Random matrices, free probability, and the enumeration of maps

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This research is in the area of random matrices. The proposal focuses on two aspects of this domain: (i) Analysis of the topological expansions. It was shown in the seventies that matrix integrals can be viewed as generating functions for the enumeration of maps, that is connected graphs which are sorted by their genus, where the dimension plays the role of genus parameter. This is the so-called topological expansion. They were first used in physics as a clever way to enumerate maps. It then became an apparatus to find invariants in many different fields. On a different point of view, the limit of random matrices is well described by free probability. Hence, free probability is useful to analyze topological expansions and conversely topological expansions and random matrices can be used to construct new concepts in free probability, with consequences in operator algebra. The first part of the proposal addresses several questions in this direction. One of the goals consists in developing a general mathematical scheme to obtain topological expansions. Another one is to use random matrices and classical probability theory to develop the theory of optimal transport in free probability, hence proving isomorphisms results in operator algebra.(ii)The second part of the proposal concerns the study of the universality classes of random matrices. In fact, after the recent breakthroughs exploring the universality class of Gaussian matrices, it is natural to study matrices which are definitely out of this class. The proposal offers to study heavy tailed matrices, and more precisely localization/delocalization of their eigenvectors as well as local fluctuations of the eigenvalues in the bulk. This research is in the area of random matrices, which are connected with many domains of mathematics and physics, for instance as a large (random) array of data, as the approximation of operators such as the Hamiltonian of large physical systems, or more exotically via the non-trivial zeroes of the Riemann zeta function. The project focuses on its applications to operator algebra, combinatorics and physics.