Algebraic Invariants of Matroids and their Applications

About This Grant

Matroids are fundamental combinatorial objects that simultaneously capture the notion of independence in topics as diverse as graph theory, linear algebra, and field extensions. Recently, geometry has entered the scene and has provided new perspectives for tackling long-standing problems in matroid theory. Behind the resolution of each of these conjectures is an algebraic invariant that exists for any matroid, but whose properties are predicted by geometry. The broad aim of this project is to study these algebraic invariants with a focus on uncovering their impact and applications throughout combinatorics. This project includes mentoring graduate students, creating a learning community at NC State for graduate students interested in combinatorics, and initiating a middle-school outreach activity in Raleigh, NC. The first project aims to give a simpler and more general definition of the intersection cohomology module of a matroid, a module that played a key role in the proof of Dowling–Wilson's top heavy conjecture from 1975. This new definition will work over positive characteristic fields and thus will lead to positive characteristic analogues of Kazhdan–Lusztig polynomials of matroids. The project will also introduce a matroidal analogue of Stanley's local h-polynomials with the aim of shedding light on classical problems in graph colorability and matroid realizability. The second project aims to understand to what extent there might be a "Chow theory" for arbitrary partially ordered sets. In the settings of matroids, polytopes, and Coxeter groups, the Chow polynomials of posets have interpretations in terms of well-studied notions in the three respective areas. The guiding theme is to prove results for arbitrary graded bounded posets by discovering commonalities among these three settings; and, on the other hand, to interpret results that hold for arbitrary posets in these three specific settings. The third project aims to understand log-concavity phenomena in algebraic combinatorics and commutative algebra. On the one hand, new techniques from commutative algebra will be introduced into the study of log-concavity of polynomials with the goal of investigating log-concavity conjectures for Schur-like polynomials; and, on the other hand, the theory of Lorentzian polynomials will be used to investigate log-concavity questions for natural polynomials arising in commutative algebra. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.