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Organizing Committee
Australian National University
University of Minnesota
North Dakota State University
University of Southern California
Abstract
Webs are diagrammatic tools for representing complex calculations graphically. These diagrams first arose from the representation theory of classical groups, and they have since become important in disparate areas of mathematics. In representation theory, they encode morphisms of quantum groups. In topology, webs give rise to powerful link invariants. In algebra and geometry, Kuperberg's sl3 web bases have important relationships with the theory of cluster algebras and affine buildings. In combinatorics, they explain certain dynamics on Young tableaux. Recent work by Gaetz--Pechenik--Pfannerer--Striker--Swanson introduced an sl4 web basis that has exploited and extended exciting connections between webs, plabic graphs, and crystals. There are further connections to total positivity, duality conjectures for cluster algebras and mirror symmetry.
Recently, there have been major developments in the theory of webs and web bases from multiple perspectives, such as recent work of Bodish--Elias--Rose--Tatham in type C motivated by applications to link homology. We believe that webs may serve as a nexus for disparate communities of mathematicians to meet. This includes not only fostering connections between researchers in algebraic combinatorics, Schubert calculus, and representation theory, but also building bridges with researchers in algebraic geometry via mirror symmetry and cluster geometry as well as researchers in topology and knot theory via the diagrammatic method. The goal of this workshop is thus to bring together experts from these communities and cross-fertilize these diverse subjects by spreading knowledge of recent developments and perspectives on our shared interest in webs.
This workshop will also provide an opportunity for distinct research groups to share current code functionality as well as future needs and desires, with an aim towards building collaborative and open-source computing capabilities.
Recently, there have been major developments in the theory of webs and web bases from multiple perspectives, such as recent work of Bodish--Elias--Rose--Tatham in type C motivated by applications to link homology. We believe that webs may serve as a nexus for disparate communities of mathematicians to meet. This includes not only fostering connections between researchers in algebraic combinatorics, Schubert calculus, and representation theory, but also building bridges with researchers in algebraic geometry via mirror symmetry and cluster geometry as well as researchers in topology and knot theory via the diagrammatic method. The goal of this workshop is thus to bring together experts from these communities and cross-fertilize these diverse subjects by spreading knowledge of recent developments and perspectives on our shared interest in webs.
This workshop will also provide an opportunity for distinct research groups to share current code functionality as well as future needs and desires, with an aim towards building collaborative and open-source computing capabilities.
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Workshop Schedule
Monday, December 08, 2025
8:50 - 9:00 AM EST
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EST
Quantum representations and webs
11th Floor Lecture Hall
Speaker
Julianna Tymoczko, Smith College
Session Chair
Patricia Hersh, University of Oregon
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 $n=3$.
In this talk, we sketch some of the historical evolution of webs, including work of Kuperberg, Khovanov, Fontaine, Cautis-Kamnitzer-Morrison, Fraser-Lam-Le, and others, as well as connections to algebraic geometry and combinatorics. We also describe a new approach, joint with Heather M. Russell, that uses a set of colored paths called \emph{strands} generalizing graph-theoretic and combinatorial notions from smaller dimensions to $n \geq 4$, and that also give global information about webs.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Webs as hourglass plabic graphs
11th Floor Lecture Hall
Speaker
Christian Gaetz, UC Berkeley
Session Chair
Patricia Hersh, University of Oregon
Abstract
I will introduce the ongoing research program, joint with Oliver Pechenik, Stephan Pfannerer, Jessica Striker, and Josh Swanson, in which we reinterpret webs as "hourglass plabic graphs" whose trip permutations align with the promotion permutations of tableaux. I will describe the application of this perspective to the construction of an SL(4) web basis generalizing Kuperberg's SL(3) basis, and, time permitting, will mention further applications to geometric embeddings of webs generalizing those of Fontaine-Kamnitzer-Kuperberg and to more general web bases.
