RUI: Algebra and Geometry of Matroids and Polytopes

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The field of combinatorics has grown and deepened enormously in the last 50 years, in response to the mathematical needs of modern computing, physics, and biology, and the computational needs of all areas of mathematics. This research project in combinatorics is driven by two related philosophies: many mathematical objects are best understood by studying the rich discrete structures underlying them, and many of those discrete structures are best understood by building geometric models for them. This grant constitutes the academic backbone of the San Francisco State University-Colombia Combinatorics Initiative, a vibrant research and training collaboration featuring research-based courses, research projects, and summer schools. Since 2007 the initiative has trained more than 250 pre-Ph.D. students; more than half of the US participants are members of underrepresented groups in mathematics, and more than 80 have gone on to Ph.D. programs. The initiative’s practices of excellence and inclusion are shared broadly by the PI and many others, and they serve as a model for other math and science programs nationwide and beyond.This project studies two research directions at the intersection of combinatorics and geometry: 1. We construct and study several geometric models of a matroid. Using tools from tropical geometry, Hodge theory, and intersection theory, we derive novel combinatorial properties of matroids that do not seem accessible purely combinatorially. 2. Measuring polytopes is very difficult in general, but it can be done when the polytopes have sufficient combinatorial or algebraic structure. We construct and measure families of polytopes arising in various mathematical settings, and use those measurements to compute algebraic and geometric quantities of interest. In both of these research directions, relevant objects are often built from a configuration of vectors – usually a root system. Polytopes and matroids offer a powerful toolkit to study such configurations. The proposed projects will further our understanding of fundamental questions in combinatorics, discrete geometry, representation theory, and algebraic geometry.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.