Conference speaker abstracts:
Grant Barkley
Title: Extended weak order
Abstract: Extended weak order is the poset of biclosed sets in a positive root system, introduced by Matthew Dyer. It is an invariant of the underlying Coxeter group W, and contains the weak order on W as an order ideal. There are many open conjectures asserting that the extended weak order on an infinite Coxeter group behaves like the weak order on a finite Coxeter group. For instance, Dyer conjectures that the extended weak order is a lattice, which is a famous property of the weak order when W is finite. We will discuss some of these conjectures and recent progress on them.
Amanda Burcroff
Title: Eventual sign coherence for quivers
Abstract: The theory of cluster algebras has shown that many important spaces in math and physics have beautiful fundamental properties, such as the Laurent phenomenon, positivity, and sign coherence. This last property says that the combinatorial operation of mutation preserves some local structure in certain quivers, and the only known proofs rely on heavy tools from representation theory or algebraic geometry. Gekhtman and Nakanishi posed the Asymptotic Sign Coherence Conjecture for arbitrary quivers, which says sign coherence should eventually emerge in any sufficiently generic infinite mutation sequence. We prove, using purely combinatorial methods, that this conjecture holds with probability 1 for a random mutation sequence and that it holds in full for many families of quivers. This is joint work with Scott Neville.
Sergey Fomin
Title: Expressive curves
Abstract: A real plane algebraic curve C is called expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject to a mild technical condition), describe several constructions that produce expressive curves, and relate their study to the combinatorics of plabic graphs, their quivers, and links. This is joint work with Eugenii Shustin.
Christian Gaetz
Title: Combinatorial invariance for the coefficient of q in Kazhdan–Lusztig polynomials
Abstract: I will describe joint work with Grant Barkley and Thomas Lam in which we study the Combinatorial Invariance Conjecture (CIC), which asserts that Kazhdan–Lusztig polynomials depend only on the combinatorics of Bruhat order. Motivated by the cluster structure on Richardson varieties, we prove the combinatorial invariance of the coefficient of q in KL polynomials for arbitrary Coxeter groups. We also prove the Gabber–Joseph conjecture for the second-highest Ext group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group. No background on Kazhdan–Lusztig theory will be assumed.
Allen Knutson
Title: Matrix positroid varieties
Abstract: Fulton defined matrix Schubert varieties in 1992 by taking the preimage of Schubert varieties in GLn and taking the closure in Matn. Now in affine space, one can study these using Gröbner degeneration, and obtain polynomial representatives for their equivariant cohomology classes (fundamental and Chern–Schwartz–MacPherson). Recently similar representatives were found for the classes of positroid varieties [Fan–Guo–Su–Xiong]. Fulton’s trick doesn’t work out of the box, as I’ll explain, but can be tweaked to give a parallel story (and geometric meaning to the FGSX formula). This work is joint with Paul Zinn-Justin.
Jacob Matherne
Title: Log-concavity in algebraic combinatorics and representation theory
Abstract: In this talk, we will present a survey of log-concave sequences appearing in algebraic combinatorics and representation theory. On the one hand, we will discuss log-concavity properties of Schur polynomials and their relatives, and on the other hand log-concavity properties of representation-theoretic objects in type A such as irreducible finite-dimensional representations, Verma modules, and parabolic Verma modules. This talk is based on joint work with Yairon Cid-Ruiz, June Huh, Apoorva Khare, Yupeng Li, Karola Mészáros, and Avery St. Dizier.
Isabella Novik
Title: Lower bounds on face numbers
Abstract: In this talk, I will discuss several approaches to studying the face numbers of simplicial complexes, with a particular focus on obtaining lower bounds. These approaches include classical rigidity theory, Stanley–Reisner rings, and higher‑dimensional stress spaces. I will also describe a number of both classical and recent results on face numbers that have been obtained using these tools.
Vincent Pilaud
Title: Building lattices via flat scaffolding projections
Abstract: The talk will present a new method for proving that certain posets are lattices and will illustrate it through both classical examples and more recent constructions. Joint work with Daria Poliakova.
Martha Precup
Title: Dimension stability for Hessenberg varieties
Abstract: Hessenberg varieties are subvarieties of the flag variety parametrized by conjugacy classes of matrices and a choice of a weakly increasing sequence of positive numbers. This family of varieties includes Springer fibers, the Peterson variety, and permutohedral variety as special cases, and plays an important role in geometric representation theory and combinatorics. Outside of special cases, basic questions about the geometry of Hessenberg varieties remain wide open: What is their dimension? When is a Hessenberg variety irreducible?