11:30 AM - 12:15 PM EST
Rotation-invariant webs in higher ranks
11th Floor Lecture Hall
Speaker
Oliver Pechenik, University of Waterloo
Session Chair
Patricia Hersh, University of Oregon
Abstract
Kuperberg's SL3 web basis is importantly rotation invariant, meaning that rotations of basis webs are themselves basis webs. In recent work, we extended this construction to a rotation invariant basis of SL4 webs. The guiding principle is that basis webs should naturally biject with tableaux in a way that intertwines web rotation with tableau promotion, while trips in webs (in the sense of Postnikov's plabic graphs) should correspond with promotion paths in tableaux. The main problem is to extend this framework to arbitrary rank. We show partial progress in this direction. In particular, we extend the correspondence to all two-column tableaux and separately to all acyclic webs. Further confirmation of the naturality of these constructions comes from connections to the geometry of corresponding Springer fibres. This talk is based on work with Ron Cherny, Mike Cummings, Christian Gaetz, Stephan Pfannerer, Jessica Striker, and Josh Swanson.
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Parabolic induction for Springer fibers of type C
11th Floor Lecture Hall
Speaker
Mee Seong Im, Johns Hopkins University
Session Chair
Jesse Kim, University of Florida
Abstract
Nilpotent orbits are important objects in geometric representation theory since they appear in Springer’s construction of Weyl group representations, associated varieties of primitive ideals of enveloping algebras, conical symplectic singularities, and modular representation theory of Lie algebras, to name a few. The theory of Lusztig-Spaltenstein induction is a geometric process for constructing nilpotent orbits in the Lie algebra of a reductive group from the nilpotent orbits in Levi subalgebras. I will discuss some progress made involving the parabolic induction for Springer fibers of type C. This is joint with Neil Saunders and Arik Wilbert.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
q-rationals and web categories
11th Floor Lecture Hall
Speaker
Hoel Queffelec, CNRS
Session Chair
Jesse Kim, University of Florida
Abstract
In dealing with quantum groups, q-deformed integers play a prominent role, capturing the notion of graded dimension. Morier-Genoud and Ovsienko recently introduced the concept of q-deformed rationals: a rational number r/s gets deformed into a rational fraction in the variable q. This definition, originally related to the notion of continued fraction expansion, has attracted quite a lot of attention, and connections have been made in particular to the Burau representation of braids.
I will start by giving definitions closer to that latter, topological world. Then I will present ongoing work with Perrine Jouteur, where we use these q-rationals in the context of web categories, providing interpolating families of categories that should be thought of as deformations of Deligne's Rep(Gl(t)) categories.
5:00 - 6:30 PM EST
Reception
11th Floor Collaborative Space
Tuesday, December 09, 2025
9:00 - 9:45 AM EST
Annular web algebras
11th Floor Lecture Hall
Speaker
Mikhail Khovanov, Johns Hopkins University
Session Chair
Leigh Foster, University of Waterloo
Abstract
The talk will discuss a recent work of R.Akhmechet, M.Zhang and the author on annular equivariant web algebras for SL(2) and SL(3).
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Branching patterns and web bases
11th Floor Lecture Hall
Speaker
Ben Elias, University of Oregon
Session Chair
Leigh Foster, University of Waterloo
Abstract
This talk will explain the philosophy behind web bases (like the double ladders basis) which are derived from branching rules. One is interested in studying morphisms between tensor products of generating representations. A branching graph is a generalization of the Young graph, with vertices associated to simple representations, and an edge whenever one simple is a summand of another simple tensored with a generating representation. These graphs are usually infinite. There is a tautological argument that one can construct bases for morphism spaces associated to paths in the branching graph, given data associated to each edge. We explain what a lifting of this data is, and how liftings give rise to web bases. We also explain the concept of branching patterns, which allows one to construct liftings for infinitely many edges (all of them) from a finite amount of data. Time permitting, we discuss the problem of clasp computation.
11:30 AM - 12:15 PM EST
Webs and Buildings in Type A
11th Floor Lecture Hall
Speaker
Emily McGovern, University of Oregon
Session Chair
Leigh Foster, University of Waterloo
Abstract
In this talk, we describe an embedding of Type A Webs, a diagrammatic monodical category, into a class of graph planar algebras. These graph planar algebras arise from affine type A buildings, a combinatorial structure developed by Tits in the 1970s. The relationship between finite projective geometries and affine buildings is key in establish the existence of an embedding functor in positive characteristic graph planar algebras.
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Webs and the tropical Grassmannian
11th Floor Lecture Hall
Speaker
Thomas Lam, University of Michigan
Session Chair
Jessica Striker, North Dakota State University
Abstract
The tropical Grassmannian Trop Gr(2,n) of two-planes has an important and well-known combinatorial model as phylogenetic, or metric, trees. I will discuss a proposed combinatorial model for higher planes based on webs.
