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Organizing Committee
University of Oregon
UC Berkeley
Columbia University
University of New South Wales
Abstract
Many classical results connecting representation theory and combinatorics can be succinctly understood through the concept of categorification. Two celebrated examples are Frobenius’s 1900 theorem showing that the ring of symmetric functions is categorified by the representation theory of symmetric groups, and the Kazhdan-Lusztig conjecture which relates category O for a complex semisimple Lie algebra to the Hecke algebra of the corresponding Weyl group. In these and other examples, positivity properties observed at a combinatorial level are explained conceptually by the existence of a categorification. The original proofs of the Kazhdan-Lusztig conjecture found in 1981 were geometric in nature, involving perverse sheaves and D-modules on the underlying flag varieties. Elias and Williamson found an alternative, purely algebraic proof in 2014, exploiting a diagrammatic description of the monoidal category of Soergel bimodules.
The diagrammatic approach has its origins in the discovery of new quantum invariants of knots and links in the 1980s. Crane and Frenkel raised the idea of categorifying quantum groups, hence, link invariants, already in the 1990s, and this vision prompted the development of powerful link homology theories such as Khovanov-Rozansky homology. The categorification of quantum groups involves Khovanov-Lauda-Rouquier algebras, which are often presented diagrammatically. They are a building block for Kac-Moody 2-categories, which categorify Lusztig’s modified integral form for quantized enveloping algebra. There have been many remarkable developments in this direction in the last few years, including the introduction by Webster of more general algebras categorifying tensor products, DG versions of these algebras which categorify Verma modules, and p-DG versions which are being used to categorify link invariants at roots of unity.
The field of diagrammatic categorification is still in its early stages, but it has already had a significant impact on more traditional mathematics. This workshop aims to unite both established experts and emerging scholars across various domains of diagrammatic categorification, including representation theory, combinatorics, and link homology.
The diagrammatic approach has its origins in the discovery of new quantum invariants of knots and links in the 1980s. Crane and Frenkel raised the idea of categorifying quantum groups, hence, link invariants, already in the 1990s, and this vision prompted the development of powerful link homology theories such as Khovanov-Rozansky homology. The categorification of quantum groups involves Khovanov-Lauda-Rouquier algebras, which are often presented diagrammatically. They are a building block for Kac-Moody 2-categories, which categorify Lusztig’s modified integral form for quantized enveloping algebra. There have been many remarkable developments in this direction in the last few years, including the introduction by Webster of more general algebras categorifying tensor products, DG versions of these algebras which categorify Verma modules, and p-DG versions which are being used to categorify link invariants at roots of unity.
The field of diagrammatic categorification is still in its early stages, but it has already had a significant impact on more traditional mathematics. This workshop aims to unite both established experts and emerging scholars across various domains of diagrammatic categorification, including representation theory, combinatorics, and link homology.
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Workshop Schedule
Monday, October 20, 2025
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:50 AM EDT
Quantum loop algebras, simple modules, and q-characters
11th Floor Lecture Hall
Speaker
Andrei Negut, EPFL
Session Chair
Anna Romanov, University of New South Wales
Abstract
I will survey recent developments on the construction of simple modules of quantum loop algebras (of arbitrary Kac-Moody type) with an eye toward categorification.
10:00 - 10:20 AM EDT
Coffee Break
11th Floor Collaborative Space
10:20 - 11:10 AM EDT
Elliptic Hall algebra and its categorifications
11th Floor Lecture Hall
Speaker
Eugene Gorsky, UC Davis
Session Chair
Anna Romanov, University of New South Wales
Abstract
I will review various presentations of the elliptic Hall algebra by generators and relations, including the recently discovered relation with the double Dyck path algebra of Carlsson and Mellit. I will then explain how these generators appear in various categorical setups such as the derived category of the commuting stack and the trace of the affine Hecke category, and state several conjectures about them. The talk is based on joint works with Andrei Negut, Nicolle Gonzalez and Jose Simental.
11:30 - 11:35 AM EDT
Soergel algebras as bound-quiver algebras: the infinite dihedral case
Lightning Talks - 11th Floor Lecture Hall
Speaker
Theo Pinet, McGill University
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
The blob algebra TLbN is a finite-dimensional cellular quotient of the Hecke algebra of type B. It appears naturally in statistical physics and admits a notoriously intricate representation theory.
