REU Site in Algebra, Analysis, Geometry, Matrix Theory and Statistics

中密歇根大学REU站点项目的学生将面临几个数学领域的开放性问题，包括代数、分析、几何、矩阵理论和统计学。只需要在大二阶段有一定的数学背景，就可以理解由两到三个学生组成的团队在指导老师的合作下进行的研究项目。我们提供了几个项目主题的简要概述。在组合矩阵理论中，有一种自然的方法将图与对称矩阵或厄米矩阵联系起来。一般问题是确定与给定图相关联的各类矩阵之间的最小秩。在线性代数领域，我们提出了一个名为“有限维框架”的项目。帧是冗余的生成集。悬而未决的问题是找到一种方法来构造具有理想性质的框架，如对称或稀疏性，或它们的角度和长度的给定分布。几何项目研究曲面与其关联图之间的关系，以及其嵌入射影空间的性质。基于试验变换曲线上的总时间对参数寿命分布的危险灵活性进行分类，并对已知的一形和二形参数寿命分布进行排序。学生们将学习超出他们研究工作所需的数学知识。例如，学生将接触到框架的一些应用。当信号在信道上传输时，有可能在传输过程中丢失一些信息，并且无法恢复原始信号。帧中的冗余使得“重建”原始信号成为可能。在几何问题中，学生将认识到难以理解的黎曼曲面与易于理解的组合对象（如图）之间的深刻联系。统计项目将帮助学生了解如何建模数据和如何估计某些参数。这个REU网站的智力价值是基于学生将尝试解决的具有挑战性的数学问题。学生将解决未解决的问题，其解决方案（或部分解决方案）将提高人类对数学结构及其应用的理解。11名教员将积极参与这个为期三年的项目。每年夏天，四位教师导师将协调这项工作。学生将以两到三人为一组，由一名教师指导。学生们将在每周的研讨会上展示他们的进步。每个学生将在课程结束时提交一份书面报告。学生们将被鼓励发表他们的作品，并在专业会议上做报告。拟议的REU站点的更广泛影响包括培训本科生进行基础数学研究，增加妇女和少数民族对科学和数学活动的参与，并使本科生作为年轻专业人员参与更广泛的学科领域。

The students in the REU Sites program at Central Michigan University will be presented with open questions in several areas of mathematics, including algebra, analysis, geometry, matrix theory, and statistics. Only a modest mathematical background at the sophomore level will be needed to understand research projects that will be investigated by teams of two or three students in collaboration with a faculty advisor. We provide a brief overview of several of the project topics. In the area of Combinatorial Matrix Theory, there is a natural way to associate graphs to symmetric or Hermitian matrices. The general problem is to determine the minimum rank among various classes of matrices associated with a given graph. In the area of Linear Algebra, we present a project called "Frames in Finite Dimensions." Frames are redundant spanning sets. The open question is to find a method to construct frames with desirable properties such as symmetry or sparsity or a given distribution of their angles and lengths. A project in geometry studies the relationship between a surface and its associated graph, as well as properties of its embedding into a projective space. A project in statistics aims to classify the hazard flexibility of parametric lifetime distributions based on total time on test transform curve and to rank the well known one- and two-shape parameter lifetime distributions. The students will learn mathematics beyond what is needed for their research work. For example, students will be exposed to some of the applications of frames. When a signal is transmitted over a channel it is possible that some information may be lost during transmission and the original signal cannot be recovered. The redundancy in frames makes it possible to "reconstruct" the original signal. In the geometry problem students will come to know the deep connection between Riemann surfaces, which are difficult to understand, and combinatorial objects such as graphs, which are easier to understand. Statistics projects will help students understand how to model data and how to estimate certain parameters. The intellectual merit of this REU site is based on the challenging mathematical problems students will attempt to solve. The students will tackle unsolved problems whose solutions (or partial solutions) will improve human understanding of the very structure of mathematics and its applications. Eleven faculty members will be active in the three-year program. Four faculty mentors will coordinate the effort each summer. Students will work in groups of two or three with one faculty mentor. Students will make presentations of their progress during weekly seminars. Each student will submit a written report at the end of the program. Students will be encouraged to publish their work and make presentations at professional meetings. The broader impact of the proposed REU site include training undergraduates in research in fundamental mathematics, increasing the participation of women and minorities in science and math activities, and involving undergraduates as young professionals in the broader disciplinary field.