In this talk, we will discuss how to use tableaux combinatorics to prove Hessenberg varieties satisfy a surprising dimension stability condition. Recalling that a sheet is a certain union of conjugacy classes of equal dimension, we focus on Hessenberg varieties defined over a fixed sheet of matrices, showing that all such varieties have equal dimension. This talk is based on joint work with Goldin, and also with Harada and Robichaux.
Nathan Reading
Title: Theta functions in acyclic affine type
Abstract: Scattering diagrams are discrete-geometric objects that come from the study of mirror symmetry in algebraic geometry. Given a scattering diagram, one can, in principle, compute a theta function for each integer vector, defined as a sum indexed by broken lines (piecewise linear curves that bend on the walls of the scattering diagram). In practice, it may be hard to find the broken lines and much harder still to prove that one has found all possible broken lines. Gross, Hacking, Keel, and Kontsevich defined a cluster scattering diagram for each exchange matrix and showed that the cluster monomials are the theta functions indexed by vectors in the g-vector fan. In many cases, the set of all theta functions constitutes a basis for the (upper) cluster algebra. Outside of finite type, there are additional theta functions that are not cluster monomials. We characterize these additional theta functions in affine type and compute some of the structure constants for multiplying them. One of the structure constant computations gives new “imaginary” exchange relations among cluster variables. The theta functions for vectors in the boundary of the g-vector fan span a subalgebra of the cluster algebra that we call the imaginary subalgebra. It is a tensor product of generalized cluster algebras of finite type C.
This work, joint with Salvatore Stella, is the culmination of a 15-year effort to make cluster algebras of affine type “well understood” in the same senses that cluster algebras of finite type are well understood (and that, I believe, all cluster algebras of finite mutation-type eventually will be). Our proofs use tools developed in work with Speyer, with Stella, and most recently with Rupel and Stella, including doubled Cambrian fans, an affine almost-positive roots model, combinatorial models for cluster scattering diagrams of affine type, mutation-symmetries of the exchange matrix, neighboring seeds, and dominance regions. Thus there are “a lot of moving parts”, but in this talk I will assume minimal background and try to define each new object up to “what kind of object it is”.
Colleen Robichaux
Title: Deciding Schubert positivity
Abstract: Schubert coefficients count the number of points in a generic intersection of Schubert varieties. Using the framework of computational complexity, we discuss the problem of determining when a given Schubert coefficient is positive. Then we present an algorithmic solution to the Schubert positivity problem and highlight a connection to Polynomial Identity Testing. This is joint work with Igor Pak.
Anne Schilling
Title: q-deformations of the Tsetlin library
Abstract:The Tsetlin library is a random shuffling process on permutations of n letters, where each letter i can be interpreted as a book; book i is brought to the front of the bookshelf with an assigned probability xi. We define a q-deformation of the Tsetlin library by replacing the symmetric group action on permutations by the action of the type A Iwahori–Hecke algebra. We compute the stationary distribution and spectrum of this Markov chain by relating it to a Markov chain on complete flags over the finite field vector space 𝔽qn and applying techniques from semigroup theory. We prove that for a natural choice of xi the total variation distance mixing time of the q-Tsetlin library on permutations of n is O(n) compared to Θ(nlogn) for the Tsetlin library at q=1, which demonstrates a phase transition. We also generalize the q-Tsetlin library to words (with repeated letters), and compute its stationary distribution and spectrum. This is based on joint work with Arvind Ayyer, Sarah Brauner and Jan de Gier (https://arxiv.org/abs/2601.21195).
George Seelinger
Title: Flagged LLT polynomials and their applications
Abstract: LLT polynomials are a family of symmetric functions that serve as a q-deformation of a product of (skew) Schur functions. They can be defined using tableaux combinatorics and have a surprising way of showing up in many positivity problems in symmetric function theory with connections to representation theory and algebraic geometry. Some notable examples include the shuffle theorem, the Haglund–Haiman–Loehr formula for modified Macdonald polynomials, and a relationship with chromatic symmetric functions associated unit interval orders. Recently, in joint work with Blasiak, Haiman, Morse, and Pun, we develop the theory of a nonsymmetric analogue of LLT polynomials we call flagged LLT polynomials, which can be described combinatorially in terms of certain flagged tableaux. We will survey a few of their nice combinatorial and algebraic properties and discuss how flagged LLTs can be used to give nonsymmetric generalizations of certain results stated with symmetric functions.
