Away from Independence: geometrically, algebraically, and physically motivated random matrix ensembles

远离独立：几何、代数和物理驱动的随机矩阵系综

• 批准号: 1612589
• 负责人: Elizabeth Meckes
• 金额: $ 15.92万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612589&HistoricalAwards=false
• 关键词: Away; Independence; geometrically; algebraically; physically

On the face of it, a matrix is just a box of numbers. But the encoding of information in the form of matrices is one of the most powerful ideas of modern mathematics; the language of matrices underpins nearly every area of pure and applied mathematics, and anchors the frameworks of physics, engineering, and other sciences. The study of randomly generated matrices is one of the most exciting areas of mathematics today, with theoretical advances giving us an ever-growing picture of what "typical" matrices are like, and providing us with important applications in areas like image compression and the analysis of the Big Data that surrounds us. In this project, the PI will continue her research program investigating the behavior of very large random matrices, and how that behavior depends on the way they are generated; this will advance the theory, which will help better understand the applications to some of the areas mentioned above. She will also complete her monograph "The Random Matrix Theory of the Compact Classical Groups", which gives an overview of what has been learned about randomly generated matrices with a particular relationship to high-dimensional geometry; this activity will disseminate difficult technical knowledge about random matrices to a broad audience. The PI is active in teaching at all levels, and particularly in mentoring and encouraging girls and young women interested in mathematical careers; those training and outreach activities will have great impact on the diversity of those entering STEM fields.The monograph mentioned above will describe the remarkable advances in the theory of Haar-distributed random matrices in recent years. Topics will include central limit theorems for the entries of principal sub-matrices and for general lower-dimensional projections, quantitative limit results for the empirical spectral measures, self-similarity patterns within the eigenvalue distributions, concentration of measure with geometric applications, including new proofs of the Johnson-Lindenstrauss lemma and Dvoretzky's theorem, and results on the distribution of the characteristic polynomial, with connections to the Riemann zeta function. New research projects include the study of random matrices whose distributions are invariant under rotations in matrix space, continued work on quantitative approximations of empirical spectral measures for various classes of random matrices, and in various metrics, and problems in random topology connected with topological data analysis.

从表面上看，矩阵只是一个数字的盒子。 但是，以矩阵的形式对信息进行编码是现代数学中最强大的思想之一;矩阵语言支撑着几乎所有的纯数学和应用数学领域，并支撑着物理学、工程学和其他科学的框架。 随机生成矩阵的研究是当今数学中最令人兴奋的领域之一，随着理论的进步，我们对“典型”矩阵的了解越来越多，并在图像压缩和分析我们周围的大数据等领域为我们提供了重要的应用。 在这个项目中，PI将继续她的研究计划，调查非常大的随机矩阵的行为，以及这种行为如何取决于它们的生成方式;这将推进理论，这将有助于更好地理解上述一些领域的应用。她还将完成她的专著“随机矩阵理论的紧凑的经典群体”，它给出了一个概述了什么已经学到了随机生成的矩阵与一个特殊的关系，以高维几何;这项活动将传播困难的技术知识随机矩阵的广大观众。 PI积极参与各级教学，特别是指导和鼓励对数学职业感兴趣的女孩和年轻女性;这些培训和推广活动将对进入STEM领域的人的多样性产生重大影响。上述专著将描述近年来Haar分布随机矩阵理论的显着进展。 主题将包括主要子矩阵和一般低维投影的条目的中心极限定理，经验谱测度的定量极限结果，特征值分布内的自相似模式，测度与几何应用的浓度，包括约翰逊-林登施特劳斯引理和Dvoretzky定理的新证明，以及特征多项式分布的结果，与黎曼zeta函数的联系 新的研究项目包括随机矩阵的研究，其分布是不变的旋转下在矩阵空间，继续工作的定量近似经验谱措施的各类随机矩阵，并在各种度量，和问题随机拓扑连接拓扑数据分析。