RUI: Exploring the Algebraic Geometry of Matroids

Swarthmore College, Swarthmore PA

Investigators

Abstract

There have been recent breakthroughs in parts of combinatorics related to matroid theory coming from the importation of tools and ideas from algebraic geometry, which is the study of systems of polynomial equations and their solutions. The PI will further develop the bridge between algebraic geometry and matroid theory, thereby allowing insight and progress in one subject to carry over to the other. This paves the way for well-known applications of matroids in topics such as optimization and network theory to draw from the heavily developed, and computationally powerful, machinery of algebraic geometry. The PI will also run a summer enrichment program, focusing on interactions between math and law, for disadvantaged minority students at a nearby underfunded public high school. The PI's program of bringing more algebraic geometry into matroid theory consists of three projects. (1) Matroids can be viewed as modules over the two-element Boolean semifield or as tensors in a tropical exterior algebra. By extending this framework the PI will geometrize various combinatorial constructions and results in the matroid literature. Motivating problems focus on transversal matroids and matroid irreducibility. (2) The maximal torus action on the Grassmannian beautifully brings together topology, combinatorics, representation theory, and algebraic geometry. The PI is extending portions of this story to diagonal subtori. This opens several paths of investigation, including the possibility of extending the K-theoretic interpretation of the Tutte polynomial from matroids to their multiset generalization, discrete polymatroids. (3) The PI is applying the tropical scheme theory he co-developed to construct a functorial moduli space of matroids. This entails a careful understanding of how matroids vary algebraically in families and could lead to an eventual Schubert calculus on the Dressian. 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.