The principal scientific aim of the Network is to create a fertile research environement for scientists (especially Ph.D. students and Post/docs) working in the fields of Mathematical Physics and Numerical Applications thereof by means of a steady and intense flow of personnel exchanges between the European and the Overseas nodes of the Network, as well as the organization of Network Workshops. 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 research programme is structured along three scientific Work Plans.
- (Near-to) Integrable PDEs and applications
- Isomonodromic deformations, Painlevé equations and random objects
- Numerical approaches
The primary research objectives are the following:
- Further understanding of the Mathematics of Integrability by completing the classification of the systems of spatially one-dimensional integrable evolutionary Partial Differential Equations (PDEs) that admit hydrodynamic limit (WP A).
- Understanding of the analytical aspects of asymptotic integrability of systems depending on a parameter, especially in the realm of the theory of water waves equations arising from suitable reductions of Euler flows (WP A).
- Understanding of the general universality classes appearing in critical regimes of Matrix models as well as the computation of significant properties (WP B).
- Express the tau function associated with the general solution of a system of isomonodromic deformations via generalisations of the Painlevé functions as a fredholm determinant (WP B).
- Implementation of geometric and symplectic integrators for integrable and almost integrable PDEs (WP C)
- Creation of an open source numerical library for Painlevé equations (WP C).
Approch and Methodology
The overall methodological approach relies on combining analytical, geometrical and numerical techniques, so far often confined within different areas of Mathematics and Physics. The research problems will be at firrst addressed by the approach of pure mathematics, based on the careful analysis of intrinsic connections between the mathematical constructions under consideration, on the precise formulation of the arising conjectures and on proving them as rigorous theorems. The power of efficient numerical tools will be 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, bihamiltonian 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 will be employed in the pursue of our goals.