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开展为期两年的合作伙伴关系，旨在进一步研究纯数学在刘易斯堡学院的方式，既增加了研究产出在主要是本科院校（PUI）和包容性之间的历史上代表性不足（UR）的学生。这种伙伴关系将支持PRIMES的目标，即实现，建立和发展刘易斯堡学院（FLC），少数民族服务机构和美国数学研究所（AIM），DMS支持的数学科学研究所之间的合作。 通过支持FLC的研究和外联工作，该项目不仅将推动PUI的卓越研究，而且还将增加多样性并促进包容性，因为FLC的学生群体高度多样化。FLC的历史使命是美国印第安人和阿拉斯加原住民（AI/AN）学生群体的教育，第一代大学生占学生总数的近一半。 与AIM的合作将集中在卓越的研究和努力，促进一年级UR学生的保留，特别是在STEM学科，谁可能在学术上和大学归属感的斗争。该项目的数学重点领域是广泛的图逆特征值问题（IEPG）的子问题，其要求确定其非对角项与图的邻接矩阵的零模式匹配的矩阵的所有可能的谱。 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通过扩展他们以前对度最多为4的正则图的表征，可能在几个方向上，来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。 除了考虑特征值之外，PI和其他合作者还研究了与图相关联的矩阵的零向量（以及特征向量）的稀疏性。 矩阵的火花的概念（在压缩感测领域中流行）适用于图的火花。 通过对本科生研究人员的指导，还探索了强正则图的最小不同特征值数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.