Soft Riemann-Hilbert problems and hyperbolic Fourier series

Riemann-Hilbert problems (RHP) are problems that involve jump conditions along interfaces of holomorphic functions. They are useful for the analysis of solutions of integrable nonlinear PDEs using the inverse scattering method. RHPs also apply to give asymptotics for orthogonal polynomials on the line with exponentially varying weights.

The emerging interest in determinantal point processes in the planar setting leads naturally to the study of orthogonal polynomials in the plane with exponentially varying weights.

Following the breakthrough insight of Hedenmalm and Wennman (Acta Math, 2021), the PI suggested a different and much simpler approach called soft Riemann-Hilbert problems (SRHP).

These SRHPs are actually dbar-problems, but but set up so that the main "action" takes place along an interface, assumed to be a smooth loop.

There is a clear analogy with the microlocal method of Berman, Berndtsson and Sjöstrand (2008), where the analysis is localized to a point. We would like to connect the SRHP method to nonlinear PDEs via inverse scattering.

In addition, we would like to build further on the methodto calculate effectively the polynomial Bergman kernel.Hyperbolic Fourier series is a concept derived from work by the PI and Montes-R (Annals 2011).

It is directly connected with the study of the Klein-Gordon equation, but also with the work of Radchenko and Viazovska (Fields medal 2022) on Fourier interpolation. Many issues remain to be investigated here.