In this talk, we use Soergel bimodules of affine type A1 to give a bound-quiver realization for blocks of TLbN. We then use this novel realization to deduce surprising results about the endomorphism algebras of projective TLbN-modules.
The talk is based on joint work with Alexis Leroux-Lapierre and Yvan Saint-Aubin.
11:35 - 11:40 AM EDT
Highest weight theory for representations of general linear groups in Verlinde cateories.
Lightning Talks - 11th Floor Lecture Hall
Speaker
Aleksandra Utiralova, University of Oregon
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
Verlinde category Ver_p serves as an analog of super vector spaces for Tannakian reconstruction of tensor categories in characteristic p. This means that any nice enough tensor category (Frobenius exact, of moderate growth) linear over a field of char p is equivalent to the category of representations of some affine group scheme in Ver_p.
I will talk about the classification of representations of GL(X) for X an object in Ver_p and the highest weight theory for them.
11:40 - 11:45 AM EDT
A diagrammatic approach to reflection functors
Lightning Talks - 11th Floor Lecture Hall
Speaker
Haruto Murata, the University of Tokyo
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
For an arbitrary symmetrizable generalized Cartan matrix, the category of modules over the KLR algebra (Quiver Hecke algebra) provides a categorification of the associated quantum group. In this talk, I will present a construction of reflection functors that categorify Lusztig’s braid group action, from the viewpoint of 2-representation theory. Similar functors have recently been constructed independently, by Kashiwara-Kim-Oh-Park, using a different approach.
11:45 - 11:50 AM EDT
Higher idempotent completion for some diagrammatic categories
Lightning Talks - 11th Floor Lecture Hall
Speaker
Isabela Recio, Universität Hamburg
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
What is an good notion of an idempotent map inside a 2-category? What would a categorified notion of idempotent completion look like? Following constructions introduced by Reutter-Douglas and Gaiotto-Johnson--Freyd, we explore some of the applications of higher idempotent completion in the case of Soergel bimodules, relying heavily on their diagrammatic incarnation, and the deformed foam categories underlying combinatorial formulations of deformed Khovanov-Rozansky homology.
11:50 - 11:55 AM EDT
Khovanov-Rozansky Homology, Affine Springer Fibers, and q,t-Catalans
Lightning Talks - 11th Floor Lecture Hall
Speaker
Joshua Turner, UC Davis
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
We’ll briefly talk about the relationship between homology of affine Springer fibers and Khovanov-Rozansky homology of links, and the q,t-Catalan combinatorics that arise from both.
11:55 AM - 12:00 PM EDT
Action of the Witt algebra on categorified quantum groups
Lightning Talks - 11th Floor Lecture Hall
Speaker
Jernej Grlj, University of Southern California
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
In a recent paper, joint with A. Lauda, we constructed an action of the positive half of the Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. We will present this action. We will specialize the action to the type A case. In this case the presented action induces an action of the positive Witt algebra on \mathfrak{gl}_n-foams, recovering the action of Qi, Robert, Sussan, and Wagner.
12:00 - 12:05 PM EDT
2-Categorifical affine symmetries and quantum boson algebras
Lightning Talks - 11th Floor Lecture Hall
Speaker
Sam Qunell, UCLA
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
Representations of KLR (quiver Hecke) algebras categorify the positive part of the quantum group associated to any symmetrizable Cartan matrix. This categorical perspective makes certain symmetries more natural to study. For example, the induction and restriction functors between categories of KLR algebra modules play an important role in the theory. A closer investigation of these functors reveals surprising new symmetries. In this talk, I explain how the induction and restriction functors for KLR algebras can be used to obtain a 2-representation of the corresponding affine positive part in type A. I also describe a new categorification of a closely related algebra, the q-boson algebra, in all symmetrizable Kac-Moody types.
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:20 PM EDT
Categorified braid actions on tensor products and perverse equivalences
11th Floor Lecture Hall
Speaker
Iva Halacheva, Northeastern University
Session Chair
Jose Simental Rodriguez, Universidad Nacional Autónoma de México
Abstract
In the setting of categorified quantum group representations, Chuang and Rouquier introduced certain complexes, known as Rickard complexes, which lift Lusztig’s braid group action to the categorical level. Subsequently, Webster constructed a categorical braid group action lifting the R-matrix action on tensor products of representations. I will discuss these two constructions in the setting of perverse equivalences - known results about the former and work in progress about the latter joint with Erika Beserra.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:50 PM EDT
Temperley-Lieb categorification
11th Floor Lecture Hall
Speaker
Chris Bowman, University of York
Session Chair
Jose Simental Rodriguez, Universidad Nacional Autónoma de México
Abstract
We discuss new and old results on the Temperley—Lieb algebra, its categorification and representation theoretic structure.
