Operators and Free Probability

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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.