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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.
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.
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)
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).
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.