Finite Factors, Operators, and Free Probability

这个项目集中在研究自由随机变量和相关领域中出现的算符理论和函数论的几个方面。所涉及的工具在一定程度上是这些领域固有的，但也涉及其他领域的输入，如组合学和代数几何。一个重要的主题是利用组合Littlewood-Richardson规则来研究Hilbert空间上紧算子的特征值问题或有限von Neumann代数中的自伴元素的特征值问题。这一规则(及其连续扩展)在某些算子的不变子空间的分类方面也起到了不同方向的作用。第二个主要主题是研究自由概率的弱极限定律和强极限定律，以及自由卷积的正则性问题。这里有许多问题，经典函数论的方法产生了有趣的，有时甚至是意想不到的结果。其他要考虑的算子论问题包括谱的Nevanlinna-Pick问题及其类似物、超不变子空间和p-熵。在这些问题中，Nevanlinna-Pick问题受到控制理论问题的启发，可能有实际应用，而p-熵在理论计算机科学中有应用。这个项目的目的是通过强调它们与数学领域的联系来解决泛函分析中的几个问题，这些领域乍一看似乎并不相关。一个例子是使用代数几何的方法来解决分析问题。这个项目的一些研究领域在工程上有潜在的应用，特别是在控制理论(特别是大型结构的控制)方面，主要研究人员的早期成果实际上已经被纳入工程项目。除了纯粹的科学价值外，这笔赠款支持的活动预计将通过培训博士后和博士后学生以及编制新的课程材料产生影响，这些教材将纳入该项目的一些研究成果。首席调查员培养了一些博士生，其中三名是女性，一名是西班牙裔。这些学生的工作对他们进入的领域(算子理论、自由概率、控制理论和偏微分方程式)产生了重大影响。这位首席研究员还指导了几位后来事业成功的博士后同事。最后，首席调查员指导了印第安纳大学REU项目的两名本科生。如上所述，本科生、研究生和博士后阶段的辅导活动将在该项目中发挥不可或缺的作用。

This project focuses on several aspects of operator theory and function theory that arise in the study of free random variables and related areas. The tools involved are partly intrinsic to these areas, but also involve input from other areas, such as combinatorics and algebraic geometry. One important theme is the use of the combinatorial Littlewood-Richardson rule in the study of eigenvalue problems for compact operators on a Hilbert space or for self-adjoint elements in a finite von Neumann algebra. This rule (and its continuous extensions) also plays a role in a different direction concerning the classification of invariant subspaces of certain operators. A second major theme is the study of weak and strong limit laws in free probability, as well as regularity questions for free convolutions. There are many problems here where methods of classical function theory yield interesting, and sometimes unexpected results. Other problems of operator theory to be considered treat the spectral Nevanlinna-Pick problem and its analogues, hyperinvariant subspaces, and p-entropies. Of these questions, the Nevanlinna-Pick problem is inspired by control theory questions and may have practical applications, while p-entropies have applications in theoretical computer science.The aim of this project is to solve several problems in functional analysis by highlighting their connections to areas of mathematics that do not seem, at first glance, to be related. An example is the use of methods of algebraic geometry in the solution of analysis problems. Some of the areas of study in this project have potential applications in engineering, especially in control theory (specifically, the control of large structures), and earlier results of the principal investigator have actually been incorporated into engineering projects. Besides their purely scientific merit, the activities supported by this grant are expected to have an impact through the training of doctoral and postdoctoral students and the creation of new course materials that will incorporate some of the research findings from the project. The principal investigator has trained a number of Ph.D. students, of whom three are women and one is Hispanic. The work of these students has had a significant impact on the fields that they entered (operator theory, free probability, control theory, and partial differential equations). The principal investigator has also mentored several postdoctoral associates who went on to successful careers. Finally, the principal investigator has directed two undergraduates in the Indiana University REU program. As indicated, mentoring activities at the undergraduate, graduate, and postdoctoral levels will play an integral role in this project.