Free probability, von Neumann algebras, subfactors and random matrices

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AbstractShlyakhtenkoThe aim of the proposal is to study connections between four areas of mathematics: von Neumann algebras; subfactor theory; free probability theory and random matrix theory. The main tools for the proposed research come from a synthesis of ideas from these four subjects. This involves free stochastic analysis and the theory of square-integrable cohomology in free probability theory; Popa's deformation-rigidity theory in von Neumann algebras; combinatorial structure of random matrices related to random multi-matrix models with a potential; and the planar algebra approach to subfactor theory, as pioneered by Jones. All of the four areas mentioned above are amazingly rich mathematically. For example, Jones' subfactor theory has led to the discovery of a novel knot invariant, which has uses in diverse areas of mathematics and beyond (including the study of structure of DNA). Research in random multi-matrix models has engineering applications such as cell phone design. The focus of the present research is on the interplay between these four areas with the aim towards developing tools and techniques that are likely to impact all of the areas involved.