Positive Vector Bundles in Combinatorics

Matroids are combinatorial objects that appear in many important areas of mathematics because they capture the notion of “independence” in diverse mathematical constructs, such as graphs, field extensions, hyperplane arrangements, matchings, and discrete optimizations. Using various techniques developed in recent years, the PI aims to deepen the interaction between combinatorics and algebraic geometry, both in matroid theory and in contexts beyond matroids such as Coxeter combinatorics and algebraic statistics. The PI plans to involve undergraduate and graduate students in the project. The PI will work on several projects in matroid theory are proposed using the new framework of "tautological classes of matroids." Many of these projects concern new properties for numerical invariants of matroids that have implications to some long-standing conjectures in matroid theory. These projects also inspire projects in Coxeter combinatorics and algebraic statistics. Completion of these projects will reveal new structural properties of combinatorial objects such as polymatroids and delta-matroids, and will inform the complexity of maximum-likelihood problems in algebraic statistics. 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.