Operator Theory, Free Probability, and Related Problems

0307166-H Bercovici(印第安纳大学)作者计划在确定自由卷积的正则性的意义上研究自由调和分析；措施的自由卷积似乎表现出比它们的经典对应的更大的正则性。他还计划研究自由过程的极限分布，特别是在倍增理论中出现的那些。作者还将研究算子论中的相关问题，特别是von Neumann代数的自伴元素和的特征值的行为，以及这些问题与Littlewood-Richardson系数的关系。算子理论中的其他问题将包括Hilbert空间上的等距线族的分类。本文的研究首先旨在增加对当前算子代数和算子理论领域工作中出现的问题的理解。这位提出者过去曾研究过操作者和控制理论之间的联系，他提出的工作中的结果可能再次与更多的应用领域有关。特别是，一些提出的插补问题是由控制论中的问题引起的，而自由概率论确实与物理学有关。提名者将让研究生参与他的工作，从而继续为培养新一代数学家做出贡献。他还将参与博士后助理的培训，从而为印第安纳大学成功的Vigre项目做出贡献。

ABSTRACTDMS 0307166- H Bercovici (Indiana Univ)The proposer plans to study free harmonic analysis in the sense of determining regularity properties of free convolutions; free convolutions of measures seem to exhibit greater regularity than their classical counterparts. He also plans to investigate limit distributions of free processes, particularly those arising in the mutliplicative theory. The proposer also will look at related problems in operator theory, particularly at the behavior of eigenvalues of sums of selfadjoint elements of a von Neumann algebra, and the relationship of these questions with Littlewood-Richardson coefficients. Other problems in operator theory will include the classification of families of isometries on a Hilbert space.The research in this proposal is aimed first of all at increasing the understanding of questions arising in current work in the fields of operator algebras and operator theory. The proposer has in the past investigated connections between operators and control theory, and the results in his proposed work may again have connections with more applied areas. In particular, some of the proposed interpolation problems arose from questions in control theory, and free probability theory does have connections with physics. The proposer will involve graduate students in his work, thus continuing to contribute to the training of a new generation of mathematicians. He will also participate in the training of postdoctoral associates, thus contributing to a successful VIGRE program at Indiana University.