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.

Pi Chavez与纯粹和应用数学研究所（IPAM）之间的合作伙伴关系奖，结合了Matroid理论的纯正和应用方面，同时在加利福尼亚州圣玛丽学院（SMC）（SMC），西班牙裔和小学下级和小学下学机构的本科研究经验中。两个主要目的是：1）正式化SMC和IPAM之间的合作伙伴关系，包括PI参与2024年春季的几何形状，统计方法和可集成性长期计划，以及2）提高PI的科学及其对本科生成功的影响，通过支持本科研究助理和教学年份，以成功地支持这些目标。 PI将进行多个研究项目，其中一些将包括本科生的研究贡献，以加强Matroid理论的纯正和应用方面之间的联系。此外，PI致力于增强STEM的多样性和包容性，并将继续进行高中宣传，支持SMC学生团体，并参与强调少数数学家声音的项目。该项目包括以下科学活动。 （1）通过经典和热带的几何镜头研究KP-Soliton溶液与标志贴素之间的关系。这个项目进一步扩大了这种贴素的应用解释与我们在热带几何环境中的理解之间的联系。 （2）在部分定居者上扩展电流多面体和几何结果，并使用一种新的方法来描述经典定居者的三角形。这为经典对象提供了新的见解，为广义定居者家族添加了有用的信息，并且是本科研究的绝佳环境。 （3）使用多面有方法来解决有关cotitiveroid，Polypository和Flag cotiveroid不变性的问题。朝这个方向的两个项目是：（a）证明积极的多面是ehrhart的阳性，（b）描述了国旗阳性的ehrhart多项式。这些结果将加深多面体，阳性和非负grassmannian的组合联系。在经典意义和应用意义上，Matroid理论的重要性不断扩大，这表明了跨学科研究如何弥合这两个领域。该奖项反映了NSF的法定任务，并通过使用基金会的智力优点和更广泛的影响来评估NSF的法定任务，并通过评估诚实地支持了审查标准。