Operators and Free Probability

Hari Bercovici将致力于算子理论和函数论的几个方面的工作，这些理论出现在自由随机变量的研究和其他领域。他的工具是这些领域固有的，但也涉及其他领域的投入，如组合学。一个重要的主题是利用组合Littlewood-Richardson规则研究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。这一规则及其连续推广在某些算子的不变子空间的分类方面也起到了不同的作用。另一个重要的主题是研究自由概率和单调概率论中的弱极限定律和强极限定律。这里有许多问题，经典函数理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涉及谱的Nevanlinna-Pick问题及其类似物、超不变子空间、对偶代数和p-熵。这个项目将拓宽特定理论领域的数学知识，并寻求在数学、控制理论和计算机科学的相关领域中的应用。Hari Bercovici将寻求研究生的参与，如果可能的话，本科生也会参与。这将有助于培养未来的科学家。

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.