Knot invariants, and Soergel bimodules

August 10-14, 2020
University of Oregon

Speakers Registration Schedule Abstracts Zoom information Slides etc


Titles and abstracts for the minicourse, led by You Qi (University of Virginia):

I. Introduction to Hopfological algebra.
Hopfological algebra, introduced by Khovanov, is a generalization of homological algebra which can be associated to any finite dimensional Hopf algebra. Traditional homological algebra (chain complexes, homotopies, dg-algebras, the derived category, etcetera) is a special case, and all of these concepts can be generalized. Another special case is the category of p-complexes, which categorifies the cyclotomic ring at a prime root of unity. In this talk we discuss the motivation behind categorification at a root of unity, and the basics of Hopfological algebra.

II. Categorified quantum sl(2) at a root of unity.
To categorify a module over the cyclotomic ring, one needs to produce a p-dg category, which is the analogue of a dg-algebra for this kind of Hopfological algebra. When there is already a graded category which categorifies an algebra at generic parameter q (such as the Khovanov-Lauda-Rouquier category for the positive half of the quantum group, or Lauda's categorification of all of quantum sl(2)), one might hope to equip that category with a p-dg structure. However, determining the p-dg Grothendieck group is still difficult, and requires understanding a new kind of direct sum decomposition. This talk focuses on this issue in the context of categorified quantum sl(2) at a root of unity.

III. Towards tensor product representations.
This lecture will focus on categorified tensor product representations of quantum sl(2) at a prime root of unity. In particular, we introduce some new tools which help to transfer categorical actions from one setting to another via a Soergel-like functor.

Nicolle Gonzalez (UCLA): A skein theoretic Carlsson-Mellit algebra

Abstract: In their proof of the shuffle conjecture Carlsson and Mellit introduced an algebra, known as the Aq,t algebra. In this talk I will discuss joint work with Matt Hogancamp entailing a skein-theoretic formulation of the Aq,t algebra specialized at t = q-1, that recovers the original algebraic formulation and provides the prime structure for categorifying its polynomial representation.

Matt Hogancamp (Northeastern University): Curved Hecke categories III

Abstract: In this talk I will discuss a certain category of curved complexes (aka the curved Hecke category) whose construction conjecturally possesses a Koszul self-symmetry which is evidently lacking in the usual category of Soergel bimodules (aka the Hecke category). This curved Hecke category is a generalization of a category constructed in type A in joint work with Eugene Gorsky.

The goal of the talk will indicate the relation (known and conjectured) between the curved Hecke category and the following categories: (1) the usual Hecke category, (2) its Koszul dual, and (3) graded category O. Along the way we will learn the "why" of curved complexes. This is based on joint work with Shotaro Makisumi.

Joel Kamnitzer (University of Toronto): Parabolic restriction for Coulomb branch algebras and categorical g-actions for truncated shifted Yangians

Abstract: Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians. Motivated by the geometric Satake correspondence, we define a categorical g-action on modules for these truncated shifted Yangians. Our main tool is the study of how the Coulomb branch algebra changes when we pass from G, V to L, U, where L is a Levi in G and U is the invariants for a coweight defining L.

Mikhail Khovanov (Columbia University): Frobenius extensions, link homology, and foam evaluation

Abstract: We'll discuss two recent papers, one joint with Louis-Hadrien Robert, the other with Nitu Kitchloo, on a cube of Frobenius extensions, respectively, on a deformation of the Robert-Wagner evaluation.

Shotaro Makisumi (Columbia University): Curved Hecke categories I and II

Abstract: The Hecke algebra admits an involution which preserves the standard basis and exchanges the canonical basis with its dual. This involution is categorified by "monoidal Koszul duality" for Hecke categories, studied in previous joint work with Achar, Riche, and Williamson. In this talk I will explain the following rough statement: "The Koszul dual of the Hecke category is equivalent to the derived category of bimodules for a particular Koszul complex in the Hecke category of the Langlands dual." This is motivated by the curved Koszul duality of Positselski and Burke. Based on joint work with Matt Hogancamp.

Louis-Hadrien Robert (Universite de Geneve): Symmetric Khovanov-Rozansky homology.

Abstract: The aim to this talk is to present a categorification of the symmetric MOY calculus. I'll explain how to derive from this some link homology theories categorifying Reshetikhin--Turaev link invariants associated with symmetric powers (such as the colored Jones polynomial). The main ingredients are some special foams called vinyl foams. Along the way we will see connection with Soergel bimodules and their Hochschild homology groups. (joint with Emmanuel Wagner)

David Rose (University of North Carolina): Webs in type C.

Abstract: A fundamental question one can ask is for a presentation of a given algebraic object via generators and relations. The advent of quantum topology suggested considering this question in the case of categories of quantum group representations, where its resolution gives insight into link invariants and TQFT, and serves as a starting point for categorification. We will discuss results of Kuperberg in rank < 3 and Cautis-Kamnitzer-Morrison in type A, and then turn to recent work of the speaker and collaborators that give the first results on this problem for rank > 2 outside type A. This is based on joint work with Tatham, and time permitting might also touch on work in progress with Bodish, Elias, and Tatham.

Joshua Sussan (CUNY): p-DG theory and relatives of the zigzag algebra.