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, October 21, 2025
9:00 - 9:50 AM EDT
How to compute many decomposition numbers for symmetric groups (… in principle, on a big computer, using code that doesn’t yet exist).
11th Floor Lecture Hall
Speaker
Geordie Williamson, University of Sydney
Session Chair
Alexander Kleshchev, University Of Oregon
Abstract
It is a very interesting question as to what the large-scale structure of decomposition numbers for symmetric groups looks like. Does order or chaos reign? One way to approximate an answer to this question is to compute many examples, and see if one can see any patterns. Over many years, we (Elias, Libedinsky, Riche, Jensen,…) developed a way of computing much further out than was previously possible. I then spent a year implementing this and running it on a high-powered computer for 8 months, which led to the billiards conjecture (with Lusztig) in the 3-row case. One sees a beautiful pattern in a region (“billiards”) but we couldn’t guess what happens beyond. Interestingly, my implementation suffers from some unexpected difficulties, which blocks further progress. In more recent work with Riche (and inspired by work of Hazi) we worked out a way that these difficulties can be circumvented in principle. I’ll try to outline this new approach. Now all that remains is to actually implement it … I’ll report back in four years time.
10:00 - 10:20 AM EDT
Coffee Break
11th Floor Collaborative Space
10:20 - 11:10 AM EDT
The Temperley-Lieb tower and the Weyl algebra
11th Floor Lecture Hall
Speaker
Peter Samuelson, University of California, Riverside
Session Chair
Alexander Kleshchev, University Of Oregon
Abstract
Khovanov's Heisenberg category (and its later generalizations) acts on the category of modules over the tower of symmetric groups (or Hecke algebras). The generating objects act via compositions of induction and restriction functors, and generating morphisms map to certain natural transformations between these compositions. In joint work, Harper and I defined the "Weyl category,'' whose starting point is Temperley-Lieb algebras instead of Hecke algebras. In this talk, I explain the construction of this category, its action, and what we know and/or expect about its Grothendeick group and trace. One new feature is the existence of "idempotent bubbles."
11:30 - 11:35 AM EDT
Kazhdan-Lusztig equivalence via Soergel bimodules
Lightning Talks - 11th Floor Lecture Hall
Speaker
Jonathan Gruber, University of York
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
Abstract
The category of finite-dimensional representations of a complex simple Lie algebra admits two canonical non-semisimple deformations: One is the parabolic BGG category O for an affine Lie algebra, the other is the category of representations of a quantum group at a root of unity. In a landmark result from the 1990s, Kazhdan and Lusztig have established an equivalence between these non-semisimple abelian categories. More recently, Gaitsgory has proposed a conjectural extension of the Kazhdan-Lusztig equivalence to a derived equivalence between the (non-parabolic) BGG category O of the affine Lie algebra and the category of representations of a ""mixed"" quantum group. In this talk, I will establish a variant of the derived equivalence conjectured by Gaitsgory, involving the principal blocks of the aforementioned categories, by relating the derived categories of these principal blocks to categories of Soergel bimodules. I will also explain how this derived equivalence can be used to recover a (non-derived) variant of the Kazhdan-Lusztig equivalence, again involving the principal blocks.
This is joint work with Peter Fiebig.
11:35 - 11:40 AM EDT
Interpolating Feigin-Frenkel Duality at the Critical Level
Lightning Talks - 11th Floor Lecture Hall
Speaker
Andrew Riesen, Masschusetts Institute of Technology
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
11:40 - 11:45 AM EDT
A Diagrammatic Quantization of the Geometric Satake Equivalence in Type A
Lightning Talks - 11th Floor Lecture Hall
Speaker
Koppara Philip Thomas, University of Oregon
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
Abstract
In arXiv:1403.5570, the author constructed a functor from Rep(U_q(sl_n)) to a certain subcategory of singular Soergel bimodules (for a q-deformed realization of the affine permutation group) when n=2, 3 using diagrammatic replacements for both categories. We generalize this result for n>3. This is joint work with Ben Elias.
