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

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，旨在通过增加以本科为主的机构(PUI)的研究产出和历史代表性不足(UR)学生的包容性，促进刘易斯堡学院在纯数学方面的研究。这一合作伙伴关系将支持PRIMES目标，即支持、建立和加强刘易斯堡学院(FLC)和美国数学研究所(AIM)之间的合作，前者是一个少数民族服务机构，后者是DMS支持的数学科学研究所。通过支持FLC的研究和外展努力，该项目不仅将促进PUI的研究卓越，而且鉴于FLC高度多样化的学生群体，还将增加多样性和促进包容性。FLC的历史使命是教育美国印第安人和阿拉斯加原住民(AI/AN)学生群体，第一代大学生占学生总数的近一半。与AIM的合作将专注于提高一年级UR学生的研究水平和努力，特别是在STEM学科，他们可能在学业和大学归属感方面都有困难。该项目的数学重点领域是图的广义逆特征值问题(IEPG)的子问题，该问题要求确定其非对角线输入与图的邻接矩阵的零模式匹配的所有可能的谱。IEPG是许多图论家和组合矩阵理论家感兴趣的难题。PI和她的合作者的目标是增加关于由图描述的对称矩阵上不同特征值的最小数目的知识，方法是扩展他们以前对正则图的刻画，最多四次，可能在几个方向上。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(从而特征向量)的稀疏性。矩阵火花的概念(在压缩传感领域很普遍)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不同特征值的最小数量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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.