Thank you for your interest in our learning seminar on categorification.
Email any of the organizers if you'd like to be added to the Google group and receive weekly notifications about the upcoming talks and related activity. You'll also receive the Zoom link, title, and abstract through email.
Starting September 4, 2020, Weekly talks will be given over Zoom on Friday, 10 am Pacific US time / 1 pm Eastern US time / evening times in Europe.
Talks will be 35minutes + 5 min break + 20 minutes followed by 10 minutes of questions, but feel free to stay afterwards to chat if you'd like.
We plan to follow the usual Zoom etiquette: Keep your microphone muted during the talk. If you have questions, please type them into the chat, and one of the organizers will notify the speaker of questions during a natural pause. Questions and discussion are encouraged!
Below you'll find our schedule of talks, as well as a Google calendar with the same information. If video of a talk is recorded, it will be posted under the speaker's abstract in the schedule of talks.
The organizers,
Ben Elias, Nicolle Gonzalez, Mikhail Khovanov, and Melissa Zhang
GNR conjectures and categorified Young idempotents, a survey
In this series of 2 talks, I intend to give an overview of the interconnected subjects of Gorsky-Negut-Rasmussen conjectures and categorified Young idempotents, including nuggets of y-ification and categorical diagonalization. The first talk will focus on the GNR conjectures from the perspective of categorified Young idempotents, including several concrete open problems. The subject of the second talk will depend somewhat on interest, but the plan will be to carefully describe the categorified symmetrizing idempotent and its role in performing computations with Soergel bimodules in type A, including Khovanov-Rozansky homology of torus links.
Computing bases in linear 2-categories using rewriting theory
In representation theory, several families of linear monoidal categories admitting diagrammatic presentations by generators and relations emerged. One of the important questions for such a linear category is to compute bases for every set of morphisms, and to that purpose various methods are developed. In this talk, we will discuss how these questions can be tackled using rewriting theory. Rewriting approaches in linear contexts are well known for associative, commutative and non-commutative algebras, relating some classical algebraic concepts such as Gröbner bases and the Bergman diamond lemma, but are more complicated for higher-dimensional linear categories. We will describe the framework of rewriting with string diagrams in linear 2-categories, and explain how to reach two fundamental computational properties: termination and confluence. We will introduce extensions of these constructions to the case of rewriting modulo isotopies in pivotal linear 2-categories. As an illustration, we will show how these methods apply on the 2-categorification of quantum groups introduced by Khovanov, Lauda and Rouquier.
Extra structures on Khovanov-Rozansky homology
Interpretation of the triply graded Khovanov-Rozansky homology of a positive link as the cohomology of a certain braid variety suggests existence of certain extra structures. I will explain how y-ification and an action of tautological classes naturally appear in this context. A proof of a conjecture of Dunfield-Gukov-Rasmussen on an interesting symmetry in Khovanov-Rozansky homology of knots will be a consequence.
Coherent categorification of quantum loop sl(2)
We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpotent cone of type A. This is a first attempt to try to understand coherent categorification of quantum loop algebras in terms of quiver Hecke algebras. If time permits, we will also explain an application to relating two different monoidal categorifications of the cluster algebra structure on the open quantum unipotent cell of affine type A_1. This is based on joint work with Michela Varagnolo and Eric Vasserot.
Notes of talk
Representations of Quiver Hecke algebras
Quiver Hecke algebras describe the categorification of the enveloping algebra of the positive part of Kac-Moody algebras. I will explain how they arise from the geometry of quiver varieties. I will introduce some classes of modules over quiver Hecke algebras, in relation with different bases of quantum groups. I will discuss the cluster structures related to functions on the positive part of a Kac-Moody group. This will be illustrated with the case of affine sl2.
Slides of talk
Stated Skein Modules and Algebras
This talk will explore the stated skein algebras of surfaces and the stated skein modules of marked 3-manifolds. Ordinary skein modules are made of linear combinations of framed links in a manifold subject to skein relations. When generalizing to stated skein modules, we also consider arcs between boundary components with an assigned state. Our main goal will be to describe some of these algebraic structures using a few key homomorphisms. An emphasis on the connection to quantum groups will be made. This is joint work with Thang Le.
Faithfulness of 2-representations of sl(2)
A result about the representation theory of sl(2) states that an element of the enveloping algebra of sl(2) is zero if and only if it acts by zero on every finite dimensional simple module. In this talk, we will describe a categorification of this result. Associated with sl(2) is a 2-category U introduced by Rouquier and Khovanov-Lauda. The (n+1)-dimensional simple sl(2)-module lifts categorically to a 2-representation L(n) of U. We will prove that a complex of 1-morphisms of U is null-homotopic if and only if its image in L(n) is null-homotopic for every non-negative integer n.
