Operator Theory, Free Probability, and Related Problems

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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.