RANDOM MATRICES IN FUNCTIONAL ANALYSIS

这一建议的主要研究目标涉及无限维分析中的问题：在基本上是无限维的空间上进行泛函分析，例如与布朗运动或无限维正交变换组相关的随机变量空间。在这里，通过随机矩阵理论的镜头来看待这些问题，随机矩阵理论是一个相对较新的领域，它是概率论、复分析和组合学的交汇点。随机矩阵理论在泛函分析领域提供了强大的新工具。此外，随着联系的发展，这两个领域之间的相互作用使得泛函分析能够反馈并给出与随机矩阵相关的结构的见解。拟议的研究包括三个项目，调查相互关联的主题，西格尔-巴格曼分析，随机分析，和希尔伯特空间的随机子空间。还提出了关于随机特征向量的随机矩阵理论的一个项目。在前三个案例中，在首席研究者看来，来自随机矩阵理论的工具具有巨大的潜力，可以对新旧问题提供富有成效的见解。对于后一种方案，信息流是朝着相反的方向移动的：来自算子代数中自由概率理论的复分析技术被用来给出关于一对随机矩阵的特征空间的几何的定量信息。在所有拟议的研究中，有没有可能在非常不同的领域之间存在相互作用？概率论、几何泛函分析和算子代数仅举几例。此外，后一个项目的动机是信号处理和电气工程其他部分的问题，并承诺将其应用于现实世界。在科学的所有分支中，数组(也称为矩阵)是记录数据的一种常见的有效方式。在这些阵列中寻找意义是一个从平凡到高度复杂的企业。一个相关的例子在信号处理中很常见，在信号处理中，将时变信号数字化以产生矩形幅度阵列。滤除噪声(或解码信号)意味着找到识别矩阵中有序模式和随机模式的方法。利用从20世纪50年代开始发展的物理学成分，80年代的统计数据，以及最近20年来的复杂和功能分析，现在有了丰富而强大的工具集合，用于这种信号到随机噪声的分离。该提案包括由这些工具及其在功能分析中的基础推动的几个项目。首席研究员是与随机矩阵理论相关的多个领域的专家，他希望通过探索纯数学和应用科学之间的新旧联系，取得丰硕的成果。在至少一个提议的项目中，实际应用于信号处理问题(对于大型天线阵列)具有重要的前景。这项建议涉及不同程度的复杂程度的研究问题，因此本科生、研究生、博士后和教职研究人员可能会参与其中。

The principal research goals of this proposal deal with questions in infinite-dimensional analysis: functional analysis on spaces that are fundamentally infinite-dimensional, such as the space of random variables associated to a Brownian motion or an infinite-dimensional group of orthogonal transformations. Herein, these questions are viewed through the lens of random matrix theory, a relatively new field at the confluence of probability theory, complex analysis, and combinatorics. Random matrix theory provides powerful new tools in the arena of functional analysis. Moreover, as connections have grown, interplay between the two fields has allowed functional analysis to feedback and give insights into structures associated to random matrices. The proposed research includes three projects, investigating the interconnected themes of Segal-Bargmann analysis, stochastic analysis, and random subspaces of a Hilbert space. Also proposed is a project in random matrix theory, concerning random eigenvectors. In the former three cases, tools from random matrix theory have, in the view of the principal investigator, great potential to give fruitful insights into problems both old and new. For the latter project, the flow of information moves in the opposite direction: the complex analytic techniques from free probability theory in operator algebras are used to give quantitative information about the geometry of the eigenspaces of a pair of random matrices. In all of the proposed research, there is the potential for interaction between very different fields ? probability theory, geometric functional analysis, and operator algebras to name a few. Moreover, the latter project is motivated by, and promises real-world application to problems in signal processing and other parts of electrical engineering.Arrays of numbers (also known as matrices) are a common efficient way to record data, in all branches of science. Finding meaning in those arrays is an enterprise that runs from the mundane to the highly sophisticated. A relevant example is common in signal processing, where a time-varying signal is digitized to produce a rectangular array of amplitudes. Filtering noise out (or decoding signals) means finding ways to recognize ordered versus random patterns within the matrix. Using ingredients developed in physics starting in the 1950s, statistics in the 1980s, and more recently complex and functional analysis in the last two decades, there is now a rich, robust collection of tools for such signal-to-random-noise separation. This proposal includes several projects motivated by those tools and their foundations within functional analysis. The principal investigator is an expert in a number of fields related to random matrix theory, and expects fruitful results to follow from exploring old and new connections between pure mathematics and applied science. In at least one proposed project, there is significant promise of actual practical applications to signal processing problems (for large arrays of antennas). This proposal involves research problems at varying levels of sophistication, and so undergraduate students, graduate students, and postdoctoral and faculty researchers may participate.