Invited Speakers:
Jonathan Leake (University of Waterloo)
Title: Log-concavity, High-dimensional Expanders, and Lorentzian Polynomials
Abstract:
In this talk, we demonstrate a connection between log-concavity statements and sampling algorithms via high-dimensional expanders and Lorentzian polynomials. To do this, we first discuss two conjectures which were resolved about 5-10 years ago: one on the log-concavity of independent sets of matroids (due to Brändén-Huh and Anari-Liu-Oveis Gharan-Vinzant), and one on efficiently sampling bases of matroids (due to Anari-Liu-Oveis Gharan-Vinzant). From there we will present some new results on generalized graph colorings which extend these and other previous results. In particular, we will show how this can be used to obtain log-concavity statements and sampling algorithms for linear extensions of posets. No knowledge of matroids will be required for this talk. Joint work with Kasper Lindberg and Shayan Oveis Gharan.
Colleen Robichaux (University of California, Los Angeles)
Title: Deciding Schubert Positivity
Abstract:
Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation. In this talk we discuss the closely related problem of determining the positivity of Schubert coefficients from a computational complexity perspective. This is joint work with Igor Pak.
Chris Eur (Carnegie Mellon University)
Title: h*-vectors for matroids
Abstract:
The Hodge theory of matroids by Adiprasito--Huh--Katz resolved outstanding conjectures in matroid theory by establishing positivity properties for Chow rings of matroids. In algebraic geometry, a notion complementary to the Chow ring is the K-ring of a variety. We establish positivity properties for K-rings of matroids. Our results give a new proof of the 20-year-old f-vector conjecture of Speyer and resolve the conjecture of Tohaneanu that higher order Orlik--Terao algebras are Cohen--Macaulay.
Susan Morey (Texas State University)
Title: Extremal ideals and Bounding Invariants of Square-free Monomial Ideals
Abstract:
This talk will focus on a concretely defined class if ideals, called {\it extremal ideals}, with an emphasis on demonstrating how techniques from a variety of mathematical disciplines can be employed to gain useful information for these ideals. More specifically, for each positive integer $q$, there is an extremal ideal with $q$ generators. Understanding algebraic properties of an extremal ideal provides information for all square-free monomial ideals with $q$ generators. Multiple invariants and algebraic properties, such as the betti numbers and integral closures, can be studied for all square-free monomial ideals with $q$ generators by finding or bounding the desired information for extremal ideals. The definition of extremal ideals is highly symmetric, which has proven useful in applying techniques such as linear programming and discrete geometry.
Contributed Talks:
Ethan Partida (Brown University, Providence, R.I.)
Title: Graded Ehrhart Theory of Unimodular Zonotopes
Abstract:
Graded Ehrhart theory is a new q-analogue of Ehrhart theory based on theorbit harmonics method. We study the graded Ehrhart theory of unimodular zono-topes from a matroid-theoretic perspective. Generalizing a result of Stanley (1991), weprove that the graded lattice point count of a unimodular zonotope is a q-evaluationof its Tutte polynomial. We conclude that the graded Ehrhart series of a unimodularzonotope is rational and obeys graded Ehrhart—Macdonald reciprocity. In an algebraicdirection, we prove that the harmonic algebra of a unimodular zonotope is a coordinatering of its associated arrangement Schubert variety. Using the geometry of arrange-ment Schubert varieties, we prove that the harmonic algebra of a unimodular zonotopeis finitely generated and Cohen–Macaulay. We also give an explicit presentation of theharmonic algebra of a unimodular zonotope in terms of generators and relations. Ourwork answers, in the special case of unimodular zonotopes, two conjectures of Reinerand Rhoades (2024).
Leigh Foster (University of Waterloo)
Title: The Squish Map and the SL_2(C) Double Dimer Model
Abstract:
A plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a "coarser" plane partition, made of cubes of side length 2. Indeed, there are two such approximations obtained by "rounding up" or "rounding down" to the nearest cube. We relate this coarsening (or downsampling) operation to the squish map, and exhibit a related measure-preserving map between the dimer model on the honeycomb graph, and the SL_2 double dimer model on a coarser honeycomb graph; we compute the most interesting special case of this map, related to plane partition q-enumeration with 2-periodic weights. As an application, we specialize the weights to be certain roots of unity, obtain novel generating functions (some known, some new, and some conjectural) that (−1)-enumerate certain classes of pairs of plane partitions according to how their dimer configurations interact.
Tomaz Kosir (University of Ljubljana, Slovenia)
Title: A proof of the Box Conjecture for commuting pairs of matrices.
Abstract:
We will review the Box Conjecture of Anthony Iarrobino and his collaborators on commuting pairs of nilpotent matrices, and sketch a proof of the Conjecture. The proof hinges naturally on the Burge correspondence between the set of all partitions anda set of binary words, and on Shayman’s results on invariant subspaces of a nilpotent matrix. This is joint work with John Irving and Mitja Mastnak.
Moriah Elkin (Cornell University, Ithaca, NY, USA)
Title: Open quiver loci, CSM classes, and chained generic pipe dreams
Abstract:
In the space of type A quiver representations, putting rank conditions on the maps cuts out subvarieties called "open quiver loci." These subvarieties are closed under the group action that changes bases in the vector spaces, so their closures define classes in equivariant cohomology, called "quiver polynomials." Knutson, Miller, and Shimozono found a pipe dream formula to compute these polynomials in 2006. To study the geometry of the open quiver loci themselves, we might instead compute "equivariant Chern-Schwartz-MacPherson classes," which interpolate between cohomology classes and Euler characteristic. I will introduce objects called "chained generic pipe dreams" that allow us to compute these CSM classes combinatorially, and along the way give streamlined formulas for quiver polynomials.
