Integrable systems in Geometry, Asymptotics and Inverse Problems

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Integrable systems consist in a special class of overdetermined sets of partial differential (or difference) equations. They appear in several contexts in slightly different guises, including Random Matrix Theory, Moduli Spaces of Riemann surfaces and connections, Stochastic Processes and Inverse problems. A common thread to all these instances is the possibility of reformulation in terms of a particular boundary value problem for matrix-valued analytic functions, or what is now commonly referred to as a Riemann-Hilbert problem (RHP). The proposed research seeks to both advance the general understanding of RHPs as well as their application to several outstanding problems. ***Some of the specific goals include the study of the Poisson geometry underlying deformation theory of RHPs, extending also to RHPs on Riemann surfaces.***In the context of intersection numbers on the moduli space of curves (Gromov-Witten invariants), the generating function can be associated also to a RHP and this has the benefit of leading to a rigorous asymptotic analysis. The generating function of intersection numbers between fundamental classes in the moduli space of curves is obtained from a matrix integral (Kontsevich); the integral is known to provide a formal solution of the KdV hierarchy. From works of G. Moore's, the formal connection with isomonodromic deformations (thus, indirectly, RHP) was observed.****However an analytic approach is still missing. It is a goal to complete this description in terms of a suitable RHP for a matrix of fixed size of the Kontsevich integral and generalizations thereof. This in particular will shed light on non-formal properties of the generating function, such as the nonlinear Stokes' phenomenon (analytic resummation).*** Another application of RHPs is in solving inverse problems. Here the project focuses on the inverse (and forward) scattering theory of the nonlinear Schroedinger equation: in the semiclassical limit as Planck's constant is sent to zero is considered also in the study of elongated phases of Bose--Einstein condensates and in oceanography, where it has been proposed as describing the underlying mechanism of formation of the so-called rogue waves and for the ``three sister'' rogue waves.*** A second outstanding inverse problem originates in the area of medical imaging (tomography); in order to reduce irradiation of patient's tissue, the (ill-posed) problem of inversion with partial data must be addressed. Then the open question is the degree of instability of the reconstruction, which is translated into a question about asymptotic behavior of singular values and singular functions for a certain integral operator.********