RUI: Algebraic and Geometric Aspects of Matroids, Polytopes, and Arrangements

San Francisco State University, San Francisco CA

Investigators

Abstract

This research project is driven by the philosophy that many objects and relationships in mathematics are best understood by studying the rich discrete structures underlying them. 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. This project studies central questions in algebra and geometry where the combinatorial structure of high-dimensional geometric objects plays a crucial role, and offers unexpected, innovative approaches to long-standing problems. This research program constitutes the academic backbone of the San Francisco State University-Colombia Combinatorics Initiative, 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 the initiative has trained more than 200 pre-Ph.D. 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 helps train mathematicians worldwide through the distribution of course videos, lecture notes, and research projects. This project studies important questions in various fields of mathematics, such as representation theory (Kostant partition functions), enumerative geometry (moduli spaces of curves), polyhedral number theory (Ehrhart polynomials), Brunn-Minkowski theory (valuations of polytopes), Hopf-Lie theory (algebraic structures on polytopes), tropical geometry (tropical linear spaces, moduli spaces, and Hodge theory), total positivity (positroids and their algebraic structure), and combinatorial commutative algebra (Fröberg's conjecture). These subjects are related in surprising ways, and the techniques from one field become powerful tools in the others. At the heart of many of these questions lies a configuration of vectors -- often a root system -- that plays an essential role. The combinatorial theories of polytopes, (Coxeter) matroids, and hyperplane arrangements are designed to study these configurations, and the powerful toolkit that they offer is the unifying thread of this project. Solutions to the problems under study will have a strong impact in combinatorics and discrete geometry, and will further our understanding of central questions in algebra and geometry.