Combinatorics in Geometry

PI将通过拟阵、凸几何和类似结构的组合研究，探索热带几何、舒伯特微积分和凸性的三个相关项目。热带线性空间，也许是最简单和最基本的热带变种，有一个非常有趣的离散结构，还远未被理解。第一个项目是这些和其他相关对象的组合研究，如热带超平面安排和定向拟阵。第二个项目是研究Littlewood-Richardson数的标志品种，由开放的问题，计算他们组合，并确定何时消失。第三个项目探讨凸几何及其在三个方面的作用：非正曲率的度量空间，热带几何，以及涉及修剪程序的随机过程。这个建议是由这样一种哲学驱动的，即通过研究它们背后丰富的离散结构，可以最好地理解纯数学和应用数学中的许多对象，关系和程序。PI将把这种哲学应用于各种几何和组合问题，这些问题深深植根于纯数学，并且经常受到优化和遗传学的具体应用的启发。该研究项目是旧金山弗朗西斯科州立大学-哥伦比亚组合数学计划的学术骨干，该计划是这两个地点的学生之间正在进行的教学和研究合作，其中许多人是数学代表性不足的群体的成员。通过联合课程和研究项目，学生参加他们的第一次国际学术经验，同时作出重大的科学贡献。

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.