PRIMES: Matroids, Polyhedral Geometry, and Integrable Systems

PRIMES：拟阵、多面体几何和可积系统

• 批准号: 2332342
• 负责人: Anastasia Chavez
• 金额: $ 30.41万
• 依托单位: Saint Mary's College of California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-02-01 至 2026-01-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2332342&HistoricalAwards=false
• 关键词: PRIMES; Matroids; Polyhedral; Geometry; Integrable

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.

这个PRIMES奖PI查韦斯和纯粹与应用数学研究所（IPAM）之间的合作伙伴关系结合了拟阵理论的纯粹和应用方面，同时推进本科生的研究经验在圣玛丽的加州学院（SMC），西班牙裔服务，主要是本科院校。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推进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，加强拟阵理论的纯和应用方面之间的联系。此外，PI致力于提高STEM的多样性和包容性，并将继续与高中外展，支持SMC学生团体，并参与突出少数民族数学家声音的项目。该项目包括以下科学活动。(1)通过经典和热带几何透镜研究KP孤子解与旗状正粒的关系。这个项目进一步的应用解释positroids和我们的理解之间的联系在热带几何设置。(2)推广了部分置换面体上现有的多面体和几何结果，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，并且是本科生研究的一个很好的环境。(3)使用polytopal方法来解决关于positroid，polypositroid和flag positroid不变量的问题。这方面的两个方案是：（a）证明正拟阵多胞形是Ehrhart正的，（B）描述旗正拟阵的Ehrhart多项式。这些结果将深化多面体，正拟阵，非负格拉斯曼的组合连接。拟阵理论在经典和应用两方面的重要性不断扩大，这表明跨学科研究对弥合这两个领域是多么必要。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。