Partial Differential Equations (PDEs) 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 the discovery of the recurrence behaviour of soliton solutions, the Lax pair formulation, and the infinite number of conservation laws. Over the last 40 years refined analytical geometrical as well as numerical tools have been developed to study integrable or nearly integrable systems.
The idea that an integrable behaviour persists in real (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 predictive power of numerics and scientific computing is used both as a testing tool for theoretical models and as a generator of new conjectures.
Random matrices, introduced in the study of the statistical properties of quantum systems with a great 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.
Just like the class of dispersive PDEs mentioned above, these random “objects” display in suitable asymptotic scalings, a completely integrable behaviour, 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 grows indefinitely.
One of the principal scientific aim of the Network is to create a fertile research environment for younger scientists (especially Ph.D. students and Post-docs) working in the fields of Mathematical Physics by means of a steady and intense flow of personnel exchanges between the European and the Overseas nodes of the Network. It is expected that the broad interdisciplinary basis and intertwining of methods of Geometry and Mathematical Physics lead to solutions of hard geometrical problems and to formulations of new mathematical models of physical phenomena.
The overall methodological approach relies on combining analytical, geometrical and numerical techniques, so far often confined within different areas of Mathematics and Physics. The power of efficient numerical tools is instrumental to dismiss wrong conjectures, to formulate new ones and even to support some steps for their analytical proof. General tools of the theory of integrable systems such as Riemann – Hilbert boundary value problems, bi-Hamiltonian geometry, Frobenius manifolds, as well as analytical tools such as general complex analysis, potential theory in the complex plane, Deift-Zhou steepest descent method for oscillatory integrals are being used in the pursue of our goals.
