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

从表面上看，矩阵就是一盒数字。但是以矩阵的形式编码信息是现代数学中最强大的思想之一；矩阵语言几乎支撑了纯数学和应用数学的每一个领域，并锚定了物理、工程和其他科学的框架。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”矩阵有了越来越多的了解，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，以及这种行为如何依赖于它们的生成方式；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“紧经典群的随机矩阵理论”，其中概述了与高维几何具有特定关系的随机生成矩阵的知识；这项活动将向广大受众传播有关随机矩阵的困难技术知识。PI积极参与各级教学，特别是指导和鼓励对数学事业感兴趣的女孩和年轻妇女；这些培训和外展活动将对进入STEM领域的人员的多样性产生重大影响。上面提到的专著将描述近年来haar分布随机矩阵理论的显著进展。主题将包括主子矩阵项和一般低维投影的中心极限定理，经验谱测度的定量极限结果，特征值分布内的自相似模式，几何应用的测度集中，包括Johnson-Lindenstrauss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ζ函数的联系。新的研究项目包括研究在矩阵空间旋转下分布不变的随机矩阵，继续研究各种类型的随机矩阵和各种度量的经验谱测度的定量近似，以及与拓扑数据分析相关的随机拓扑问题。

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.