Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank r on n elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants. In this talk, we will first sketch the background that is needed to understand their polytope and valuative invariants. We will then provide the needed characterizations of many of the matroids that Ferroni and Fink conjectured to yield vertices, we will give additional examples of such matroids, and we will mention some open problems.
Order polytopes of posets have been a very rich topic at the crossroads between combinatorics and discrete geometry since their definition by Stanley in 1986. The h^*-polynomials of order polytopes of graded posets are known to be gamma-nonnegative by work of Brändén, but it is currently unknown whether they are real-rooted (this is a special case of the Neggers-Stanley conjecture, that was disproved in its general form by work of Brändén and Stembridge).
In the present talk we will introduce an equivariant version of Brändén's gamma-nonnegativity result, using the tools of equivariant Ehrhart theory (introduced by Stapledon in 2011). Namely, we prove that order polytopes of graded posets are always gamma-effective, i.e., that the gamma-polynomial associated with the equivariant h^∗-polynomial of the order polytope of any graded poset has coefficients consisting of actual characters. This is joint work with Akihiro Higashitani.
Computing linear extensions and order polynomials of posets is in general a hard problem with no explicit formulas and nice structure and properties. When the poset is a Young diagram of a straight shape, the number of linear extensions is given by the hook-lnegth formula, yet the corresponding number of plane partitions (order polynomial of that poset) does not have a product formula. In the absence of such nice formulas we will show a general approach to proving that the order polynomials of posets corresponding to skew Young diagrams have positive coefficients. We will also discuss some applications to matroids. Joint work with Luis Ferroni and Alejandro Morales.
Zaslavsky’s theorem says that the number of regions in the complement of a real hyperplane arrangement is equal to an evaluation of the characteristic polynomial of the arrangement. This relation generalises to all oriented matroids and can be categorified using the Orlik-Solomon algebra and three seemingly different filtrations of the tope space of an oriented matroid: the dual Varchenko–Gelfand degree filtration, Kalinin’s spectral sequence, and Quillen’s augmentation filtration. In this talk, we show that all of these filtrations coincide. The Varchenko–Gelfand filtration has the advantage that it is defined over the integers. We also show that the Varchenko-Gelfand approach can be used to filter the so-called Z-sign cosheaf on fan of the underlying matroid. The filtration of the Z/2-sign cosheaf has previous applications to the topology of real algebraic varieties via patchworking.
This talk is based on joint work with Chi Ho Yuen.
The Chow ring of a matroid was introduced by Adiprasito, Huh, and Katz as the Chow ring of the toric variety associated to the Bergman fan. This ring possesses several interesting properties such as satisfying Hard Lefschetz and (conjecturally) having a real-rooted Hilbert—Poincaré series.
In this talk, we introduce the singular cohomology ring of a matroid, which generalizes the Chow ring. It is defined as the singular cohomology ring of the toric variety associated to the Bergman fan. After giving an overview of the singular cohomology of smooth toric varieties, we draw out some of the analogies between the singular cohomology ring and the Chow ring of a matroid. In particular, we show for uniform matroids that these singular cohomology rings exhibit Lefschetz and real-rootedness properties similar to those of the Chow ring. This is based partially on joint work with Lorenzo Vecchi.
We consider the relative homology of an affine hyperplane arrangement and its Poincaré dual, the cohomology at finite distance. I will present two descriptions of this latter, the first in terms of the Orlik—Solomon algebra and the second in terms of logarithmic forms having vanishing residues at infinity. If time permits, I will also explain generalizations of these results to toric arrangements.