Rigorous approaches to universality results in random matrix theory, integrable systems and nonlinear integrable wave equations

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The proposed project focuses on the study of universal behaviors in critical phenomena in the areas of 1) random matrix theory; 2) random point processes; 3) nonlinear wave equations (Korteweg de Vries, Nonlinear Schroedinger, etc.) These areas are tied with the theory of Painleve' equations, isomonodromic systems and Riemann-Hilbert problems (i.e. non-Abelian boundary value problems). The critical phenomena we address occur in the study of asymptotic properties with respect to a small (or large) parameter; for example in the area of integrable wave equations like the NonLinear Schroedinger equation (NLS) the small parameter is Planck's constant. The equation is advocated in the study of special elongated phases of Bose-Einstein condensates and the study of rogue waves in oceanography. The behavior of the generic solution exhibits radically different behaviors in well defined regions of spacetime which we can think of as different phases; the notion of universality refers to the behavior in a scaling region around points on the boundary separating different phases, e.g. the points of gradient catastrophe. Such a behavior is independent of the initial data and (conjecturally) stable under certain perturbations of the dynamics as well. Depending on the nature of the transition, the phenomena display different scales w.r.t. the small parameter. In the case of Random Matrices, the universality typically refers to the asymptotic behavior of the statistics of eigenvalues at edges or in the bulk (e.g. Tracy-Widom distribution, Sine-kernel, etc); these behaviors depend only on symmetry properties of the model. Along the same lines the transition of certain random point processes displays similar characteristics (for example the transition from the Pearcey to the Airy point process in self-avoiding random walks). Recent result include a (partial) proof of B. Dubrovin conjecture on the behavior of small--dispersion NLS in the neighborhood of the point of gradient catastrophe and its relation to the theory of Painleve' equations: a novel link with rogue waves (Peregrine solutions of NLS) has been made in the oscillatory regions.