We will give two applications of this result to the Rickard complex T. The Rickard complex is a complex of 1-morphisms of U categorifying the simple reflection of SL(2) and providing derived equivalences on 2-representations of U by work of Chuang-Rouquier. We will prove that T is invertible in the homotopy category of U and that there is a homotopy equivalence between TE and FT[-1], where E and F are the Chevalley generators of U."
Doubly Periodic Tableaux, DAHA Representations, and Quantum Groups at Roots of Unity
In this talk, we begin by introducing the set of doubly periodic tableaux (DPT). These are like standard Young tableaux, but instead of filling a Young diagram with integers, we fill the entire plane and are subject to certain periodicity conditions. These tableaux generate a vector space that naturally carries a representation of the type-A doubly affine Hecke algebra (DAHA). This restricts to a useful action of the type-A affine Hecke algebra (AHA). Each DPT determines a periodic lattice path, and restricting to those DPT for a fixed lattice path, we find the AHA action is isomorphic to another AHA action on certain intertwiner spaces for quantum groups at roots of unity. Thus, using the combinatorics of DPT, we are able to give formulas for dimensions of such intertwiner spaces. Finally, if time permits, we discuss the relationship to a conjecture of Morton-Samuelson involving the skein algebra on the torus, and outline a possible proof. This is joint work with Lea Bittmann, Anton Mellit, and Chiara Novarini.
Working with curvature (with applications to geometric modular representation theory)
Abstract: Homological algebra is commonly considered to revolve around the equation d^2 = 0. In several recent works in modular geometric representation theory, there has appeared a kind of homological algebra where d^2 does not vanish but is described in terms of a distinguished "curvature" element. The main goal of my talk will be to explain how curvature arises naturally in homological algebra, and to define some basic concepts to allow you to work effectively with it. In particular, I will discuss the notions of cdg (curved dg) category, cdg (co)algebra and (co)modules, and Koszul duality in this setting. I will briefly mention some applications: re-interpretation of modular nearby cycles of Achar, of modular free-monodromic tilting sheaves of Achar--M--Riche--Williamson, and a particularly nice form of the Koszul duality phenomenon for the Hecke category (joint with Matt Hogancamp).
Verlinde rings, eigenfunctions and DAHA actions
In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A.
I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.
Fourier matrices for G(d,1,n) via quantum general linear groups
In the classification of unipotent characters of finite reductive groups, Lusztig has introduced some N-modular data, which can be interpreted in terms of Drinfeld double of some finite groups. These modular data depends only on the Weyl group of the reductive group and not on the field of definition of the finite reductive group. A combinatorial notion of unipotent characters for the complex reflection group G(d,1,n) has been developed by Malle, and to each family of unipotent characters, he associated a Z-modular datum. The goal of this talk is to derive a categorification of these modular data from the representation of some quantum groups. We will first recall what categorifying an N-modular datum means, and then explain how to generalize to Z-modular data using slightly degenerate categories. Since the building blocks of these modular data are the exterior powers of the (renormalized) character table S of the cyclic group of order d, we will focus on these matrices and explain how the representations of U_q(gl(m))$ give a categorification of the m-th exterior power Wedge^m(S), the parameter being a root of unity of order 2d.
Singular Hodge theory for matroids II: Kazhdan—Lusztig polynomials
Kazhdan—Lusztig polynomials of matroids are analogous to Kazhdan—Lusztig polynomials of Coxeter groups. In both theories, the polynomials have a purely combinatorial recursive definition of the polynomials, and it is not at all clear from the definition that they will have non-negative coefficients. When a Coxeter group is a Weyl group, KL positivity can be proved by studying the intersection cohomology groups of Schubert varieties. In the general case, there are no varieties, but you can use the theory of Soergel bimodules as a substitute. A similar story holds in the matroidal setting, using some of the same structures that were introduced in last week’s talk. I’ll tell you all about it! Like last week, this is joint work with Tom Braden, June Huh, Jacob Matherne, and Botong Wang.
