The 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.
Research funded by this grant centered on researching connections between three mathematical subjects: free probability theory, random matrix theory and subfactor theory. Random matrix theory aims to describe the behavior of large generic matrices. Such matrices appear naturally in many questions, including, for example, in mathematical approaches to improving cell phone antennae, or in the analysis of statistical correlations of a large number of variables. In the case that several matrices are involved, free probability theory provides the analysis tools. For example, free stochastic calculus allows one to formulate stochastic equations in a way that already takes into account the large size of matrices. Finally, subfactor theory provides a rich source of symmetries that can simplify computations. The principal scientific outcomes of this work involve the establishment of the analytical framework to deal with random matrices posessing suitable symmetries. The resulting advances in free stochastic calculus and the resulting analysis have had impact in von Neumann algebra theory, free probability theory, as well as subfactor theory. Several scientific publications funded by this grant give the precise statements of these results. The PI has reported these results at several scientific conferences, including an invited sectional talk at the International Congress of Mathematicians in 2010. At the same time, the grant provided research opportunities to several graduate students, resulting in several publications. One of the students supported by the grant has graduated with a PhD degree, and two will be graduating in a year. Additionally, the proposal funded two successful workshop, each bringing together over 25 participants working in the fields related to the subject of the proposal. These workshops resulted in new perspectves and research collaborations.
