Applications of random matrix theory to probabilistic aspects of operator algebras

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Operator Algebra Theory is the mathematical framework for the study of quantum mechanics.It has many applications to various parts of mathematics and renewed interaction with other scientific areas (physics, quantum information...). The question of studying the structure of operator algebras has been the object of tremendous research and progresses during the last decades. The object of my research is to focus on 'asymptotic probabilistic' aspects of these algebras. Very important questions have been left open for many years in Operator Algebra Theory, such as Connes' embedding problem. One of my leitmotiv would be to try to figure out which operator algebras can be approximated by matrices in the sense of moments, and how good can these approximations be. In the same vein, I plan to investigate which non-commutative random variables can be approximated by random matrices, and how this extends at the level of continuous time processes. The method proposed relies mainly on probabilistic techniques inspired from Random Matrix Theory and on representation theoretic methods. In particular, Matrix Integral theory is a tool I intend to make a fundamental use of. Also, I plan to make use of diagrammatic expansion (Feynman diagrams) of relevant matrix integrals. An other method of potential crucial use is asymptotic combinatorics following methods of Biane and Speicher, and recent breakthroughs in the Horn problem. I expect that the operator algebraic motivations of this proposal will provide new applications and tools to random matrix theory. I also believe that interdisciplinary collaborations with physicists could arise from my research program, as it has already arosen in the past (with theoretical physicists and astronomers).