Combinatorics in Geometry

PI将通过对拟阵、凸几何和类似结构的组合研究，探索热带几何中的三个相关项目，舒伯特微积分和凸性。热带线性空间，也许是最简单和最基本的热带变种，有一种非常有趣的离散结构，远未被理解。第一个项目是对这些物体和其他相关物体，如热带超平面排列和定向拟阵的组合研究。第二个项目是研究旗帜变种的Littlewood-Richardson数，这是由如何组合计算它们并确定它们何时消失的未决问题推动的。第三个项目探索凸几何及其在三种背景下的作用：非正曲率度量空间、热带几何和涉及修剪过程的随机过程。这一建议是基于这样一种哲学，即通过研究它们背后的丰富离散结构，可以最好地理解纯数学和应用数学中的许多对象、关系和过程。PI将把这一哲学应用于各种几何和组合问题，这些问题牢固地植根于纯数学，并且经常受到最优化和系统发育学中具体应用的启发和使用。这一研究项目是旧金山州立大学-哥伦比亚组合数学倡议的学术支柱，该倡议是这两个地点的学生之间持续的教学和研究合作，其中许多学生是数学领域代表性不足的群体的成员。通过联合课程和研究项目，学生们在参与他们的第一次国际学术经历的同时，做出了重大的科学贡献。

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.