Abstracts

Daniel Bernstein Algebraic matroids in rigidity theory and matrix completion

Given an irreducible subvariety V of C^S, the coordinate projections of C^S preserving the dimension of V give the spanning sets of a matroid on ground set S called the algebraic matroid underlying V. Certain problems in matrix completion and rigidity theory motivate the study of the algebraic matroids underlying determinantal varieties and Cayley-Menger varieties. In this talk, I will give a broad overview of our current state of knowledge about these matroids with particular attention to open problems.

Justin Chen Matroids in Macaulay2

Matroids are a powerful combinatorial abstraction of notions like independence and submodularity, but there are many interesting examples which are concrete and amenable to computation! I will discuss/demo the Matroids package in Macaulay2, which is a general-purpose package for creating, testing, and exploring with matroids. Possible topics include converting between different representations of matroids, formation/detection of minors, computing deletion-contraction invariants, and matroid isomorphism. Audience participation (e.g. requests for favorite matroids) is strongly encouraged.

Galen Dorpalen-Barry Whitney Numbers for Poset Cones

Hyperplane arrangements dissect R^n into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes x_i=x_j inside R^n for the symmetric group, thought of as the type A_{n-1} reflection group. Here,

• cones correspond to posets,
• chambers within the cone correspond to linear extensions of the poset,
• the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset.

We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation.

Max Kutler The motivic zeta function of a matroid

We associate to any matroid a motivic zeta function. If the matroid is representable by a complex hyperplane arrangement, then this coincides with the motivic Igusa zeta function of the arrangement. We show that this zeta function is rational and satisfies a functional equation. Moreover, it specializes to the topological zeta function introduced by van der Veer. We compute the first two coefficients in the taylor expansion of this topological zeta function, answering two questions of van der Veer. This is joint work with David Jensen and Jeremy Usatine.

Connor Simpson Simplicial generators for Chow rings of matroids

We introduce a new presentation for the Chow ring of a matroid in terms of variables whose action can be interpreted combinatorially as matroid quotients. Applications of this property include a new proof of Poincar\'e duality for the Chow ring, a combinatorial interpretation for a Groebner basis of Feichtner and Yuzvinsky, a formula for the volume polynomial of the Chow ring, and a new proof of the Heron-Rota-Welsh conjecture.

Jeremy Usatine Hyperplane arrangements and compactifying the Milnor fiber

Milnor fibers are invariants that arise in the study of hypersurface singularities. A major open conjecture predicts that for hyperplane arrangements, the Betti numbers of the Milnor fiber depend only on the combinatorics of the arrangement. I will discuss how tropical geometry can be used to study related invariants, the virtual Hodge numbers of a hyperplane arrangement’s Milnor fiber. This talk is based on joint work with Max Kutler.

Cynthia Vinzant Log-concave polynomials, matroids, and expanders

Complete log-concavity is a functional property of real multivariate polynomials that translates to strong and useful conditions on its coefficients. I will introduce the class of completely log-concave polynomials in elementary terms, discuss the beautiful real and combinatorial geometry underlying these polynomials, and describe applications to random walks on simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.