PRIMES: Matroids, Polyhedral Geometry, and Integrable Systems

这个PRIMES奖是为了表彰PI Chavez和纯粹与应用数学研究所（IPAM）之间的合作，将矩阵理论的纯粹和应用方面结合起来，同时推进加州圣玛丽学院（SMC）的本科研究经验，这是一所西班牙裔服务的主要本科机构。两个主要目标是：1)正式建立SMC和IPAM之间的伙伴关系，包括PI参与2024年春季几何，统计方法和可积性长期计划，以及2)通过支持本科生研究助理和资助一年的教学休假来成功实现这些目标，从而推进PI的奖学金及其对本科生成功的影响。PI将开展几个研究项目，其中一些将包括本科生的研究贡献，以加强拟阵理论的纯粹方面和应用方面之间的联系。此外，PI致力于加强STEM领域的多样性和包容性，并将继续开展高中外展活动，支持SMC学生团体，并参与突出少数族裔数学家声音的项目。本项目包括以下科学活动。(1)通过经典几何透镜和热带几何透镜研究了kp孤子解与旗子正子的关系。该项目进一步将正极体的应用解释与我们对热带几何环境的理解联系起来。(2)推广了目前关于部分多面体的多面体和几何结果，并采用一种新的方法来描述经典多面体的三角剖分。这为经典对象提供了新的见解，为广义复面体家族增加了有用的信息，并且为本科生的研究提供了一个很好的环境。(3)利用多面体方法解决关于正极、多正极和标记正极不变量的问题。在这个方向上的两个项目是：(a)证明正子多边形是Ehrhart正的，(b)描述flag正子多边形的Ehrhart多项式。这些结果将加深多面体、正极体和非负格拉斯曼体的组合联系。在经典意义和应用意义上，拟阵理论的重要性不断扩大，这表明跨学科研究是连接这两个领域的必要条件。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This PRIMES award for a partnership between PI Chavez and the Institute for Pure and Applied Mathematics (IPAM) combines the pure and applied sides of matroid theory while advancing undergraduate research experiences at Saint Mary’s College of California (SMC), a Hispanic serving and primarily undergraduate institution. Two major aims are: 1) formalize a partnership between SMC and IPAM, including the PI's participation in the Spring 2024 Geometry, Statistical Methods, and Integrability long program, and 2) advance the PI's scholarship and its impact on undergraduate success by supporting undergraduate research assistants and funding a year of teaching leave to successfully attain these goals. The PI will conduct several research projects, some of which will include undergraduate student research contributions, that strengthen the connection between the pure and applied sides of matroid theory. Additionally, the PI is dedicated to enhancing diversity and inclusion in STEM and will continue with high school outreach, supporting SMC student groups, and engaging with projects that highlight voices of minoritized mathematicians.The project includes the following scientific activities. (1) Investigate the relationship between KP-soliton solutions and flag positroids through classical and tropical geometric lenses. This project furthers the connection between this applied interpretation of positroids and our understanding in the tropical geometric setting. (2) Extend current polyhedral and geometric results on the partial permutahedron and use a new approach to describe triangulations of the classical permutahedron. This offers new insight on a classical object, adds useful information to the family of generalized permutahedra, and is a great setting for undergraduate research. (3) Use polytopal methods to address questions about positroid, polypositroid, and flag positroid invariants. Two projects in this direction are: (a) prove positroid polytopes are Ehrhart positive, and (b) describe Ehrhart polynomials of flag positroids. These results will deepen the combinatorial connection of polytopes, positroids, and the nonnegative Grassmannian. The expanding importance of matroid theory in both the classical and applied sense shows how necessary cross-disciplinary research is to bridging these two areas.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.