Melissa Sherman-Bennett
Title: Unexpected toric Richardson varieties
Abstract: This talk focuses on Richardson varieties in the complete flag variety Fl(n), which are intersections of a Schubert variety with an opposite Schubert variety. Richardson varieties are indexed by intervals [u,v] in the Bruhat order on the symmetric group. In joint work with E. Gorsky and S. Kim, we study the question: when is a Richardson variety a toric variety? All Richardson varieties admit an action by an (n-1)-dimensional torus T, which also acts on Fl(n); the Richardson varieties which are toric varieties with respect to T were classified independently by Anderson and Tsukerman–Williams. We find that there are many additional toric Richardsons, which we call “unexpected”. We give a classification of toric Richardsons using the combinatorics of Bruhat intervals and investigate their moment polytopes.
Hunter Spink
Title: The quasisymmetric flag variety.
Abstract: Homology classes in the flag variety complete flag variety GLn/B correspond to linear functionals on the ring of symmetric coinvariants. In this talk I will describe recent work with Nantel Bergeron, Lucas Gagnon, Philippe Nadeau, and Vasu Tewari, on how linear maps between flag varieties corresponding to “setting variables to zero” give rise to an unusually combinatorially Schubert positive subvariety of the flag variety — the “quasisymmetric flag variety”.
Anna Weigandt
Title: Weakorder on alternating sign matrix varieties
Abstract: Alternating sign matrices (ASMs) form the MacNeille completion of the strong Bruhat order on the symmetric group. There is a natural interpretation of this poset as the containment order on ASM varieties, which are generalized determinantal ideals. In 2018, Hamaker and Reiner defined weak Bruhat order on ASMs, which when restricted to the symmetric group, is the usual weakorder. We initiate a geometric study of weakorder on ASMs varieties, focusing on how combinatorial properties of this poset describe geometric properties of ASM varieties. This is joint work with Laura Escobar and Patricia Klein.
Alex Yong
Title: RSK as a linear operator
Abstract: The Robinson–Schensted–Knuth correspondence (RSK) is a bijection between nonnegative integer matrices and pairs of Young tableaux. Viewing RSK as a linear operator on the coordinate ring of matrices leads to questions about its eigenvalues and diagonalizability. We give a diagonalizability criterion involving the ADE Dynkin diagrams and E9. This is joint work with Ada Stelzer.
Graduate student lightning talk abstracts:
Mackenzie Bookamer
Title: Springer fibers and connections to Kazhdan–Lusztig theory
Abstract: Springer fibers play a key role in geometric representation theory and have deep connections to Kazhdan–Lusztig theory. In this talk, we focus on two-row Springer fibers, where irreducible components are indexed by standard Young tableaux and represented combinatorially with standard non-crossing matchings. We are particularly interested in looking at the intersections of these components, and explore how their geometry reflects the structure of Kazhdan–Lusztig left cells. This talk aims to highlight the interplay between geometry, combinatorics, and representation theory.
Cameron Chang
Title: Fence complexes and positroid varieties
Abstract: We describe a polyhedral complex associated to a positroid variety, called fence complexes. The Ehrhart polynomial of these complexes gives the Hilbert polynomial of the positroid variety. Moreover, the fence complexes fit together to give a polyhedral presentation of the regular CW complex structure on the totally nonnegative Grassmannian. This is joint work with Pranav Enugandla and Josephine Hlavinka.
Ariana Chin
Title: Classification of Zamolodchikov periodic cluster algebras
Abstract: Zamolodchikov periodicity is a property of certain discrete dynamical systems and was one of the primary motivations for the creation of cluster algebras. It was first observed by Zamolodchikov in his study of thermodynamic Bethe ansatz for simply-laced Dynkin diagrams, and was proved by Keller to hold for tensor products of two Dynkin diagrams. More recently, Galashin–Pylyavskyy classified the Zamolodchikov periodic quivers. In this talk, we discuss the classification of all Zamolodchikov periodic cluster algebras, with connections to W-graphs, root systems, and maximal green sequences.
