RANDOM MATRICES IN FUNCTIONAL ANALYSIS

该提案的主要研究目标是处理无限维分析中的问题：基本上是无限维的空间上的函数分析，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，这些问题被认为是通过随机矩阵理论，一个相对较新的领域，在概率论，复杂的分析和组合学的汇合透镜。 随机矩阵理论在泛函分析的竞技场中提供了强有力的新工具。 此外，随着联系的增长，这两个领域之间的相互作用使泛函分析能够反馈并深入了解与随机矩阵相关的结构。 拟议的研究包括三个项目，调查Segal-Bargmann分析，随机分析和希尔伯特空间的随机子空间的相互关联的主题。 还提出了随机矩阵理论中的一个项目，涉及随机特征向量。 在前三种情况下，从随机矩阵理论的工具，在首席研究员的观点，巨大的潜力，使富有成效的见解，无论是旧的和新的问题。 对于后一个项目，信息的流动在相反的方向移动：复杂的分析技术，从自由概率论在算子代数被用来给定量信息的几何特征空间的一对随机矩阵。 在所有拟议的研究中，有可能在非常不同的领域之间进行互动？概率论、几何泛函分析和算子代数等等。 此外，后一个项目的动机是，并承诺在信号处理和电气工程的其他部分的问题的实际应用。数组的数字（也称为矩阵）是一种常见的有效的方式来记录数据，在所有的科学分支。 在这些数组中寻找意义是一项从平凡到高度复杂的事业。一个相关的例子在信号处理中很常见，其中时变信号被数字化以产生幅度的矩形阵列。 滤除噪声（或解码信号）意味着找到识别矩阵中有序模式与随机模式的方法。使用20世纪50年代开始的物理学，20世纪80年代的统计学以及最近20年的复杂函数分析，现在有一个丰富，强大的工具集合用于这种信号与随机噪声分离。 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。首席研究员是随机矩阵理论相关领域的专家，并期望通过探索纯数学和应用科学之间的新旧联系取得丰硕成果。 在至少一个拟议的项目中，有实际的实际应用信号处理问题（大型天线阵列）的显着承诺。 这项建议涉及不同复杂程度的研究问题，因此本科生，研究生，博士后和教师研究人员可以参与。

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.