Exact and asymptotic methods in Random Matrix Theory and Integrable Systems

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Random matrix models were introduced by Wigner to model nuclear interactions and have witnessed a renewed interest later on due to unexpected (at the time) ties with a widespread range of areas in both theoretical physics (string theory and two-dimensional quantum gravity), pure and applied mathematics (number theory, combinatorics, statistical modeling of finance) fluid dynamics of incompressible viscous media (Hele-Shaw cells)  and the theory of completely integrable dynamical systems in infinite dimensions. The ongoing project of research focuses on this latter  relationship between random matrix theory (RMT)  and certain dynamical systems known under the names of Toda, Volterra, Relativistic Toda, Ablowitz-Ladik, 2-Toda lattices etc. A minor component of this projects is devoted to the relation with the so-called Laplacian growth problems of incompressible fluids. In random matrix theory the main interest is to describe the statistical properties of the spectrum of the matrices (the eigenvalues), in particular when the size of the matrix is large.  The probability densities and all correlation functions between these eigenvalues can be expressed in terms of certain polynomials (orthogonal polynomials and variations thereof like biorthogonal polynomials). The importance of these statistical functions in the original motivation for nuclear physics was that an eigenvalue represent an energy of the quantum system and knowing their distributions allows to predict on a statistical basis the spectra in emission and absorption of samples of such nuclei. One of the main findings of the past three years has been the relevance of a "spectral curve" associated naturally to these polynomials. It is one of the main aims of this project to extend the spectral curve-technique to increasingly larger areas of RMT and to use the algebraic structure to infer otherwise unaccessible  nonperturbative asymptotic informations (such as -e.g- the maximal genus of the asymptotic spectral curve etc.)