This talk is based on joint work in progress with Nick Early.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 5:00 PM EST
Wednesday, December 10, 2025
9:00 - 9:45 AM EST
Webs and 3-row increasing tableaux
11th Floor Lecture Hall
Speaker
Rebecca Patrias, University of St Thomas
Session Chair
Amanda Burcroff, MIT
Abstract
How can we find the correct notion of a web corresponding to a 3-row increasing tableaux that has the property that web rotation corresponds to K-promotion of the increasing tableaux? This talk goes through our journey to answer this question, including joint works with Oliver Pechenik, Jessica Striker, Chris Fraser, and Jesse Kim.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Plabic Tangles and Cluster Promotion Maps
11th Floor Lecture Hall
Speaker
Lauren Williams, Harvard University
Session Chair
Amanda Burcroff, MIT
Abstract
Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps called `promotions' between products of Grassmannians. Our central conjecture is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. We describe the underlying operad structure, and relate promotion maps to vector-relation configurations and to the degree of the amplituhedron map on positroid varieties. Finally we give an example (the `4-mass box') which points to new positivity properties for non-rational maps beyond cluster algebras. This is based on joint work with Chaim Even-Zohar, Matteo Parisi, Melissa Sherman-Bennett, and Ran Tessler.
11:30 AM - 12:15 PM EST
Clasped Webs, Trip Permutations, and Promotion on Tableaux
11th Floor Lecture Hall
Speaker
Stephan Pfannerer-Mittas, University of Waterloo
Session Chair
Amanda Burcroff, MIT
Abstract
This is a joint project with Elise Catania and Jesse Kim.
Classical $U_q(\mathfrak{sl}_r)$-webs provide a diagrammatic framework for the invariant space of tensor products of copies of the vector representation. To model invariant spaces of tensor products involving arbitrary representations Kuperberg introduced \emph{clasped webs} for $\mathfrak{sl}_3$. His construction uses webs satisfying a \emph{minimal cut property} to form a basis of these clasped spaces. We give a new characterization of clasped basis webs in terms of \emph{trip permutations} $trip_\bullet$, a property we call \emph{self-trip-avoiding}. This viewpoint naturally generalizes to higher rank via the \emph{hourglass plabic graph framework} of Gaetz--Pechenik--Pfannerer--Striker--Swanson. A key observation in their framework is that basis webs for $\mathfrak{sl}_4$ can be indexed by rectangular standard Young tableaux $T$ satisfying $trip_\bullet(W) = prom_\bullet(T)$, where $prom_\bullet$ is the tuple of promotion permutations. We extend this by showing that, in arbitrary rank, SYT satisfying a \emph{self-prom-avoiding} property index a basis of the clasped space. Finally, for $r = 3$, we connect promotion on non-rectangular SYT of size $n$ to a notion of \emph{rotation} on clasped webs where one is taking $n$ copies of $V$ together with an additional clasp for $V^\mu$ for some weight $\mu$.
12:25 - 12:30 PM EST
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
The deep locus of a cluster variety
11th Floor Lecture Hall
Speaker
David Speyer, University of Michigan
Session Chair
Joshua Swanson, University of Southern California
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:15 PM EST
Software Demo: Evaluation of webs in Python
Lightning Talks - 11th Floor Lecture Hall
Speaker
Hoel Queffelec, CNRS
Session Chair
Joshua Swanson, University of Southern California
Abstract
We have developed code to compute the x-polynomial from my talk on Monday, before we knew it could be computed by evaluation. The core of the code is an implementation of webs and (some) web relations, that, apparently, suffice to evaluate any closed webs in the HOMFLY-PT and all Reshetikhin-Turaev settings. I can give a quick tour of how webs are handled in practice. The result is a very slow code, but that hopefully should be easily tweakable to suit other persons purposes.
This is joint work with Perrine Jouteur.
4:15 - 4:30 PM EST
Software Demo: Webs and the Tropical Grassmannian
Lightning Talks - 11th Floor Lecture Hall
Speaker
Nicholas Early, Institute for Advanced Study
Session Chair
Joshua Swanson, University of Southern California
Abstract
The tropical Grassmannian TropG(2,n) is famously parameterized by phylogenetic, or metric trees. In on-going joint work with Thomas Lam, we have generalized this construction to higher dimensional spaces Trop+G(3,n) using webs. I will share Mathematica code developed before and during the course of our project to compute, manipulate and visualize the structures which arise.
