Operators and Free Probability

算子和自由概率

基本信息

批准号：
0600562
负责人：
Hari Bercovici
金额：
$ 26.38万
依托单位：
Indiana University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2006
资助国家：
美国
起止时间：
2006-07-01 至 2012-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600562&HistoricalAwards=false
关键词：
Operators Free Probability

项目摘要

Hari Bercovici will work on several aspects of operator theory and function theory, which arise in the study of free random variables and other areas. His tools are intrinsic to these areas, but also involve input from other areas, such as combinatorics. 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 selfadjoint 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. Another important theme is the study of weak and strong limit laws in free probability, as well as in monotonic probability theory. 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 concern the spectral Nevanlinna-Pick problem and its analogues, hyperinvariant subspaces, dual algebras, and p-entropies.This project will broaden mathematical knowledge in specific theoretical areas, as well as seek applications in related areas of mathematics, control theory, and computer science. Hari Bercovici will seek the participation of graduate and, when possible, undergraduate students. This will contribute to the training of future scientists.

哈里·贝科维奇（Hari Bercovici）将在操作者理论和功能理论的几个方面工作，这是在自由随机变量和其他领域的研究中产生的。他的工具对这些领域是固有的，但也涉及其他领域的投入，例如组合制剂。一个重要的主题是在研究希尔伯特空间上紧凑型操作员的特征值问题或在有限的von Neumann代数中的自助元素元素的特征值问题的研究中使用组合。该规则及其连续扩展也在不同的方向上发挥了作用，该方向与某些操作员的不变子空间的分类有关。另一个重要主题是研究自由概率以及单调概率理论中弱和强限制定律。这里有许多问题，经典功能理论的方法产生有趣的结果，有时是意外的结果。操作者理论的其他问题被认为是涉及频谱Nevanlinna-Pick问题及其类似物，超变变子空间，双代数和P-凝管。该项目将在特定理论领域扩大数学知识，并在数学，控制理论和计算机科学的相关领域中寻求应用。哈里·贝科维奇（Hari Bercovici）将寻求研究生和本科生的参与。这将有助于对未来科学家的培训。