11:45 - 11:50 AM EDT
c-Birkhoff polytopes
Lightning Talks - 11th Floor Lecture Hall
Speaker
Jianping Pan, Arizona State University
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
Abstract
The volume of the Birkhoff polytope was very hard to compute -- it was done, but with a lot of effort and there is a very complicated formula. It turns out that certain Birkhoff subpolytope has extremely easy volume formula in poset-theoretic information.
11:50 - 11:55 AM EDT
Local Models of Ariki--Koike Blocks
Lightning Talks - 11th Floor Lecture Hall
Speaker
Dinushi Munasinghe, National and Kapodistrian University of Athens
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
Abstract
Steadied quotients of (weighted) KLR algebras were introduced by Webster and are obtained by "tilting" the usual cyclotomic relation. This quotient can be visualized by assigning the strands in the diagrammatic presentation the tendency to move in certain directions, and looking at which idempotents stabilize. We present steadied quotients of KLR algebras as representatives of Morita equivalence classes of blocks of Ariki--Koike algebras. This is based on joint work with Ben Webster.
11:55 AM - 12:00 PM EDT
A Diagram Calculus for a few Type-A Lusztig–-Vogan Categories
Lightning Talks - 11th Floor Lecture Hall
Speaker
Victor Zhang, UNSW
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
Abstract
The Kazhdan--Lusztig conjecture (1979) connects bases of the Hecke algebra to decomposition numbers of important representations of complex semisimple Lie algebras. As an approach to proving the conjecture, a complete diagrammatic language was developed for the category of Soergel bimodules, following the effort of various people throughout the 2010s. This diagrammatic theory culminated in an algebraic proof of the conjecture, whose existing proofs rely on the heavy machinery of D-modules, in a more general setting. In the analogous world of real reductive Lie groups, a similar question was asked by Lusztig and Vogan (1983) and proven with geometry. Now, the obvious question is this: can we use diagrammatics here? In view of an answer, we give the beginnings of a diagram calculus, for some real Lie groups of type A.
12:00 - 12:05 PM EDT
A Crystal Temperley--Lieb Category
Lightning Talks - 11th Floor Lecture Hall
Speaker
Moaaz Alqady, University of Oregon
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
Abstract
By considering a suitable renormalization of the Temperley--Lieb category, we study its specialization to the case q=0. Unlike the q≠0 case, the obtained monoidal category, TL_0(𝕜), is not rigid or braided. We provide a closed formula for the Jones--Wenzl projectors in TL_0(𝕜) and give semisimple bases for its endomorphism algebras. We then describe a coboundary structure on TL_0(𝕜) and show that its idempotent completion is coboundary monoidally equivalent to the category of 𝔰𝔩2-crystals. This gives a diagrammatic description of the commutor for 𝔰𝔩2-crystals defined by Henriques and Kamnitzer and of the resulting action of the cactus group.
12:05 - 12:10 PM EDT
Quiver presentations for KLR algebras via Hecke category diagrammatics
Lightning Talks - 11th Floor Lecture Hall
Speaker
Alice Dell’Arciprete, University of York
Session Chair
Stephen Griffeth, Instituto de Matematica y Fisica Universidad de Talca
Abstract
A central aim of representation theory is to describe the submodule structure of important algebras. In this talk, I will focus on Khovanov–Lauda–Rouquier (KLR) algebras and explain how their structure can be understood via a close connection with the diagrammatic Hecke categories of maximal parabolics in finite symmetric groups. A key ingredient in this story is combinatorics: Dyck tableaux not only allow us to compute graded decomposition numbers but also give us a way of determining the extensions between them. This perspective turns out to be rich enough to completely determine the Ext-quiver and relations presentation of these algebras. I will illustrate the ideas with explicit examples, showing how combinatorial data and diagrammatic methods combine to give an explicit description of these algebras.
