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

路易斯堡学院和美国数学研究所将进行为期两年的合作，旨在进一步推进路易斯堡学院的纯数学研究，以增加主要本科院校（PUI）的研究成果和历史上代表性不足的学生（UR）的包容性。这种伙伴关系将支持PRIMES的目标，即在少数民族服务机构刘易斯堡学院（FLC）和dms支持的数学科学研究所美国数学研究所（AIM）之间建立、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的卓越研究，还将增加多样性和促进包容性，因为FLC的学生群体非常多样化。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的合作将侧重于卓越的研究和提高大学一年级学生的保留率，特别是在STEM学科，这些学生可能在学业和大学归属感上都很挣扎。该项目的数学重点领域是广义图的反特征值问题（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矩阵的所有可能谱。IEPG是许多图论理论家和组合矩阵理论家感兴趣的一个难题。PI和她的合作者的目标是通过扩展他们之前最多四次的正则图的表征，可能在几个方向上，增加关于图所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了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（在压缩感知领域很流行）适用于图的火花。通过对本科生的指导，本文还探讨了强正则图的显著特征值最小数与有限框架理论之间的联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.