Abstract: The zigzag algebra is one of the simplest examples of an algebra admitting a categorical braid group action. Via deformations and Koszul duality, there are closely related algebras which also give rise to categorical braid group actions. We will describe these examples before incorporating aspects of p-DG theory into these constructions.

Anne-Laure Thiel (Universite de Caen): A Soergel category for cyclic groups

Abstract: The category of Soergel bimodules plays an essential role in (higher) representation theory and for the construction of homological invariants in knot theory. After having briefly recalled the definition and some facts about this category, the aim of this talk will be to present some of its generalizations. I will focus on a Soergel-like category attached to a cyclic group. I will give a complete description of this category through a classification of its indecomposable objects and study its split Grothendieck ring. This gives rise to an algebra which is an extension of the Hecke algebra of the cyclic group, that can be presented by generators and relations. If time permits, I will mention some partial results about a diagrammatic description of this category: how Catalan numbers appear in this context and how the Temperley-Lieb algebra can describe certain morphism spaces in this category. This is joint work with Thomas Gobet.

Daniel Tubbenhauer (Universitat Zurich): On categories of tilting modules

Abstract: In this talk I will report on the progress in the project of trying to understand categories of tilting modules as categories, meaning the morphisms in these categories and their relations, with the focus being on SL(2) and SL(3). This is joint work with Paul Wedrich.

Emmanuel Wagner (IMJ-PRG): Categorification of 1 and of the Alexander polynomial

Abstract: I'll give a combinatorial and down-to-earth definition of the symmetric gl(1) homology. It is a (non-trivial) link homology which categorifies the trivial link invariant (equal to 1 on every link). Then I'll explain how to use this construction to categorify the Alexander polynomial. Finally, if time permits, I will relate this construction to the Hochschild homology of Soergel bimodules (joint with L-H. Robert).

Ben Webster (Perimeter Institute and University of Waterloo): Howe to translate Gelfand-Tsetlin

Soergel bimodules have natural manifestations in 3 different contexts: combinatorial (i.e. diagrammatic calculus), geometric (i.e. perverse sheaves on the flag variety) and representation theoretic (i.e. Harish-Chandra bimodules/category O).

In each of these contexts, there are generalizations that might interest you: - on the combinatorial side, there is a categorification of the kth tensor power of C^n via KLRW algebras, studied by Khovanov-Lauda-Sussan-Yonezawa in the context of "categorical symmetric Howe duality"; they propose that the action of S_k on this tensor power categories to an action of Soergel bimodules, and prove this for n=2. - on the geometric side, you can replace a flag by a sequence of maps V_1 -> V_2 -> ... -> V_m -> C^n (considered up to isomorphism) without requiring injectivity. You can convolve B-equivariant sheaves on this space X with perverse sheaves on B\G/B. - on the representation theoretic side, you can replace category O by Gelfand-Tsetlin modules (the modules over U(gl_n) which are locally finite under the Gelfand-Tsetlin subalgebra S). Like category O, these carry an action of translation functors, which are effectively a copy of Soergel bimodules.

In fact, all of these generalizations are the same! The modules over appropriate KLRW algebras are a graded lift of the category of Gelfand-Tsetlin modules, and Koszul dual to the category of B-equivariant perverse sheaves on X, and all of these equivalences are compatible with the actions of Soergel bimodules. I'll try to explain this result, and how whatever your perspective, none of the objects involved are as scary as you might think.

Paul Wedrich (Universitat Bonn): Invariants of 4-manifolds from Khovanov-Rozansky link homology

Abstract: Ribbon categories are 3-dimensional algebraic structures that control quantum link polynomials and that give rise to 3-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the gl(N) quantum link polynomial, to obtain a 4-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth 4-manifolds. The technical heart of this construction is the functoriality of Khovanov-Rozansky homology for links in the 3-sphere. Based on joint work with Scott Morrison and Kevin Walker.

Geordie Williamson (University of Sydney): Miraculous Treumann-Smith theory and geometric Satake

Abstract: This talk will be about geometric approaches to the representation theory of reductive algebraic groups in positive characteristic p. A cornerstone of the geometric theory is the geometric Satake equivalence, which gives an incarnation of the category of representations as a category of perverse sheaves on the affine Grassmannian. It is surprising that several "easy" algebraic facts have no explanation on the geometric side. This is frustrating, as several deep conjectures appear to point to a clear relation with the geometry of the affine Grassmannian. One dreams of using geometric Satake as a fundamental localisation theorem (akin to Beilinson-Bernstein localisation for complex semi-simple Lie algebras) from which one can deduce structural results in representation theory. There are now several pieces of evidence in the Langlands program for the viability of this philosophy.

I will explain a recent step towards this dream. One of the "easy" algebraic facts alluded to above is the linkage principle, which decomposes the category of representations into "blocks" controlled by the affine Weyl group. In joint work with Simon Riche, we explain this decomposition via a certain mod p version of hyperbolic localization, known as Treumann-Smith theory. The theory has its roots in Smith's study of Z/pZ actions on spheres in the 1930s, and was upgraded to sheaves a few years ago by Treumann. We also deduce a new proof of the Lusztig character formula (for large p) and a conjecture of ours on characters of tilting modules (for all p).