# Background

Partial Differential Equations (PDEs) undoubtedly are among the main tools for an efficient modelling of physical phenomena. Integrable PDEs, exhibiting infinite-dimensional analogues of regular (integrable) behaviour displayed by finite dimensional systems, began to be studied in the middle of the XX century in fluid dynamics, field theory and plasma physics.

It was observed that, in specific models and regimes, a balance between non-linearity and dispersion leads to the formation of stable patterns. The modern theory of integrable systems grew up around the study of the Korteweg de Vries (KdV) equation, with origins in the seminal work of Zabusky and Kruskal about the recurrence behaviour of solutions, the discovery of the Lax pair formulation, multi-soliton so-lutions and infinite number of conservation laws.

Over the last 30 years refined analytical and geometrical tools have been developed to study integrable or nearly integrable systems such as Riemann – Hilbert boundary values problems, infinite-dimensional generalizations of KAM and Nekhoroshev theories, algebraic geometry of Riemann surfaces and theta-functions, the theory of infinite-dimensional Lie algebras and techniques of differential geometry involved in the study of the Hamiltonian structures of dynamical systems and in the theory of Frobenius manifolds. The idea that an integrable behaviour persists in non-integrable systems, together with the combination of the state-of-the-art numerical methods with front-line geometrical and analytical techniques in the theory of Hamiltonian PDEs, is the leitmotiv of this research project. The asymptotic regimes leading to phase transitions display universality properties which can be analysed both numerically and analytically. The predictive power of numerics and scientific computing will be used both as a testing tool for theoretical models and as a generator of new conjectures.

Random matrices, introduced in the study the statistical properties of quantum systems with a large number of degrees of freedom (heavy nuclei), have been shown to successfully enter various fields such as number theory, combinatorics, quantum field theory, and even wireless telecommunications. One of the main reasons for their fascinating modelling power is that as the dimensions of the matrices tend to infinity the local statistical properties of the eigenvalues become independent of the probability distribution of the given matrix space. This important and long conjectured property of such a fast developing field of research was proved only recently. Like the class of dispersive PDEs mentioned above, these random “objects” display in suitable asymptotic scalings, a completely integrable behaviour. This was the clue to the conjecture, originally due to B. Dubrovin, that critical behaviour of Hamiltonian dispersive PDEs (near a gradient catastrophe of their dispersionless counterpart) is structurally independent of its dispersive “tail” and of the initial data. The critical behaviour is indeed conjectured to be universal, in the same way as the local statistical properties of random matrices become independent of the probability distribution when the size of the matrix goes to infinity.