Finite Factors, Free Probability, and Combinatorics in Operator Theory

算子理论中的有限因子、自由概率和组合学

• 批准号: 1362954
• 负责人: Hari Bercovici
• 金额: $ 29.61万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362954&HistoricalAwards=false
• 关键词: Finite; Factors; Free; Probability; Combinatorics

Problems arising in control theory, in the physical sciences, and in other areas of mathematics have led naturally to the consideration of linear maps between infinite-dimensional spaces, as well as to other objects constructed from collections of such maps. This project is intended to add to our understanding of these concepts, with applications to the areas mentioned before as possibly distant goals. The work on this project involves mathematics from apparently disparate mathematical disciplines, such as combinatorics and probability. More specifically, it engages various newer probabilistic concepts that are appropriate for a noncommutative setting. The principal investigator will involve both graduate and undergraduate students in this research. Indeed, some of the combinatorial problems can be formulated in a sufficiently elementary manner as to be accessible without too much technical background. (Previous work with undergraduate students did result in published material.)The concrete problems in this proposal fall into several categories. The first one concerns the Littlewood-Richardson rule from combinatorics and representation theory. It is hoped that appropriate formulations of this rule will be found that apply to the study of sums and products of compact operators and elements of finite von Neumann algebras. The second category is connected with the free probability theory introduced by Voiculescu. Here the problems revolve around the addition problem for collections of free random variables and its applications to random matrices (e.g., the study of outlying eigenvalues resulting from perturbations). The third category consists of several somewhat loosely connected problems from single operator theory or the closely related non-selfadjoint algebras they generate. The problems range from structure theory for commuting isometries to questions about hyper-reflexivity, invariant subspaces, and numerical ranges. (As far as applications are concerned, the second category of problems has connections to communications theory, while single operator theory is a valuable tool in control theory. These applications are not the focus of the proposed research, but earlier work of the principal investigator has had impact in applied areas.)

在控制论、物理科学和其他数学领域中出现的问题自然导致了对无限维空间之间的线性映射的考虑，以及从这些映射的集合构造的其他对象。这个项目旨在增加我们对这些概念的理解，并将其应用于前面提到的可能是遥远目标的领域。这个项目的工作涉及到明显不同的数学学科，如组合数学和概率。更具体地说，它涉及各种适用于非交换设置的较新的概率概念。主要研究者将让研究生和本科生参与这项研究。事实上，一些组合问题可以用足够基本的方式来表述，以便在没有太多技术背景的情况下进行。（以前与本科生的工作确实导致了出版材料。这项建议中的具体问题分为几类。第一个是关于组合学和表示论中的Littlewood-Richardson规则。人们希望，这一规则的适当公式将被发现，适用于研究有限冯诺依曼代数的紧算子和元素的总和和产品。第二类与Voiculescu介绍的自由概率论有关。这里的问题围绕着自由随机变量集合的加法问题及其对随机矩阵的应用（例如，对由扰动引起的异常特征值的研究）。第三类由几个有点松散连接的问题，从单算子理论或密切相关的非自伴代数，他们产生。问题的范围从结构理论的交换等距问题的超自反性，不变的子空间，和数值范围。(As就应用而言，第二类问题与通信理论有关，而单算子理论是控制理论中的一个有价值的工具。这些应用不是拟议研究的重点，但主要研究者的早期工作在应用领域产生了影响。