4:30 - 4:45 PM EST
Software Demo: Hourglass Plabic Graphs in Sage
Lightning Talks - 11th Floor Lecture Hall
Speaker
Stephan Pfannerer-Mittas, University of Waterloo
Session Chair
Joshua Swanson, University of Southern California
Abstract
In [GPPSS25], we introduced hourglass plabic graphs (HPGs) as new avatars for webs. Modeled after Postnikov's plabic graphs, HPGs are carrying combinatorial data such as trip permutations and separation labelings. Moreover, there are operations on webs, called moves, that preserve the trip structure. We have implemented code in Sagemath to compute this data on HPGs and to perform moves on webs. Additionally, we have developed a browser-based user interface that enables the construction and analysis of HPGs interactively.
4:45 - 5:00 PM EST
Open discussion for software-based graphical calculi
Open Discussion - 11th Floor Lecture Hall
Session Chair
Joshua Swanson, University of Southern California
Abstract
Interested participants will brainstorm about existing and future software to support computation with graphical calculi.
Thursday, December 11, 2025
9:00 - 9:45 AM EST
Skein and cluster quantizations of the moduli space of decorated G-local systems
11th Floor Lecture Hall
Speaker
Tsukasa Ishibashi, Tohoku University
Session Chair
Ian Le, Australian National University
Abstract
I will discuss a comparison of the two quantization approaches to the moduli space in the title: the quantum cluster algebra and the skein algebra. If time permits, I will also explain how this relationship provides a topological perspective on the quantum Fock–Goncharov duality.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Skein Algebras and Quantum Groups
11th Floor Lecture Hall
Speaker
Thang Le, Georgia Institute of Technology
Session Chair
Ian Le, Australian National University
Abstract
Skein Algebras and Quantum Groups. The sl_n-skein algebra of a surface provides a quantization of the SL_n(C)-character variety. For surfaces with boundary, this framework extends naturally to the stated skein algebra. We demonstrate how various aspects of quantum groups admit simple and transparent geometric interpretations through the lens of stated skein algebras. In particular, we will present a geometric realization of the dual canonical basis of O_q(sl_3) using skeins."
11:30 AM - 12:15 PM EST
Webs and their intersections
11th Floor Lecture Hall
Speaker
Zhe Sun, University of Science and Technology of China
Session Chair
Ian Le, Australian National University
Abstract
G-Webs as certain trivalent graphs on the surfaces, appear naturally in the G-skein algebra and its classical limit the regular function ring of the G character variety. I will explain how we introduce their intersection number to parameterize a basis of the algebra for SL3 (joint work with Daniel Douglas, Linhui Shen, Daping Weng) and Sp4 (joint work with Tsukasa Ishibashi, Wataru Yuasa).
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Skein identities at roots of unity
11th Floor Lecture Hall
Speaker
Vijay Higgins, University of California, Los Angeles
Session Chair
Daping Weng, University of North Carolina at Chapel Hill
Abstract
The Kauffman bracket skein algebra of an oriented surface is built from knots labeled by representations of Uq(sl2). When q is generic, the irreducible representations of Uq(sl2) correspond to the Jones-Wenzl projectors from the Temperley-Lieb category. When q is a root of unity, the relationship between the TL category and Uq(sl2)-mod is less complete but is combinatorially richer. We will discuss special skein identities involving Jones-Wenzl projectors at roots of unity. We will discuss how the easiest such identity can be used to recover the Chebyshev-Frobenius homomorphism of Bonahon-Wong. This is joint work with Indraneel Tambe.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
Cluster structures on moduli space of G-local systems and Skein algebras
11th Floor Lecture Hall
Speaker
Linhui Shen, Michigan State University
Session Chair
Daping Weng, University of North Carolina at Chapel Hill
Abstract
Let G be a split semi-simple algebraic group over Q. We introduce a natural cluster structure on the moduli space of framed G-local systems on surfaces with marked points. As a consequence, these moduli spaces admit natural Poisson structures, which can be further quantized. We also investigate their intrinsic connections with the theory of skein algebras.
