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.

该提案的主要研究目标涉及无限维分析中的问题：对从根本上是无限维度的空间上的功能分析，例如与布朗运动或无限二维正交转换的随机变量的空间。 在此，这些问题是通过随机矩阵理论的镜头来查看的，这是一个相对较新的领域，在概率理论，复杂分析和组合学的汇合处。 随机矩阵理论在功能分析领域提供了强大的新工具。 此外，随着连接的发展，两个字段之间的相互作用允许功能分析以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，研究了塞加尔 - 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联系的主题。 还提出的是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中，在主要研究者的角度看来，具有富有成果的新旧问题的巨大潜力。 对于后一个项目，信息流朝相反的方向移动：操作器代数中的自由概率理论的复杂分析技术用于提供有关一对随机矩阵特征的几何形状的定量信息。 在所有提出的研究中，在非常不同的领域之间存在相互作用的潜力吗？概率理论，几何函数分析和操作符代数等。 此外，后一个项目是由动机的，并承诺对信号处理和电气工程其他部分的问题进行现实应用。数字阵列（也称为矩阵）是在所有科学分支中记录数据的常见方法。 在这些阵列中找到含义是一家从平凡到高度成熟的企业。一个相关的示例在信号处理中很常见，其中将随时间变化的信号数字化以产生矩形的振幅阵列。 过滤噪声（或解码信号）是指找到矩阵中识别有序与随机模式的方法。使用从1950年代开始在物理学中开发的成分，1980年代的统计数据，以及最近二十年中最近复杂且功能分析，现在有了丰富，强大的工具收集，用于这种信噪比，用于这种信噪比 - 噪声分离。 该建议包括在功能分析中以这些工具及其基础动机的几个项目。首席研究者是许多与随机矩阵理论有关的领域的专家，并期望从探索纯数学和应用科学之间的新旧联系中伴随着富有成果的结果。 在至少一个拟议的项目中，对信号处理问题的实际实用应用有很大的承诺（对于大型天线）。 该建议涉及在不同水平的研究问题，因此本科生，研究生以及博士后研究人员可能会参与。