Primes: matroids, polyhedral geometry, and integrable systems

This National Science Foundation (NSF) project grant under the Mathematical and Physical Sciences program (CFDA 47.049) of $304,102 awarded to Saint Mary's College of California provides funding for a partnership between the PI and the Institute for Pure and Applied Mathematics (IPAM) to advance undergraduate research experiences in matroid theory. The key activities include: 1) formalizing the partnership between Saint Mary's College and IPAM, including the PI's participation in IPAM's 2024 long program; and 2) supporting undergraduate research assistants and providing the PI a teaching leave to conduct research projects that strengthen the connection between pure and applied matroid theory. The research aims to investigate the relationship between KP-soliton solutions and flag positroids, extend results on the partial permutahedron, and use polytopal methods to address questions about positroid, polypositroid, and flag positroid invariants. The project will enhance diversity and inclusion in STEM, with the PI engaged in high school outreach and supporting student groups at Saint Mary's College.

Abstract

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.