PRIMES: Matroids, Polyhedral Geometry, and Integrable Systems

这一质数奖是为了表彰皮查韦斯与纯粹和应用数学研究所(IPAM)的合作伙伴关系，该奖项结合了拟阵理论的纯粹和应用方面，同时促进了圣玛丽学院(SMC)的本科生研究经验，圣玛丽学院是一所西班牙裔服务机构，主要是本科生。两个主要目标是：1)正式确定SMC和IPAM之间的合作伙伴关系，包括PI参与2024年春季几何、统计方法和可积长课程，以及2)通过支持本科生研究助理和资助一年的教学假来提高PI的奖学金及其对本科生成功的影响，以成功实现这些目标。PI将进行几个研究项目，其中一些将包括本科生的研究贡献，这些项目加强了拟阵理论的纯粹和应用方面之间的联系。此外，国际数学家协会致力于加强STEM的多样性和包容性，并将继续开展高中活动，支持SMC学生团体，并参与突出小数学家声音的项目。(1)通过经典几何透镜和热带几何透镜研究了KP-孤子解与FLAG正电子之间的关系。这个项目进一步加强了对正电子的这种应用解释与我们在热带几何背景下的理解之间的联系。(2)推广了部分置换面体上已有的多面体和几何结果，并用一种新的方法描述了经典置换面体的三角剖分。这提供了对经典对象的新见解，为广义置换面体家族增加了有用的信息，对于本科生的研究是一个很好的环境。(3)利用多面体方法解决正态、多态和旗形正态不变量的问题。这个方向的两个项目是：(A)证明正多面体是Ehrhart正的；(B)描述旗正多面体的Ehrhart多项式。这些结果将加深多面体、正拟阵和非负Grassmanian之间的组合联系。拟阵理论在经典意义和应用意义上日益增长的重要性表明，跨学科研究对于连接这两个领域是多么必要。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.