PRIMES: The Inverse Eigenvalue Problem for Graphs and Collaboration to Promote Inclusivity in Undergraduate Mathematics Education

PRIMES：图的反特征值问题和协作以促进本科数学教育的包容性

• 批准号: 2331072
• 负责人: Veronika Furst
• 金额: $ 21.85万
• 依托单位: Fort Lewis College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2331072&HistoricalAwards=false
• 关键词: PRIMES; Inverse; Eigenvalue; Problem; Graphs

Fort Lewis College and the American Institute of Mathematics will engage in a two-year partnership, aimed at furthering research in pure mathematics at Fort Lewis College in a way that increases both research output at a primarily undergraduate institution (PUI) and inclusivity among historically underrepresented (UR) students. This partnership will support the PRIMES goal of enabling, building, and growing collaboration between Fort Lewis College (FLC), a Minority Serving Institution, and the American Institute of Mathematics (AIM), a DMS-supported Mathematical Sciences Research Institute. By supporting research and outreach efforts at FLC, this project will not only advance research excellence at a PUI but also increase diversity and promote inclusiveness, given the highly diverse student body of FLC. FLC's historic mission is the education of American Indian and Alaska Native (AI/AN) student populations, and first-generation college students comprise nearly half of the student body. The partnership with AIM will focus on both research excellence and efforts that promote increased retention of first-year UR students, especially in the STEM disciplines, who may struggle both academically and with a sense of belonging in college.The mathematical focal area of the project is on subproblems of the broad Inverse Eigenvalue Problem for Graphs (IEPG), which asks to determine all possible spectra of matrices whose off-diagonal entries match the zero-pattern of the adjacency matrix of a graph. The IEPG is a difficult problem of interest to many graph theorists and combinatorial matrix theorists. The PI and her collaborators aim to add to the body of knowledge on the minimum number of distinct eigenvalues over symmetric matrices described by a graph, by expanding their previous characterization of regular graphs of degree at most four, possibly in several directions. In addition to considering eigenvalues, the PI and other collaborators investigate the sparsity of null vectors (and thereby eigenvectors) of matrices associated with a graph. The concept of the spark of a matrix (prevalent in the area of compressed sensing) is adapted to the spark of a graph. Through mentorship of undergraduate researchers, the connection between the minimum number of distinct eigenvalues of strongly regular graphs and finite frame theory is also explored.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.

路易斯堡学院和美国数学研究所将建立为期两年的合作伙伴关系，旨在进一步推进路易斯堡学院的纯数学研究，既提高本科院校 (PUI) 的研究成果，又提高历史上代表性不足的学校的包容性（UR）学生。此次合作将支持 PRIMES 的目标，即在少数族裔服务机构刘易斯堡学院 (FLC) 和 DMS 支持的数学科学研究机构美国数学研究所 (AIM) 之间实现、建立和发展合作。 通过支持 FLC 的研究和推广工作，该项目不仅将促进 PUI 的研究卓越，而且鉴于 FLC 学生群体高度多元化，还将增加多样性和促进包容性。 FLC 的历史使命是为美洲印第安人和阿拉斯加原住民 (AI/AN) 学生群体提供教育，其中第一代大学生占学生群体的近一半。 与 AIM 的合作将重点关注卓越研究和提高 UR 一年级学生保留率的努力，尤其是 STEM 学科的学生，这些学生可能在学术上和大学归属感上都遇到困难。该项目的数学重点领域是关于广泛的图逆特征值问题（IEPG）的子问题，该问题要求确定其非对角线条目与图邻接矩阵的零模式匹配的所有可能的矩阵谱。 IEPG 是许多图论学家和组合矩阵理论学家感兴趣的难题。 PI 和她的合作者的目标是通过扩展他们之前对最多四次正则图的描述（可能在多个方向上），来增加图描述的对称矩阵上不同特征值的最小数量的知识体系。 除了考虑特征值之外，PI 和其他合作者还研究与图相关的矩阵的零向量（以及特征向量）的稀疏性。 矩阵火花的概念（在压缩感知领域普遍存在）适用于图的火花。 通过本科生研究人员的指导，还探索了强正则图的最小数量的不同特征值与有限框架理论之间的联系。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进行评估，被认为值得支持审查标准。