Collaborative Research: Riemann-Hilbert Problems and Riemann Surfaces: Computations and Applications

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Riemann-Hilbert problems (RHPs) arise in a plethora of applications, varying from equations describing tsunamis to the understanding of nuclear energy. In its most basic form, a RHP determines a function that jumps in a prescribed way along a curve in the plane and has specified behavior far away from where the jump occurs. Such problems were first posed by Riemann and later by Hilbert at the end of the 19th century. Their study has been at the forefront of pure and applied mathematics. Until recently, little effort had been devoted to the actual computation of solutions of such problems. This research project extends recent work in carrying out numerical investigations. It is anticipated that major advances will be made in the solution of RHPs, allowing for the increased understanding of tsunamis, fast optical communication, and other physical phenomena.The goal of the project is to develop new computational tools for the solution of RHPs and their extensions. Traditionally, RHPs arise in the context of singular integral equations and the Wiener-Hopf technique. More recently, RHPs have been connected to random matrix theory, nonlinear special functions, and nonlinear wave equations. RHPs may be posed on Riemann surfaces, and nonlinear jump conditions may be specified. Recent developments involving the investigators and collaborators have led to the development of accurate and efficient numerical algorithms for the solution of RHPs, for problems posed on Riemann surfaces, and for the computation of special functions such as the Schottky-Klein prime function. However, many open problems remain, particularly concerning new applications. This project aims to develop new computational methods to solve these problems, with an emphasis on the development of fast and efficient algorithms that can deal with complicated geometries, and to deploy them in applications. The investigators, postdoctoral scholar, and collaborators bring together a unique combination of expertise in the different areas needed to successfully carry out the collaborative project.