Finite Factors, Free Probability, and Combinatorics in Operator Theory

在控制理论、物理科学和其他数学领域中出现的问题，自然导致人们考虑无限维空间之间的线性映射，以及由这些映射的集合构建的其他对象。这个项目的目的是增加我们对这些概念的理解，前面提到的领域的应用程序可能是遥远的目标。这个项目的工作涉及显然不同的数学学科的数学，如组合学和概率论。更具体地说，它采用了适合于非交换设置的各种更新的概率概念。该研究的主要研究者将包括研究生和本科生。事实上，有些组合问题可以用一种足够基本的方式来表述，不需要太多的技术背景就可以理解。（之前对本科生的研究确实发表了一些文章。）这项建议中的具体问题可分为几类。第一个是来自组合学和表示理论的Littlewood-Richardson规则。希望找到适用于有限冯·诺依曼代数的紧算子和元的和与积的适当的表述。第二类与Voiculescu提出的自由概率论有关。这里的问题围绕着自由随机变量集合的加法问题及其在随机矩阵中的应用（例如，由扰动引起的离群特征值的研究）。第三类是由几个从单算子理论或由它们产生的密切相关的非自伴随代数中得到的有些松散连接的问题组成。问题的范围从交换等距的结构理论到关于超自反性、不变子空间和数值范围的问题。(就应用而言，第二类问题与通信理论有关，而单算子理论在控制理论中是一个有价值的工具。这些应用并不是本研究的重点，但主要研究者的早期工作已经在应用领域产生了影响。)

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.)