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

REU 代数、分析、几何、矩阵理论和统计学网站

• 批准号: 1156890
• 负责人: Sivaram Narayan
• 金额: $ 28.35万
• 依托单位: Central Michigan University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-05-01 至 2015-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1156890&HistoricalAwards=false
• 关键词: REU; Site; Algebra; Analysis; Geometry

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.

中密歇根大学REU网站计划的学生将在数学的几个领域提出开放性问题，包括代数，分析，几何，矩阵理论和统计。只有一个适度的数学背景，在大二水平将需要了解研究项目，将由两个或三个学生的团队与教师顾问合作进行调查。我们提供了几个项目主题的简要概述。在组合矩阵理论领域，有一种自然的方式将图与对称矩阵或厄米特矩阵联系起来。一般的问题是确定与给定图相关联的各种类型的矩阵之间的最小秩。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。公开的问题是找到一种方法来构造具有理想性质的框架，例如对称性或稀疏性或它们的角度和长度的给定分布。 几何学中的一个项目研究曲面与其关联图之间的关系，以及其嵌入到射影空间中的性质。统计学中的一个项目旨在基于测试转换曲线上的总时间对参数寿命分布的风险灵活性进行分类，并对众所周知的单形状和双形状参数寿命分布进行排名。学生将学习数学超出了他们的研究工作所需要的。 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，在传输期间可能丢失一些信息，并且原始信号不能被恢复。帧中的冗余使得有可能“重建”原始信号。 在几何问题中，学生将了解难以理解的黎曼曲面与更容易理解的图形等组合对象之间的深层联系。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试图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名教师将积极参加为期三年的计划。每年夏天，四名教师导师将协调这项工作。学生将在一个教师导师的两个或三个小组工作。学生将在每周的研讨会上介绍他们的进展情况。每个学生将在课程结束时提交一份书面报告。学生将被鼓励发表他们的工作，并在专业会议上做演讲。拟议的REU网站的更广泛的影响包括培训本科生在基础数学的研究，增加妇女和少数民族在科学和数学活动的参与，并涉及本科生作为年轻的专业人员在更广泛的学科领域。