Schedule for: 23w5149 - Algebraic Aspects of Matroid Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Intersection theory studies how subvarieties of an algebraic variety X intersect. Algebraically, this information is encoded in the Chow ring A(X). When X is the toric variety of a simplicial fan, Brion gave a presentation of A(X) in terms of generators and relations, and Fulton and Sturmfels gave a "fan displacement rule” to intersect classes in A(X), which holds more generally in tropical intersection theory. In these settings, intersection theoretic questions translate to algebraic combinatorial computations in one point of view, or to polyhedral combinatorial questions in the other. Both of these paths lead to interesting combinatorial problems, and in some cases, they are important ingredients in the proofs of long-standing conjectural inequalities. This talk will survey a few problems on matroids that arise in combinatorial intersection theory, and a few approaches to solving them. It will feature joint work with Graham Denham, Chris Eur, June Huh, Carly Klivans, and Raúl Penaguião.

Baker and Bowler's theory of matroids with coefficients can be understood as an extension of linear algebra from fields to unwieldier objects such as partial fields and hyperfields. In this talk, we complement this theory with the notion of a morphism of matroids with coefficients, which passes through a subtle process that we call "perfection". Eventually we gain a categorical framework for matroid bundles over F1-schemes. All this stems from joint ideas with Baker, Jarra and Jin. As a sample application, we define the Tutte-Grothendieck ring of an F1-scheme, which can be seen as a "detropicalization" of algebraic K-theory. The Tutte-Grothendieck ring of the moduli space of matroids carries a "universal Tutte class" whose pullback to any matroid is the Tutte polynomial of the matroid. If time allows, we muse about how this might be used to reprove the Fink-Speyer theorem.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Orthogonal matroids provide combinatorial abstraction of maximal isotropic subspaces of 2n- dimensional spaces, and are a generalization of matroids to type D. We extend Baker-Bowler's theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, and orthogonal vector sets, and establish basic categorical properties. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. For instance, we will give a one-slide proof that an orthogonal matroid is regular if and only if it is representable over $\mathbb{F}_2$ and $\mathbb{F}_3$. This is joint work with Donggyu Kim.

The extremal function of a class of matroids maps each positive integer n to the maximum number of elements of a simple matroid in the class with rank at most n. We will present a result concerning the role of finite groups in minor-closed classes of matroids, and then use it determine the extremal function for several natural classes of representable matroids. This is joint work with Jim Geelen and Peter Nelson.

Motivated by certain toric vector bundles on a toric variety, we introduce "tautological classes of matroids" as a new geometric model for studying matroids. We describe how it unifies, recovers, and extends various results from previous geometric models of matroids. We then explain how it raises several new questions that probe the boundary between combinatorics and algebraic geometry, and discuss how these new questions relate to older questions in matroid theory.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will share some discoveries on K-rings of wonderful varieties and matroids. The main result is a Hirzebruch—Riemann—Roch-type theorem. I will also discuss applications to moduli spaces of curves. Joint work with Matt Larson, Sam Payne and Nick Proudfoot.

I will survey high-dimensional expanders (HDX) and the alternative perspective they provide on some of the recent advances in matroid theory concerning log-concave/Lorentzian polynomials. The HDX perspective has been key in solving algorithmic problems concerning sampling and/or counting in combinatorial structures, including matroids and some objects beyond matroids (such as matchings, Eulerians tours, etc.). I will formulate conjectures which, if proven, would generalize parts of the theory that has been developed for log-concave polynomials/matroids. I will then mention some results concerning fractionally log-concave and sector-stable polynomials, which provide evidence for the general conjectures.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We reinterpret the Hodge-Riemann relations in degree 1 for matroid fans as a convex-geometric phenomenon, which at the moment has a rather circuitous proof. This generalizes a certain convexity preserving property of matroid greedy algorithms for maps from convex domains to Sym^n(R).

In this talk, I will study the Poincaré polynomials of many graded objects of recent interest in the field of matroid theory. These will include the Poincaré polynomial of the graded Möbius algebra (whose coefficients are the Whitney numbers of the second kind), of the intersection cohomology module and a certain quotient of it (the Z-polynomial and Kazhdan--Lusztig polynomial of a matroid, respectively), and of the usual and augmented Chow ring of a matroid. I will give a survey of what is known about these polynomials, focusing on properties such as nonnegativity, top-heaviness, unimodality, log-concavity, gamma positivity, and real-rootedness. This talk will include joint work with Tom Braden, Luis Ferroni, June Huh, Nick Proudfoot, Matthew Stevens, Lorenzo Vecchi, and Botong Wang.

Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many matroid invariants are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid. In my talk I will present the class of (elementary) split matroids which contains strictly the large class of paving matroids. Valuations on this relatively new class of matroids behave very well. I will demonstrate this by looking at a few examples. This talk is based on work with Michael Joswig and Luis Ferroni.

