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.