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:20 PM EDT
iQuantum groups and categorification, I
11th Floor Lecture Hall
Speaker
Weiqiang Wang, University of Virginia
Session Chair
Liron Speyer, Okinawa Institute of Science and Technology
Abstract
iQuantum groups (iQG) arise from quantum symmetric pairs associated with Satake diagrams, and they form a generalization of Drinfeld-Jimbo quantum groups. In this talk, we will explain the basics of iQG, including idivided powers (=rank one icanonical basis), bilinear forms, and standardized canonical basis. We will end with discussing about categorification of rank one iQG. This is joint work with Jon Brundan and Ben Webster.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:50 PM EDT
iQuantum groups and categorification, II
11th Floor Lecture Hall
Speaker
Ben Webster, University of Waterloo
Session Chair
Liron Speyer, Okinawa Institute of Science and Technology
Abstract
I'll discuss the philosophy of relating categorifications using deformation and localization. While this can be applied in a number of settings, I'll focus on recent work which gives a categorification of iQuantum groups in the style of Khovanov and Lauda in many cases (which includes a new proof of the non-degeneracy of the original categorified quantum groups as a special case). This is joint work with Jon Brundan and Weiqiang Wang.
Wednesday, October 22, 2025
9:00 - 9:50 AM EDT
Towards a quantum algorithm for Khovanov homology
11th Floor Lecture Hall
Speaker
Aaron Lauda, University of Southern California
Session Chair
Nicolle Gonzalez, UC Berkeley
Abstract
One of the central challenges in quantum algorithm design is to identify problems of genuine computational interest that admit exponential speedups over classical approaches. Shor’s algorithm for factoring is a landmark example, and quantum algorithms for approximating the Jones polynomial have similarly demonstrated exponential advantage. In this talk, we present a quantum algorithm that estimates the ranks of Khovanov homology groups using Hodge theory. This approach can be applied to a broad class of homology problems given certain assumptions.
Based on this construction, we show that estimating the Betti numbers of Khovanov homology is universal for various models of quantum computation. While classical algorithms for Khovanov homology scale exponentially in the number of crossings of a knot, our quantum algorithm is efficient provided the corresponding Hodge Laplacian thermalizes in polynomial time and has a sufficiently large spectral gap. We give numerical and analytical evidence for both statements and conclude with an open questions regarding analytic bounds on the spectral gap for general knots.
10:00 - 10:20 AM EDT
Coffee Break
11th Floor Collaborative Space
10:20 - 11:10 AM EDT
Cherednik algebras and torus links
11th Floor Lecture Hall
Speaker
Xinchun Ma, UChicago
Session Chair
Nicolle Gonzalez, UC Berkeley
Abstract
A conjecture of Gorsky, Oblomkov, Rasmussen and Shende realizes the Khovanov-Rozansky homology of torus knots in terms of finite-dimensional representations of rational Cherednik algebras. Building on previous works of Hogancamp, Ellias, Mellit and Negut, we establish this conjecture via the geometry of D-modules on sl_n and coherent sheaves on Hilbert schemes of points on the plane. Partial extensions to the case of torus links will also be discussed.
11:30 AM - 12:20 PM EDT
The limits of knot detection: alternating links, mutation, and (categorified) quantum invariants
11th Floor Lecture Hall
Speaker
Pedro Vaz, Université Catholique de Louvain
Session Chair
Nicolle Gonzalez, UC Berkeley
Abstract
Knot invariants such as the Jones and HOMFLYPT polynomials, and their categorifications, are central tools in low-dimensional topology. But how effective are they as classifiers? In this talk I will explain a recent work with A. Lacabanne, D. Tubbenhauer, and V. Zhang, where we show that many classical and quantum invariants detect alternating links with probability zero: their distinguishing power decays exponentially due to insensitivity to oriented mutation. Large-scale computations further reveal that categorified invariants perform no better than their polynomial counterparts, highlighting intrinsic limits of knot invariants and pointing to new challenges in quantum topology.
12:30 - 12:35 PM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:35 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:20 PM EDT
Recursive computations for Khovanov-Rozansky homology
11th Floor Lecture Hall
Speaker
Mikhail Mazin, Kansas State University
Session Chair
Alistair Savage, University of Ottawa
Abstract
Khovanov-Rozansky homology of torus knots and links were computed recursively using categorified Young symmetrizers of Elias and Hogancamp in a series of papers by different combinations of Ben Elias, Matt Hogancamp, and Anton Mellit. In our joint paper with Carmen Caprau, Nicolle Gonzalez, and Matt Hogancamp, we showed that the same recursion also computes KR homology of the monotone knots of triangular partitions. Can these methods be applied to other families of knots? A natural class of knots to explore are the monotone knots of concave partitions.
