Grant: Discovery Grants Program - Individual

The theory of Integrable Systems sprung from a special class of Dynamical Systems that can be "solved" explicitly in the sense of classical mechanics; it surprisingly lies at the cross-road between a number of seemingly distant areas and has found applications in Geometry (differential and algebraic), Combinatorics, Statistics to name the main ones. A common thread in the methods of Integrable Systems is the formulation of an auxiliary linear problem that depends on infinitely many parameters (called "times") in such a way that, depending on the context, certain properties of the auxiliary problem remain invariant: for example the spectral data (or "energy levels" in a physical context), or -in the case of differential equations- the monodromy data, i.e., a suitable notion of boundary value conditions. The project's overarching goal is the advancement and refinement of methods stemming from that area, with particular emphasis on geometric and analytic aspects: in many if not most instances one can formulate all these auxiliary systems in terms of a peculiar type of boundary value problem for a matrix- valued function in the complex plane; this formulation is referred to as a "Riemann--Hilbert Problem", as a reference to the twenty- first problem in the celebrated list of David Hilbert's presented at the International Congress of Mathematics in 1900. The areas where the methods are applicable comprise the theory of wave equations, the geometry of the deformation spaces of many geometric structures (Riemann Surfaces, vector bundles etc.), combinatorics of covering maps of the Riemann sphere ("Hurwitz numbers"), statistical properties of large stochastic matrices as well as random point fields, spectral theory of integral operators with application for example to the stability theory of the image reconstruction in tomography. The proposed project advances several fronts of the research of the candidate, where the common methodology is the formulation of an appropriate Riemann--Hilbert problem:  Orthogonality and boundary value problems  on Riemann Surfaces,  Symplectic geometry of (wild) character varieties, asymptotic behaviour of enumerative geometric problems,  matrix orthogonal polynomials and  the relationship between anharmonic quantum oscillators and Painlevé equations.