Singular Hodge theory for matroids I: the Top-Heavy Conjecture
A theorem of de Brujin and Erdős says that a collection of n lines in a projective plane intersect in at least n points. This is a special case of the more general "Top-Heavy Conjecture" of Dowling and Wilson (1974). This conjecture was formulated for all matroids and was proven for hyperplane arrangements (realizable matroids) by Huh and Wang in 2017. A key idea in their proof is to use the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement. For arbitrary matroids, no such variety exists; nonetheless, I will discuss a proof of the Top-Heavy Conjecture for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory. This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.
Categorical braid group actions, the cactus group, and crystals
Let $\mathfrak{g}$ be a semisimple, simply-laced Lie algebra, and $C$ an abelian categorification of an integrable $U_q(\mathfrak{g})$-representation $V$. Chuang and Rouquier construct a collection of equivalences on $D^b(C)$, known as Rickard complexes, which are subsequently shown by Cautis and Kamnitzer to give an action of the braid group on $D^b(C)$. I will discuss the Rickard complex corresponding to the longest element of the Weyl group and show it is a perverse equivalence, generalizing a result of Chuang and Rouquier in the $\mathfrak{sl}_2$ case. I will also explain how this result allows us to recover an action of the cactus group on the crystal for $V$, originally defined via generalized Schützenberger involutions. This is joint work with Tony Licata, Ivan Losev, and Oded Yacobi.
The seminar will be on hiatus and will resume January 29th, 2021.
Green’s theory of cells in categorification
What have Kazhdan–Lusztig cells, cellular algebras and highest weight categories in common? They all originate in Green's work on semigroups. This talk will be an introduction to Green's theory of cells with an emphasis on examples, from semigroups to algebras to categorification.
Conference website HERE
No Seminar
On the spectrification of Khovanov homology
In the first talk, we will try to give a complete construction of the stable homotopy refinement of (sl_2) Khovanov homology, by constructing a cube in the Burnside 2-category and describing one way such a cube gives rise to a stable homotopy type. In the second type, we will discuss a variety of further topics: a brief summary of the proof this spectrum is a link invariant, roughly how it relates to our original construction via flow categories, some computations, and, if time permits, some remarks about applications and work in progress on functoriality. This is joint work with Tyler Lawson and Sucharit Sarkar, and some of the constructions first appeared in work of Hu-Kriz-Kriz.
Equivariant annular Khovanov homology
Asaeda-Przytycki-Sikora defined a homology theory for links in interval bundles over surfaces. The special case of the thickened annulus is known as annular APS homology or annular Khovanov homology. I will explain how to build an equivariant version of this theory and outline some of its properties.
Y-ification of link homology and its applications
I will discuss a generalization ("y-ification") of Khovanov-Rozansky link homology which depends on one additional variable per link component. The y-ified homology has interesting link splitting properties which generalize the work of Batson and Seed in Khovanov homology. As an application, I will explicitly compute Khovanov-Rozansky homology of all (n,kn) torus links. This is joint work with Matt Hogancamp.
Higher representations and cornered Heegaard Floer homology
In the first talk I will discuss recent work with Raphael Rouquier, focusing on a higher tensor product operation for 2-representations of Khovanov's categorification of U(gl(1|1)^+).
In the second talk, I will discuss recent work with Raphael Rouquier, focusing on gl(1|1)^+ 2-representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, along with a tensor-product-based gluing formula for these 2-representations that expands on work of Douglas-Manolescu.
A categorification of the Lusztig-Vogan module
Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lusztig-Vogan in the 80's in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I'll describe a categorification of this module using Soergel bimodules, with a focus on examples.
On Kazhdan-Lusztig theory for affine Weyl group
Kazhdan-Lusztig polynomials are a big mystery. On a recent work with Leonardo Patimo (following Geordie Williamson) we were able to calculate them explicitly for affine A2. We dream of a similar description for all affine Weyl groups, but it seems like an incredibly difficult program. I will explain some new results in this direction and what we believe that is doable. Another part of this project is to produce an approach towards the following question: for a given element in an affine Weyl group, what are the prime numbers p such that the p-canonical basis is different from the canonical basis? This is a joint project with Leonardo Patimo and David Plaza.
Generalizing Rasmussen's s-invariant, and applications
I will discuss a method to define Khovanov and Lee homology for links in connected sums of copies of S1 x S2. From here we can define an s-invariant that gives genus bounds on oriented cobordisms between links. I will discuss some applications to surfaces in certain 4-manifolds, including a proof that the s-invariant cannot detect exotic 4-balls coming from Gluck twists of the standard 4-ball. If time allows, I will also discuss our new combinatorial proof of the slice Bennequin inequality in S1 x S2. All of this is joint work with Ciprian Manolescu, Marco Marengon, and Sucharit Sarkar.