In this talk I will review the recursion that computes the KR homology of torus knots and monotone knots of triangular partitions, and then explore on examples how similar methods can be applied to other knots and, perhaps, what are the limitations.
This talk is based on our joint paper with Carmen Caprau, Nicolle Gonzalez, and Matt Hogancamp, as well as recent discussions with Nicolle Gonzalez and Eugene Gorsky.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:50 PM EDT
Symmetries of link homology in characteristic p
11th Floor Lecture Hall
Speaker
Joshua Sussan, CUNY
Session Chair
Alistair Savage, University of Ottawa
Abstract
We will construct an action of sl(2) on gl(p) Khovanov--Rozansky homology in characteristic p. We will obtain several topological applications such as a new obstruction to a knot being slice.
This work is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.
Thursday, October 23, 2025
9:00 - 9:50 AM EDT
Diagrammatics of the Delannoy category
11th Floor Lecture Hall
Speaker
Mikhail Khovanov, Columbia University
Session Chair
Nicolas Libedinsky, Universidad de Chile
Abstract
Delannoy category was introduced by N.Harman and A.Snowden. It's based on convolution of certain constructible functions via the compact Euler characteristic as the integration measure. We'll explain the construction and properties of the Delannoy category, introduce diagrammatics for it, and discuss its possible use in categorification.
10:00 - 10:20 AM EDT
Coffee Break
11th Floor Collaborative Space
10:20 - 11:10 AM EDT
Categorified i-quantum groups and spin link homology
11th Floor Lecture Hall
Speaker
Elijah Bodish, Indiana University Bloomington
Session Chair
Nicolas Libedinsky, Universidad de Chile
Abstract
i-Quantum group theory gives rise to q-deformed special orthogonal algebras different than Drinfeld-Jimbo quantum groups. These i-quantum special orthogonal algebras have mysterious nonclassical representations which do not q-deform finite dimensional representations of the special orthogonal algebra. The nonclassical representations do not immediately fit into the framework of Bao-Wang’s i-canonical basis or Brundan-Wang-Webster’s categorified i-quantum groups. One key to understanding nonclassical representations is Wenzl’s recent observation that they are afforded by quantum spin Howe duality in type B. In this talk, I will discuss evidence, coming from joint work with Elias-Rose on type B webs and type B link homology, that categorified quantum spin Howe duality in type B is just ``folded” categorified skew Howe duality in type A.
11:30 AM - 12:20 PM EDT
Quantum representations and webs
11th Floor Lecture Hall
Speaker
Julianna Tymoczko, Smith College
Session Chair
Nicolas Libedinsky, Universidad de Chile
Abstract
The combinatorial spider is a diagrammatic category that encodes quantum $\mathfrak{sl}_n$ representations, and was formalized by Kuperberg. Webs correspond to the morphisms in this category, drawn as directed planar graphs with skein-type relations that indicate algebraic equivalences. Webs are well-understood in the case $n=2$, when they are essentially the Temperley-Lieb diagrams, and in the substantially more complicated case $n=3$. But despite considerable interest from knot theorists, representation theorists, and combinatorists, we lacked a similar set of graph-theoretic tools to analyze webs explicitly when $n \geq 4$.
In this talk, we sketch some of the historical evolution of webs, including work of Kuperberg, Khovanov, Fontaine, Cautis, Kamnitzer, Morrison, and others, as well as connections to algebraic geometry and combinatorics. We also describe a new approach using a set of colored paths called \emph{strands} that generalize graph-theoretic and combinatorial notions from smaller dimensions to $n \geq 4$, and that also give global information about webs. This work is joint with Heather M. Russell.
