Categorical Invariants of Matroids

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

A matroid is an object that encodes a combinatorial abstraction of the notion of linear dependence. The name derives from the word matrix: given a matrix A, there is an associated matroid that records the information of which subsets of the columns of A are linearly dependent. Not all matroids arise in this manner, but those that do form a large class of matroids that provide good intuition for matroids in general. There has been an explosion of work on matroids in the past ten years, thanks largely to the work of June Huh, which was recognized with a 2022 Fields Medal. Huh has used techniques from algebraic geometry to resolve conjectures about matroids that have been open for many years, including Welsh's log concavity conjecture from 1976, and Dowling and Wilson's top heavy conjecture from 1975. The common theme behind the various directions of this project is to upgrade various theorems, conjectures, and constructions involving numerical invariants of matroids to richer structures that take symmetries of matroids into account. Broader impacts of the project include mentoring of graduate students and workshop organization. The first project will enrich our understanding of log concavity and real rootedness in the context of matroid invariants by working equivariantly. In special cases, it allows to ask new questions about the cohomology of the configuration space of points in the plane, and about Eulerian polynomials. The second project sheds light on the mysterious notion of valuativity by recasting it in categorical terms, and provides a method for compute large classes of matroid invariants equivariantly. The third project introduces the notion of the local h-polynomial of a matroid, and has the potential to make new inroads into classical questions about matroid realizability and graph colorability. The last project involves a categorification of the graph minor theorem, with applications to the homology of configuration spaces of graphs. 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.