The main goal of the present project is to put the theory of random matrices, random permutations and random walks in the context of the theory of integrable systems. This goal will be achieved by a significant progress in the theory of integrable systems and their perturbations, including its analytic, geometric and numerical aspects. Perturbative classification of integrable partial differential equations (PDEs) and their discrete analogues, various approaches to analytically constructing their exact solutions, the analysis of singular limits of important classes of solutions, the spectral geometry of Riemann surfaces, new asymptotic methods involving Painlev\'e transcendents and theta-functions, efficient numerical approaches to solving dispersive PDEs, all these techniques will be developed and applied to the rigorous analysis of various types of models of the theory of random matrices.
Fields of science
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