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:20 PM EDT
Webs as hourglass plabic graphs
11th Floor Lecture Hall
Speaker
Christian Gaetz, UC Berkeley
Session Chair
You Qi, University of Virginia
Abstract
Kuperberg's webs give a diagrammatic calculus for spaces of tensor invariants of quantum groups. These are best understood in the cases of U_q(sl_2) and U_q(sl_3), when it is possible to choose a subset of the webs forming a distinguished basis (the non-crossing matchings for sl_2 and "non-elliptic" webs for sl_3). These were used by Khovanov in his categorification of the corresponding quantum link invariants. I will describe an ongoing research program, joint with Pechenik, Pfannerer, Striker, and Swanson, in which we interpret webs as generalizations of Postnikov's plabic graphs. This perspective allows for a unified combinatorial description of the known distinguished web bases and the discovery of the first such basis for sl_4.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:50 PM EDT
Superalgebra deformations of web categories
11th Floor Lecture Hall
Speaker
Robert Muth, Duquesne University
Session Chair
You Qi, University of Virginia
Abstract
For a Frobenius superalgebra A with even subalgebra a, we define associated web supercategories which generalize a number of symmetric web category constructions. These diagrammatic categories are equipped with asymptotically faithful functors to the category of gl_n(A)-modules and its endofunctor category, and serve as ‘thickened’ versions of the wreath and affine wreath algebras over A. For various choices of A/a these categories yield a diagrammatic presentation for RoCK blocks of Hecke algebras, and conjectural connections to blocks of Schur algebras, Sergeev superalgebras, and KLR algebras which I will discuss. This is joint work with Nicholas Davidson, Jonathan Kujawa, and Jieru Zhu.
Friday, October 24, 2025
9:00 - 9:50 AM EDT
Diagrammatic singular Soergel bimodules
11th Floor Lecture Hall
Speaker
Hankyung Ko, Uppsala University
Session Chair
Leonardo Patimo, università di Pisa
Abstract
In joint work with Ben Elias, Nicolas Libedinsky, Leonardo Patimo, we constructed a diagrammatic basis of the morphism spaces between singular Soergel bimodules. This is analogous to Elias-Williamson’s construction in regular Soergel bimodules modelled on Libedinsky's light leaves. The next step, in progress with the same coauthors, is a presentation of the (2-)category by diagrammatic generators and relations. We might call the latter diagrammatic categorification of parabolic double cosets in a Coxeter system. This talk is an introduction to singular Soergel bimodules, its diagrammatics, and combinatorics of parabolic cosets, where we also look at some singular light leaves.
10:00 - 10:20 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:20 AM EDT
Demazure operators and Switchbacks, with a hint of Spetses
11th Floor Lecture Hall
Speaker
Ben Elias, University of Oregon
Session Chair
Leonardo Patimo, università di Pisa
Abstract
This talk will build on the previous talk of Hankyung Ko, but will be more elementary. It is based on joint work with Ko, Libedinsky, and Patimo. We introduce Demazure operators associated to double cosets. Through this lens we motivate the concept of "reduced expressions for double cosets," and we observe the switchback relations. Demazure operators are the technical and algebraic heart of (singular) Soergel bimodules, and we illustrate this with a theorem: that reduced expressions for the longest element contain the "deep bimodule" as a summand. Time permitting, we discuss recent work with Juteau and Young, where the "reduced expressions for the longest element in G(m,m,n)" were computed. We discuss implications of the deep bimodule theorem for the Spetses program.
11:30 AM - 1:20 PM EDT
Lunch/Free Time
1:20 - 2:10 PM EDT
The minimal Rickard complexes of braids on two strands
11th Floor Lecture Hall
Speaker
Joshua Wang, Princeton University and IAS
Session Chair
Jonathan Brundan, University of Oregon
Abstract
The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when closure is color-compatible. For each braid on two strands with any colors, we construct a minimal complex that is homotopy equivalent to its Rickard complex. It is not obtained by laborious simplification; instead, it is defined directly by explicit formulas obtained by educated guesswork and reverse engineering a conjectural connection between colored sl(N) homology and SU(N) representations of the knot group.
2:30 - 3:20 PM EDT
Skeins on Tori
11th Floor Lecture Hall
Speaker
Monica Vazirani, UC Davis
Session Chair
Jonathan Brundan, University of Oregon
Abstract
The G-skein module for the 3-torus is finite dimensional by a result of Gunningham-Jordan-Safronov. For G= GL_N and SL_N, there is a lovely combinatorial formula for these dimensions in terms of the partitions of N.
Our formula arises from studying skeins on the 2-torus via the representation theory of the double affine Hecke algebra (DAHA) of type A and its connection to quantum D-modules.
Using the DAHA, we also prove that all tangles in the relative N-point skein algebra are in fact equivalent to linear combinations of braids, modulo skein relations.
I will discuss these results.
This is joint work with Sam Gunningham and David Jordan.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
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