Coulomb branches and cylindrical KLRW algebras
Remarkable work of Braverman-Finkelberg-Nakajima has constructed, based on some fancy quantum field theory, a fascinating collection of spaces called "Coulomb branches." The definition of these involves the geometry of affine Grassmannians, and thus is not so easy for many people think about. Luckily, the algebraic tools for understanding the most important cases of these algebras is quite broadly known in the categorification community: the diagrammatics of KLRW algebras. I'll explain how to make this connection, and the consequences for representation theory and knot homology.
A tour via examples of Beilinson-Bernstein localization
The Beilinson-Bernstein localization theorem provides an equivalence between the category of representations of a semisimple Lie algebra with a fixed infinitesimal character and a category of twisted D-modules on the associated flag variety. When it was established in 1981 as a mechanism for proving the Kazhdan-Lusztig conjectures, it demonstrated the immense power of using geometric techniques to tackle problems in representation theory. Now, 40 years later, a variety of geometric methods are widely used across representation theory, but the Beilinson-Bernstein localization theorem remains a model tool for its breadth and simplicity.
In this talk, I’ll illustrate the Beilinson-Bernstein localization theorem through a series of examples by describing what several of our favorite Lie algebra representations look like in D-module clothing. We’ll focus on the Lie algebra sl(2,C) and draw pictures of the D-modules corresponding to finite-dimensional representations, Verma modules, principal series representations of SL(2,R), and Whittaker modules.
The seminar will be on hiatus for the last two weeks of August and will resume September 4th, 2020, on the new schedule: Fridays, 1 pm Eastern US time / 10 am Western US time / evening times in Europe.
Minicourse on "Hopfological algebras and categorification at a root of unity"
On the functoriality of the Khovanov homology for tangles
In 2002 Khovanov proved his homology to be projectively functorial with respect to link (or tangle) cobordisms. Shortly after, Jacobson showed that signs do really matter, and Bar-Natan reformulated Khovanov’s construction topologically using a quotient of (an additive closure of) 2-cobordisms called Bar-Natan bicategory. Since then, many approaches were developed to fix the sign issue. All solutions propose to keep track of orientations and actually replace Bar-Natan category by something else. If you want to work over Z, than the replacement is the bicategory of GL(2)-foams. In our recent paper with M. Hogancamp, K. Putyra and S. Wehrli, we proved that Bar-Natan and GL(2)-foams constitute equivalent bicategories (by constructing an explicit bifunctor between them). This gives an explicit way to fix signs inside the Bar-Natan bicategory.
In this talk I will start with a gentle introduction for the subject, explain the proof of our theorem and, if time remains, mention an application of this result, that identifies 3 different models of the colored sl(2) homology in the quantum annular setting.
The Heisenberg category I and II
In these two talks, I will explain some foundational results about the (degenerate) Heisenberg monoidal category. This category was introduced 10 years ago by M. Khovanov (1009.3295) and encodes relations between natural transformations of the induction and restriction functors on representations of the symmetric group. It is perhaps the simplest of a number of “affine” diagrammatic monoidal categories of recent interest in representation theory, so it makes a great example pedagogically! In the first talk, I’ll try to explain the computation of its Grothendieck ring, which is a certain Z-form for the infinite-dimensional Heisenberg Lie algebra. In the second I’ll discuss the rich structure theory of Abelian module categories over the Heisenberg monoidal category. This includes the classical example of representations of symmetric groups, and many other categories from “type A” representation theory. The talks are based on two papers with A. Savage and B. Webster (1812.03255 and 1907.11988).
Derived annular Khovanov-Rozansky invariants
Recent years have seen a growing interest in annular link homology theories --- categorified invariants of links embedded in a thickened annulus and cobordisms between them. These invariants typically have more structure than their non-annular counterparts, allowing e.g. straightforward specialisation from the HOMFLYPT to the gl(N) versions, such as annular Khovanov homology for N=2.
In the first talk, I will describe how annular link homologies can be constructed from functorial tangle invariants by the completely formal construction of taking the horizontal trace of a monoidal category (actually, a braided monoidal 2-category). The focus will be on the annular link homology that categorifies the positive half of the HOMFLYPT skein of the annulus, i.e. the ring of symmetric functions. Along the way, I will discuss the type A Hecke algebra, its trace, Soergel bimodules, Rouquier complexes of braids, and I will give example computations. The ring structure of symmetric functions, the bases of Schur functions and elementary symmetric functions, and certain plethistic substitutions will play prominent roles.