I will present a proof of the Kähler properties of the Chow ring for a large class of tropical fans based on three basic operations on fans which preserve the balancing condition (orientability). In the case of matroids, this allows to circumvent some of the difficulties arising in the work by Adiprasito, Huh, and Katz. Time permitting, I will discuss generalizations both in the local and global settings, and some applications to geometric questions. Based on joint works with Matthieu Piquerez.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The characteristic polynomial of a matroid is “categorified” by the Orlik-Solomon algebra, and questions about the characteristic polynomial can be enriched to questions about the Orlik-Solomon algebra, now regarded as a graded representation of the group of symmetries of the matroid. Other polynomial invariants with natural categorifications include the Chow and augmented Chow polynomials, the Kazhdan-Lusztig polynomial, and the Z-polynomial. I will survey various results and conjectures about these categorical invariants, ending with a discussion of what it means for a categorical invariant to be valuative.

Delta-matroids are generalizations of matroids to type B. We introduce a new invariant of delta-matroids, and prove that it often satisfies log-concavity properties analogous to those of the Tutte polynomial of matroids. Joint with Christopher Eur, Alex Fink, and Hunter Spink.

Ardila and Boocher defined a class of varieties that compactify affine space and which are related to reciprocal planes. More recently, these varieties have been called matroid Schubert varieties, and they play a central role in Kazhdan-Lusztig theory for matroids and the proof of the Top Heavy conjecture. We study these varieties as equivariant compactifications and give necessary and sufficient conditions to characterize them. Our results resemble the correspondence between toric varieties and polyhedral fans. We also generalize the theory to include partial compactifications and morphisms between them.

Adiprasito, Huh, and Katz showed that the Chow ring of a finite matroid satisfies the Kähler package; in particular, its graded components have dimensions forming a sequence which is symmetric and unimodal. Inspired by this result and recent work of Liao, we strengthen this symmetry and unimodality to incorporate the action on the Chow ring of a group of matroid automorphisms. This talk is based on joint, ongoing work with Robbie Angarone and Vic Reiner.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we will explain the extension of the definitions of Chern-Schwartz-MacPherson (CSM) cycles of matroids to tropical manifolds. With this definition, we will see a correspondence theorems for the CSM classes of tropicalisations of subvarieties of toric varieties, an adjunction formula relating the CSM cycles of a tropical manifold and a codimension-one tropical submanifold and a Noether’s Formula for compact tropical surfaces. Joint work with Felipe Rincón and Kris Shaw.

Matroid symmetric powers were first introduced in their full generality by John Mason in 1981 and are exactly what one might expect: an attempt to generalize the theory of symmetric powers of linear spaces to matroids. Mason’s early work demonstrates a key obstacle in this theory by constructing matroids that lack large symmetric powers. We rediscover this theory in the context of tropical ideals — polynomial semiring ideals equipped with the structure of a finitary matroid. In this talk I will present my joint work with Felipe Rinc\'on on zero-dimensional ​``paving” tropical ideals, as well as my own investigation of the connection between matroid symmetric products and positive-dimensional tropical ideals. We will specifically see that the theory of matroid symmetric powers is, in some sense, equivalent to understanding which tropical linear spaces are the variety of a tropical ideal.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We explain how several tropical constructions can be extended to produce not necessarily realizable oriented matroids, moving beyond the classical tropicalization setting. In particular, we show how chirotopes can be glued with respect to a matroid subdivision, with subdivisions induced from triangulations of the product of two simplices as the primary example. From a topological point of view, we use a variant of Viro’s patchworking to produce a topological representation of these oriented matroids. This is joint work with Marcel Celaya and Georg Loho.

Lopez de Medrano-Rin\con-Shaw defined Chern-Schwartz-MacPherson cycles for an arbitrary matroid M and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of T(M;x,0), where T(M;x,y) is the Tutte polynomial associated with M. In this talk, we provide a direct calculation of the degree of a matroid Chern-Schwartz-MacPherson cycle by taking its stable intersection with a generic tropical linear space of the appropriate codimension and showing that the weighted point count agrees with the Gioan-Las Vergnas refined activities expansion of the Tutte polynomial.

Geometric connections have played a fundamental role in many recent approaches to matroid theory, especially as demonstrated in June Huh et. al.’s major contributions to the field through the Chow Ring of a Matroid and the development of combinatorial Hodge theory. In this talk I will present my joint work with Felipe Rinc\'on on the computation of Chern Roots, which can be used to compute other well known matroid invariants. These Chern roots were first given for matroids by Andrew Berget et. al. in the ``BEST'' paper. We simplify their formulas, and find that CSM cycles in particular can be factored in a nice way, showing that they live in a much smaller subring of the Chow ring. This in turn allows us to define new matroid invariants.

If an ordered point configuration in projective space is represented by a matrix of coordinates, the resulting matrix is determined up to the action of the general linear group on one side and the torus of diagonal matrices on the other. We study orbits of matrices under the action of the product of these groups, as well as their images in quotients of the space of matrices like the Grassmannian. The main question is what properties of closures of these orbits are determined by the matroid of the point configuration; the main result is that their equivariant K-classes are so determined. I will also draw connections to positivity and the work of Berget, Eur, Spink and Tseng. The results of mine featured here are mostly joint with Andy Berget.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will briefly describe the program to develop tropical schemes, with an emphasis on recent progress. Tropical schemes can be described as towers of (valuated) matroids, and I will focus on questions that arise at the inferface between the geometry and matroid theory.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.