Jack Chou
Title: A positive combinatorial formula for the double Edelman–Greene coefficients
Abstract: Lam, Lee, and Shimozono introduced the double Stanley symmetric functions in their study of the equivariant geometry of the affine Grassmannian. They proved that the associated double Edelman–Greene coefficients, the double Schur expansion coefficients of these functions, are positive, a result later refined by Anderson. They further asked for a combinatorial proof of this positivity. We provide the first such proof, together with a combinatorial formula that manifests the finer positivity established by Anderson. Our formula is built from two combinatorial models: bumpless pipedreams and increasing chains in the Bruhat order.
Mike Cummings
Title: Webs and smooth components of two column Springer fibers
Abstract: Webs and Springer fibers are separately important objects in representation theory. Fung’s 1997 thesis gave the first evidence of a connection between sl2 webs and Springer fibers, showing that webs naturally index and describe the components of certain “two row” Springer fibers. However, this case is known to be far from generic. We deepen this connection with a similar correspondence in the substantially more complicated “two column” case. In particular, and building on works of Fresse, Melnikov, and Sakas–Obeid, we use webs to give a clean characterization of the smooth components of two column rectangle Springer fibers and a simple description of the geometry of these smooth components.
Pranav Enugandla
Title: Clasped web bases for SL4 invariants
Abstract: In the late 90’s, Kuperberg developed a web basis for the invariant space of tensor products of irreducible modules for SL2 and SL3, providing a diagrammatic calculus for homomorphism spaces in the representation category. In 2025, a web basis for tensor products of fundamental representations for SL4 was constructed by Gaetz, Pechenik, Pfannerer, Striker, and Swanson using hourglass plabic graphs. I will talk about joint work with Christian Gaetz, in which we extend Kuperberg’s clasped web bases for invariants of tensor products of arbitrary irreducible SL4 representations.
Benjamin Grant
Title: Topologizing infinite quivers and their mutations
Abstract: We offer a topological point of view on countably infinite quivers and quiver mutation sequences. Specifically, we construct several topological spaces whose points are quivers with the natural numbers as vertices and whose topologies are understood via restrictions to certain kinds of subquivers. We show that two of these spaces are homeomorphic to the Baire space, i.e., the space of countable sequences of natural numbers. We also show that mutations provide automorphisms of these spaces, meaning one may view mutations as providing topological dynamics on these spaces. Infinite mutation sequences are also considered; a complete characterization of the density of their domains of convergence/divergence in one of these spaces is given.
Soyeon Kim
Title: Basis for H4,(3,3)(X) for certain locally acyclic cluster varieties X
Abstract: Many important algebraic varieties, such as open subvarieties of the Grassmannian and braid varieties appear to be locally acyclic cluster varieties. Lam and Speyer developed a framework for studying its cohomology. For example, we know that its cohomology has mixed Hodge structure with mixed Tate type, so one can decompose it as a direct sum of the highest weight part Hp,(p,p) and the lower weight part Hp,(q,q). The basis for this highest weight part is pretty well understood and is seemingly related to canonical forms in physics. The lower weight part is more intricate. In my talk, I will describe a basis of H4,(3,3) for certain locally acyclic cluster varieties. This is based on ongoing joint work with Tonie Scroggin.
Tuong Le
Title: Quantum bumpless pipe dreams
Abstract: Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a rich combinatorial theory. In particular, their monomial expansion is given by a bumpless pipe dream formula. Quantum double Schubert polynomials are polynomial representatives of Schubert classes in the torus-equivariant quantum cohomology of the complete flag variety. In this talk, we will first review the bumpless pipe dreams formula for double Schubert polynomials. Then we will describe a generalization of the bumpless pipe dreams called quantum bumpless pipe dreams, giving a combinatorial formula for quantum double Schubert polynomials as a sum of binomial weights of quantum bumpless pipe dreams.
Benjamin Liber
Title: An equality for balanced digraphs
Abstract: Consider a balanced directed multigraph D. An s-convergence of D is an acyclic set of arcs such that every vertex has a path to a specified vertex s. We show that for any integer k, the number of k-element s-convergences is independent of the choice of s. This generalizes classical results on spanning arborescences, acyclic orientations, and minimum feedback arc sets. Moreover, we obtain a stronger result by replacing the acyclicity condition with the requirement that the chosen set of arcs has a prescribed set of cycles. This is joint work with Darij Grinberg.
Thomas C. Martinez
Title: Affine patches of open positroid varieties
Abstract: What happens when we take an open positroid variety and impose that an additional Plücker coordinate is nonzero? In this talk, I will explain how this simple question leads to affine analogues of familiar positroid objects, including affine Deodhar diagrams, affine Richardson links, and punctured plabic graphs. These objects extend the standard positroid story and give new tools for understanding point counts and cluster structure of open positroid varieties.