Friday, December 12, 2025
9:00 - 9:45 AM EST
Diagrammatics for quantum symmetric pairs
11th Floor Lecture Hall
Speaker
Alistair Savage, University of Ottawa
Session Chair
John Lentfer, University of California, Berkeley
Abstract
Symmetric pairs consist of a complex simple Lie algebra and a subalgebra fixed by an involution. Passing to enveloping algebras, the latter becomes a Hopf subalgebra. Hence, its category of representations is naturally a monoidal category. The quantum analogue of this concept is that of a quantum symmetric pair. In the quantum setting, the subalgebra, called an iquantum enveloping algebra, is not a Hopf subalgebra. Rather, it is a coideal subalgebra. This means that the category of representations of the iquantum enveloping algebra is not monoidal. Instead, it is a module category over the category of representations of the larger quantum enveloping algebra. In this talk, we will explore the representation theory of iquantum enveloping algebras from the point of view of diagrammatic interpolating categories. This is joint work with Hadi Salmasian and Yaolong Shen.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Higher Dimers, SL_3 and SL_4 Webs and Grassmannian Cluster Algebras.
11th Floor Lecture Hall
Speaker
Kayla Wright, John Hopkins University
Session Chair
John Lentfer, University of California, Berkeley
Abstract
Two classically studied rings in algebraic geometry and representation theory are the homogeneous coordinate ring of the Grassmannian of k-planes in n-space and the ring of SL_r tensor invariants. Both admit rich combinatorial models—in particular, the focus of this workshop: webs. The rings of SL_3 and SL_4 tensor invariants possess rotation-invariant web bases, first introduced by Kuperberg for SL_3 and later extended to SL_4 by Gaetz–Pechenik–Pfannerer–Striker–Swanson. We use these web bases to explore the cluster algebra structure of the homogeneous coordinate ring of the Grassmannian. Specifically, we study the generators of our cluster algebras via SL_3 and SL_4 webs as well as higher dimer covers on certain planar bicolored graphs. By connecting these two web bases, we derive combinatorial formulas for the generators using a phenomenon called web duality, first observed in small cases by Fraser–Lam–Le. We further show that, for large families of and webs, this web duality can be realized through the transpose of standard Young tableaux, which index the basis webs. In the talk, we will present this web duality using explicit examples and focus on the cluster algebraic applications of this phenomenon.
11:30 AM - 12:15 PM EST
Kasteleyn's theorem for classical groups
11th Floor Lecture Hall
Speaker
Richard Kenyon, Yale University
Session Chair
John Lentfer, University of California, Berkeley
Abstract
This joint work with Nick Ovenhouse and Haihan Wu. Kasteleyn's theorem shows how to count dimer covers of a planar graph G with the Pfaffian of an associated adjacency-type matrix. For a graph G with connection taking values in SL(n), SO(n), or Sp(2n) we extend Kasteleyn's theorem to count traces of ""multiwebs"" in G.
12:30 - 2:00 PM EST
Lunch/Free Time
2:00 - 2:45 PM EST
Symplectic and Orthogonal webs
11th Floor Lecture Hall
Speaker
Elijah Bodish, Indiana University Bloomington
Session Chair
Gregg Musiker, University of Minnesota
Abstract
I will discuss webs for symplectic groups (type C) and orthogonal groups (type O), based on joint work with Elias--Rose--Tatham and Haihan Wu respectively. Our main results give complete sets of generators and relations for the monoidal categories generated by fundamental representations. We also find explicit bases in some special cases. The talk will begin by reviewing the rank 2 symplectic group case, which already appeared in Kuperberg's ``Spiders for rank 2 Lie algebras". Then I will explain how to generalize to higher rank and to the orthogonal setting.
3:00 - 3:45 PM EST
Type B webs and spin link homology
11th Floor Lecture Hall
Speaker
David Rose, University of North Carolina at Chapel Hill
Session Chair
Gregg Musiker, University of Minnesota
Abstract
I'll present joint work with Bodish and Elias giving a generators-and-relations presentation for the category of finite-dimensional quantum so_{2n+1} representations — the titular Type B webs. Some of the defining relations coincide with those in the orthogonal case (from Elijah's talk), but the relations involving the spin representation are instead informed by our previous work on spin link homology. This realizes a conjectural meta-principle: that webs in non-simply laced types can be obtained by ‘folding’ the categorifications of their simply laced counterparts. In the present case, this folding from type A to B is implemented via a new involution on a certain bicategory of gl_{2n} foams given in earlier joint work with Queffelec.
4:00 - 4:30 PM EST
Coffee Break
11th Floor Collaborative Space
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