The second talk explores the shortcomings of the horizontal trace construction of annular link homologies for the purpose of categorified skein theory, thus motivating the use of a derived horizontal trace. The new annular invariants take the form of twisted complexes of modules for a differential graded algebra with a new set of odd generators that encode `sliding dots around the annulus'. As a bonus, the new invariant upgrades to an invariant of links in the solid torus via automorphisms corresponding to full-twist insertion.
Based on joint work with Eugene Gorsky and Matt Hogancamp.
Khovanov homology and link detection
Khovanov homology is one of the categorified quantum invariants most often used by low-dimensional topologists. We will focus on the specific topological application of Khovanov homology to the question of link detection. In this talk, we will give an overview of Khovanov homology and link detection, mention some of the connections to Floer theoretic data used in detection results, and give a sketch of the proof that Khovanov homology detects the torus link T(2,6).
The Alexander Polynomial is a Quantum Invariant in a Different Way
If you care only about categorification, the take-home from my talk will be a challenge: Categorify what I believe is the best Alexander invariant for tangles.
Handout and additional resources: http://drorbn.net/cat20
Categorical diagonalization
Suppose you have an operator $f$ and a collection of distinct scalars $\kappa_i$ such that $\prod (f-\kappa_i) = 0$ . Then Lagrange interpolation gives a method to construct idempotent operators $p_i$ which project to the $\kappa_i$-eigenspaces of $f$ . We think of this process as diagonalization, and we categorify it: given a functor $F$ with some additional data (akin to the collection of scalars), we construct a complete system of orthogonal idempotent functors $P_i$. The categorification of Lagrange interpolation is related to the technology of Koszul duality. We illustrate this construction with simple but interesting examples involving modules over the group algebra of a cyclic group (a toy model for the Hecke category in type $A_1$). Our main application of this theory is to categorify the Young-Jucys-Murphy-centered "Okounkov-Vershik approach" to the representation theory of the symmetric group.
In the first lecture we will motivate diagonalization by discussing its uses in representation theory, and use examples from categorical representation theory to set the expectations for categorical diagonalization. Then we spend the bulk of the lecture explaining and motivating the construction of the functors $P_i$. In the second lecture, we provide some background on the Hecke category (a categorification of the symmetric group or its Hecke algebra) and its cell theory, and discuss the simultaneous diagonalization of categorified full twists.
Video Recording of talk Part 1
Video Recording of talk Part 2
Slides Part 1
Slides Part 2
Ribbon concordance and link homology theories
Given knots $K_0, K_1 \subset S^3$, a ribbon concordance from $K_0$ to $K_1$ is a smoothly embedded annulus $A \subset S^3 \times [0,1]$ bounded by $K_0 \times 0 \cup K_1 \times 1$, such that projection onto the $[0,1]$ factor has only index 0 and 1 critical points on $A$. Unlike ordinary concordance, ribbon concordance is a highly asymmetric relation; indeed, Gordon conjectured that if two knots are each ribbon concordant to each other, then they must be isotopic. In the past year and a half, there have been a tremendous number of breakthroughs using various knot homology theories to obstruct the existence of ribbon concordances, all lending support to the above conjecture. This talk will be a survey some of these results, including my own work with Ian Zemke on Khovanov homology and ribbon concordance and some more recent joint work with Onkar Singh Gujral.
Odd Khovanov homology for tangles
Khovanov categorified the Jones polynomial by constructing a corresponding link homology. His construction admits an anticommutative version (referred to as 'odd') that was developed by Ozsváth, Rasmussen and Szabó. The usual (or 'even') construction extends to tangles, taking the form of the homotopy type of a complex of bimodules over the arc algebra. The 'odd' equivalent to the arc algebra is non-associative, making it not clear what a bimodule over it should be. In the first part of the talk, I'll explain how ORS construction works in the context of chronological cobordisms, as introduced by Putyra. Then, I'll sketch how to extend this to tangles by using an odd version of arc algebras. We will quickly see that it is not naively possible to use it to construct an odd invariant, mainly because the odd arc algebra is not associative. In the second part of the talk, I'll explain how, by changing the monoidal category of vector spaces we work in, we can solve the non-associativity issue of the arc algebra. Then, we will sketch how to use it to construct an odd version of Khovanov invariant for tangles. This is a joint work with Krzysztof Putyra.
Video recording of talk Part 1
Video recording of talk Part 2