Jaewon Min
Title: Littlewood-Richardson rule of key polynomials
Abstract: Littlewood-Richardson coefficients can be computed applying branching rule to Schur polynomials. Generalized into key polynomials, the coefficients are still non-negative integers. In this talk, I will explain why this is true by multiplying particular key polynomials together. This result follows from the work done by A. Joseph and O. Mathieu, related to annihilators and filtrations concerning Demazure modules. The combinatorial description follows from the crystal graphs by M. Kashiwara and G. Lusztig.
Nutan Nepal
Title: Induced Lorentzian polynomials
Abstract: Suppose one has a party of m people, whose expertise collectively covers n topics. Given a subset T of the topics, one wishes to form a panel of |T| people from the party such that T can be covered by assigning a distinct topic to each panel member with the expertise. We show that the numbers of such panels, as T varies, form a Lorentzian polynomial. We achieve this by showing that a certain linear operator on polynomials, which we call the “inducing operator” for its connection to induced (poly)matroids, preserves Lorentzian polynomials and realizable volume polynomials.
Duy Phan
Title: Symmetry in equivariant cohomology of Pn
Abstract: We resolve a problem of Anderson and Fulton by providing a symmetric and positive product rule for the equivariant cohomology of projective space.
Zachary Slonim
Title: Stretched Schubert coefficients are eventually quasi-polynomial
Abstract: For a permutation u ∈ Sn, let N∗u ∈ SNn be the permutation with scaled Lehmer code. For given u, v, w ∈ Sn and integer N, the stretched Schubert coefficients are defined as fu,v,w(N) := cN∗u,N∗vN∗w. Our main result is that the function fu,v,w(N) is eventually quasi-polynomial. We hope to sketch the ideas of the proof which uses combinatorics of pipe dreams to show that Schubert coefficients are given as an alternating sum of the numbers of integer points in certain polytopes. These polytopes behave nicely under stretching, and we use Ehrhart theory to obtain the result.
Tanvi Thummar
Title: m is for meet-congruence
Abstract: Lattice congruences and quotients of the weak order on a finite Coxeter group are a rich source of combinatorial insight. In this talk, we will discuss meet-congruences on semilattices. We show that meet-congruences and meet-quotients play a crucial role in the combinatorics of the m-eralized weak order and m-eralized Cambrian lattice associated to a finite Coxeter group. This talk is based on joint work with Nathan Reading.
Jasper Ty
Title: Noncommutative key polynomials and Demazure atoms
Abstract: The theory of noncommutative Schur functions provides an unconventional reformulation of Schur positivity for various families of symmetric functions. One cooks up this reformulation via the classical Cauchy identity and a clever shift in perspective. This “shift” happens to be agnostic to the Cauchy identity used. For example, in 1999, Lenart used the same approach with a different Cauchy identity to compute the Schubert expansion of Grothendieck polynomials. I will discuss the foundational setup of my current project, which uses Lascoux’s nonsymmetric Cauchy identity to define noncommutative key polynomials and noncommutative Demazure atoms, both equipped to detect atom positivity and key positivity respectively.
Sienna Unter
Title: A sampling of bases for the cone of centrally symmetric generalized permutahedra
Abstract: For a centrally symmetric (CS) projective fan F, the CS deformation cone of F is the space containing all CS polytopes P whose normal fan coarsens F. In particular, if F is the braid arrangement, the CS deformation cone is the cone of all centrally symmetric generalized permutahedra (CSGP). Notably, CSGP are equivalent to connectivity functions, as defined by Robertson and Seymour in their graph minors project. Under Minkowski summation, a few classes of polytopes are known to form a basis for the cone of generalized permutahedra. Building on this work, this talk will discuss one of the classes of CSGP that form a basis for this CS deformation cone. This is joint work with Spencer Backman.
Kaitao Xie
Title: Total positivity and remarkable polyhedral spaces
Abstract: As a topic in Lie theory, total positivity has found connections to such diverse areas as combinatorics, quantum group, cluster algebra, higher Teichmuller theory, and theoretical physics. Many topological objects in total positivity appear to share some remarkable properties. In this talk, we explore these properties for the twisted flag varieties, double flag varieties, double Bruhat cells of Kac–Moody groups, and the wonderful compactifications of semisimple groups. This is based on some recent joint works with Xuhua He.