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.

刘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，旨在进一步在刘易斯堡学院的纯数学研究中进行研究，从而增加了主要是本科机构（PUI）的研究成果，并在历史上不足的学生（UR）中培养了包容性。该伙伴关系将支持少数派服务机构刘易斯大学（FLC）与美国数学研究所（AIM）之间实现，建立和不断增长的合作的素养目标（AIM）。 通过支持FLC的研究和外展工作，该项目不仅将在PUI上提高卓越的研究，而且还可以提高多样性并促进包容性，鉴于FLC的学生团体高度多样化。 FLC的历史使命是对美洲印第安人和阿拉斯加本地人（AI/AN）学生的教育，第一代大学生的教育占学生群体的一半。 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图形的邻接矩阵。 对于许多图理论家和组合矩阵理论家来说，IEPG是一个困难的问题。 PI和她的合作者的目的是通过扩展其先前对四个定期图的特征来扩展其先前的定期图表，从而在多个方向上扩展其先前的定期图表，以增加图形所描述的对称矩阵的最小特征值数量的知识。 除了考虑特征值外，PI和其他合作者还研究了与图形相关的矩阵的无效矢量（及其特征向量）的稀疏性。 矩阵的火花的概念（在压缩传感区域中流行）适应图的火花。 通过对本科研究人员的指导，还探讨了强烈规则图的不同特征值的最小特征值与有限框架理论之间的联系。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来支持的。