Polytopes and Matroids in Algebra and Geometry

San Francisco State University, San Francisco CA

Investigators

Abstract

In the last few decades combinatorics has grown and matured immensely as a field, in response to the mathematical needs of modern computing and the computational needs of all fields of mathematics. In particular, combinatorics and geometry have benefitted each other tremendously in recent years, both by the ways in which they are connected directly, and by learning from each other?s philosophies and techniques. This project studies the discrete structure of algebraic and geometric objects, and the algebraic and geometric structure of matroids and polytopes. This interdisciplinary perspective offers unexpected and effective approaches to long-standing problems. This research program constitutes the academic backbone of a vibrant research and training collaboration among primarily undergraduate institutions in the U.S. and Colombia. Through research-based courses, vertically and geographically integrated research projects, and the biannual Encuentro Colombiano de Combinatoria, students participate in a truly international cooperation while making substantial scientific contributions to combinatorics. Since 2007 this initiative has trained more than 150 pre-Ph.D. U.S. students, more than half of whom are members of underrepresented groups in mathematics, and more than 50 of whom have gone onto Ph.D. programs. The initiative also serves mathematicians worldwide through the distribution of course videos, lecture notes, and research projects. This project studies three interdisciplinary and interrelated research directions: 1. By studying the geometric structure of matroids, we obtain purely combinatorial results that seem intractable without the geometric perspective, and we establish foundational results in tropical geometry and combinatorial Hodge theory. 2. We measure polytopes of interest by recognizing that they are part of ?the right? family of polytopes, and finding a formula for the measure of all the polytopes in that family. The results have consequences in toric geometry and representation theory. 3. We explore the Hopf-algebraic structure of various families of polytopes. The resulting objects are useful in numerous combinatorial and algebraic applications, and they raise geometric questions of independent interest. At the heart of each of these three projects lies a configuration of vectors - usually a root system - that plays an essential role. The combinatorial theories of polytopes and (Coxeter) matroids are designed to study such configurations, and the powerful toolkit that they offer is the unifying thread of this project. Solutions to the proposed problems will have a strong impact in combinatorics and discrete geometry, and will further our understanding of central questions in algebra and 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.