REU Site: Matrix Analysis, Combinatorics, and Applications

威廉和玛丽本科生研究经历(REU)计划旨在引导来自不同背景的有前途的本科生从事富有成效的数学研究。这也将有助于他们对研究生学习的一部分进行有用的预览。在为期8周的课程中，精心挑选的学生将接触到矩阵分析、组合学及其应用中的高水平、有价值的问题，然后在专家的指导下为理解这些问题做出贡献。目标有两个：1)使学生能够进行足够有价值的研究，以保证在受人尊敬的媒体上发表文章；2)为有关上述主题的重要领域的知识做出实质性贡献。从历史上看，W&Amp；M REU项目在这个项目中涉及到相当多的主题。统一的主题之一是组合思想和矩阵分析之间的相互作用，通常是组合数学在理解矩阵分析问题中的作用。这一主题将继续下去。一些特定的主题领域是：1)矩阵完成问题，涉及各种类型的矩阵；2)由矩阵的图得到的对矩阵特征值的重数的限制；3)矩阵的值域；4)可以从符号模式或其条目的零/非零模式推断的矩阵的性质；5)全正矩阵；以及许多其他更具体的问题。每年，主要研究人员/导师和他们的许多合作者的积极研究计划都会收集到新的问题。这个奖项反映了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。

The William and Mary Research Experiences for Undergraduates (REU) program seeks to initiate promising undergraduate students, from a variety of backgrounds, into productive mathematical research. This will also help to give them a useful preview of part of graduate study. In an 8-week program, carefully chosen students will be exposed to high-level, worthwhile problems in matrix analysis, combinatorics, and their applications and then mentored by experts in the process of making contributions to the understanding of these problems. The goal is two-fold: 1) to enable the students to perform sufficiently valuable research to warrant publication in respected outlets, and 2) to contribute materially to the body of knowledge about important areas of the mentioned subjects. Historically, the W&M REU program has involved quite a number of topics in this project. One of the unifying themes has been the interplay between combinatorial ideas and matrix analysis, often the role of combinatorial mathematics in the understanding of matrix analytical questions. This theme will continue. Some particular topic areas have been: 1) matrix completion problems, involving a variety of classes of matrices; 2) constraints on the multiplicities of the eigenvalues of a matrix, resulting from its graph; 3) the field of values of a matrix; 4) properties of matrices that may be inferred from the pattern of signs, or the zero/nonzero pattern of their entries; 5) totally positive matrices; and many other more specific questions. Each year, new problems are gathered from the active research programs of the principal investigators/mentors and their many collaborators.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.