Combinatorics in Geometry

点击翻译按钮获取中文摘要

The PI will explore three related projects in tropical geometry, the Schubert calculus, and convexity, through a combinatorial study of matroids, convex geometries, and similar structures. Tropical linear spaces, perhaps the simplest and most fundamental tropical varieties, have a very interesting discrete structure that is far from understood. The first project is the combinatorial study of these and other related objects such as tropical hyperplane arrangements and oriented matroids. The second project is the study of the Littlewood-Richardson numbers of the flag variety, driven by the open questions of computing them combinatorially, and determining when they vanish. The third project explores convex geometries and their role in three contexts: metric spaces of non-positive curvature, tropical geometry, and random processes involving pruning procedures.This proposal is driven by the philosophy that many objects, relationships, and procedures in pure and applied mathematics are best understood by studying the rich discrete structures underlying them. The PI will apply this philosophy to various geometric and combinatorial questions which are firmly rooted in pure mathematics, and which are often motivated by and used in concrete applications in optimization and phylogenetics. This research program is the academic backbone of the San Francisco State University-Colombia Combinatorics Initiative, an ongoing teaching and research collaboration between students in these two locations, many of whom are members of underrepresented groups in mathematics. Through joint courses and research projects, students participate in their first international academic experience, while making significant scientific contributions.