RANDOM MATRICES IN FUNCTIONAL ANALYSIS

泛函分析中的随机矩阵

• 批准号: 1001894
• 负责人: Todd Kemp
• 金额: $ 12.35万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001894&HistoricalAwards=false
• 关键词: RANDOM; MATRICES; FUNCTIONAL; ANALYSIS

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.

本提案的主要研究目标是处理无限维分析中的问题：对基本上是无限维的空间的泛函分析，例如与布朗运动相关的随机变量空间或正交变换的无限维群。在这里，这些问题是通过随机矩阵理论的镜头来看的，随机矩阵理论是一个相对较新的领域，融合了概率论、复分析和组合学。随机矩阵理论为泛函分析领域提供了强大的新工具。此外，随着连接的增长，两个领域之间的相互作用使得功能分析能够反馈并深入了解与随机矩阵相关的结构。本研究包括三个项目，研究西格尔-巴格曼分析、随机分析和希尔伯特空间的随机子空间的相互关联的主题。在随机矩阵理论中提出了一个关于随机特征向量的方案。在前三种情况下，在首席研究员看来，随机矩阵理论的工具有很大的潜力，可以为新旧问题提供富有成效的见解。对于后一个项目，信息流向相反的方向移动：从算子代数的自由概率论的复杂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形状的定量信息。在所有提议的研究中，在非常不同的领域之间存在潜在的相互作用？概率论，几何泛函分析，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实际应用问题。在所有科学分支中，数字数组（也称为矩阵）是记录数据的常用有效方法。在这些数组中寻找意义是一项从平凡到高度复杂的工作。一个相关的例子在信号处理中是常见的，其中时变信号被数字化以产生一个矩形振幅阵列。滤除噪声（或解码信号）意味着找到识别矩阵中有序与随机模式的方法。利用20世纪50年代开始在物理学中发展起来的成分，80年代的统计学，以及最近二十年来的复杂和功能分析，现在有了丰富而强大的工具集来进行这种信号与随机噪声的分离。该建议包括由这些工具及其在功能分析中的基础所驱动的几个项目。首席研究员是与随机矩阵理论相关的许多领域的专家，并期望通过探索纯数学与应用科学之间的新旧联系来取得丰硕成果。在至少一个提议的项目中，对信号处理问题（大型天线阵列）的实际应用有重大的承诺。该提案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加。