John Lentfer (UC Berkeley)
Title: Diagonal supersymmetry for coinvariant rings
Abstract:
The classical coinvariant ring was generalized by Haiman (1994) to the diagonal coinvariant ring, which consists of a polynomial ring in two sets of variables quotiented by the ideal generated by polynomials invariant under the diagonal action of the symmetric group, without constant term. The (k,j)-bosonic-fermionic coinvariant rings are defined analogously for k sets of commuting (bosonic) and j sets of anticommuting (fermionic) variables. We prove the "diagonal supersymmetry" conjecture of F. Bergeron (2020), which asserts that the multigraded Frobenius series of a (k,j)-bosonic-fermionic coinvariant ring can be expressed in terms of universal coefficients, super Schur functions, and Frobenius characters.
Erica Liu (University of Waterloo)
Title: Toric Compactifications and Critical Points at Infinity in Analytic Combinatorics
Abstract:
The field of Analytic Combinatorics in Several Variables (ACSV) provides powerful tools for deriving asymptotic information from multivariate generating functions. A key challenge arises when standard saddle-point techniques fail due to the presence of critical points at infinity (CPAI), obstructing local analyses near singularities. Recent work has shown that Morse-theoretic decompositions remain valid under the absence of CPAI, traditionally verified using projective compactifications. In this talk, I will present a toric approach to compactification that leverages the Newton polytope of a generating function to construct a toric variety tailored to the function’s combinatorial structure. This refinement not only tightens classification of CPAI but also enhances computational efficiency. Through concrete examples and an introduction to tropical and toric techniques, I will demonstrate how these methods clarify the asymptotic landscape of ACSV problems, especially in combinatorially meaningful settings. This talk draws on joint work studying toric compactifications as a bridge between algebraic geometry and analytic combinatorics.
Anastasia Nathanson (University of Minnesota)
Title: Multipermutohedral Chow rings
Abstract:
The multipermutohedral Chow ring was introduced in a series of papers by Clader, Damiolini, Eur, Huang, Li, and Ramadas to study moduli spaces with cyclic symmetry. It generalizes Chow rings of permutohedral varieties, type-B Coxeter arrangements, and delta matroids. In the present work, we establish the combinatorial structure of the multipermutohedral Chow ring through an explicit Gr"obner basis, yielding a Feichtner-Yuzvinsky-type monomial basis and a formula for the Hilbert series. Using this formula, we refine the palindromicity of the Hilbert series. From a representation-theoretic perspective, we also compute the equivariant Hilbert se- ries under two natural group actions and construct combinatorial maps that establish equivariant unimodality and palindromicity.
Tianyi Yu (LACIM)
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. In this paper, 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. The proof relies on three key ingredients: a correspondence between these two models, a natural subdivision of bumpless pipedreams, and a symmetry property of increasing chains. This talk is based on joint work with Jack Chou.
Siddarth Kannan (MIT, Cambridge, MA)
Title: Pólya enumeration, wreath product symmetric functions, and moduli spaces of curves
Abstract:
We develop a calculus for calculating S_n-equivariant Euler characteristics of moduli spaces of stable curves and stable maps. Our approach involves an enrichment of Pólya's cycle index polynomial of a graph to a certain algebra of wreath product symmetric functions. This algebra is constructed as the tensor product \bigotimes_{i > 0} \Lambda(S_i), where \Lambda(S_i) is the ring of wreath product symmetric functions studied by Macdonald. Following his work, we prove that \bigotimes_{i > 0} \Lambda(S_i) may be viewed as the Grothendieck ring of polynomial functors which take symmetric sequences of vector spaces to vector spaces. This ring admits a plethystic action on the ring \Lambda of ordinary symmetric functions, which is expressed in terms of ordinary plethysm and skewing operations. In this way, each finite graph acts on the ring of symmetric functions, and this gives rise to effective computational tools for equivariant Euler characteristics.
Chi Trung Chau (Chennai Mathematical Institute, India)
Title: Second symbolic powers of squarefree monomial ideals
Abstract:
We compute sharp bounds for the Betti numbers of the second symbolic power of a squarefree monomial ideal based on its number of generators. In fact, we can construct its minimal cellular resolution explicitly. Compared to its ordinary counterpart, the resolution is not simplicial, and as the generators for higher powers are not explicitly known, there are not much hope for an analog for them at this point. This is joint work with Art Duval, Sara Faridi, Thiago Holleben, Susan Morey, and Liana Sega.
Chris McDaniel (Endicott College)
Title: Hodge-Riemann relations and a theorem of Cattani
Abstract:
In 2008, Cattani used deep results from the theory of variations of Hodge structure to prove that the so-called ordinary Hodge-Riemann relations and mixed Hodge-Riemann relations are equivalent under certain mild conditions for a class of objects he called Hodge-Lefschetz modules. We give a new elementary proof of this result in the special case where the Hodge-Lefschetz module is a graded Artinian Gorenstein algebra of codimension two.