Finite Factors, Operators, and Free Probability

有限因素、算子和自由概率

• 批准号: 1065946
• 负责人: Hari Bercovici
• 金额: $ 27.08万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1065946&HistoricalAwards=false
• 关键词: Finite; Factors; Operators; Free; Probability

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.

这个项目的重点是在自由随机变量和相关领域的研究中出现的算子理论和函数理论的几个方面。所涉及的工具部分是这些领域固有的，但也涉及其他领域的输入，如组合学和代数几何。一个重要的主题是使用组合Littlewood-Richardson规则研究希尔伯特空间上的紧算子或有限冯诺依曼代数中的自伴元素的本征值问题。这个规则（及其连续扩展）也在不同的方向上发挥作用，涉及某些算子的不变子空间的分类。第二个主要主题是研究弱和强限制法律的自由概率，以及规律性问题的自由卷积。这里有许多问题，经典函数论的方法产生有趣的，有时意想不到的结果。其他问题的算子理论要考虑治疗的频谱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。本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，乍一看，这些问题似乎并不相关。一个例子是使用方法的代数几何在解决问题的分析。该项目的一些研究领域在工程中具有潜在的应用，特别是在控制理论（具体而言，大型结构的控制）方面，并且首席研究员的早期成果实际上已被纳入工程项目中。除了纯粹的科学价值外，这笔赠款所支助的活动预计将通过培训博士和博士后学生以及编写新的课程材料产生影响，这些材料将纳入该项目的一些研究成果。主要研究者培养了多名博士。学生，其中三名是妇女，一名是西班牙裔。这些学生的工作产生了重大影响的领域，他们进入（算子理论，自由概率，控制理论和偏微分方程）。首席研究员还指导了几位博士后同事，他们继续成功的职业生涯。最后，首席研究员指导了印第安纳州大学REU项目的两名本科生。如前所述，本科生、研究生和博士后的指导活动将在本项目中发挥不可或缺的作用。