Abstract: In its simplest form, a Riemann-Hilbert problem consists of determining a function so that it is analytic everywhere except possibly on a curve that splits the complex plane given some boundary information on the curve. The methodology for solving these types of problems can be thought of as a generalization of the Cauchy integral formula. Not surprisingly, a wide range of problems in mathematics and physics can be recast as Riemann-Hilbert problems. In particular, Riemann-Hilbert problems are closely related to singular integral equations which arise int the study of wing airfoils, material elasticity, the Radon transform, and the inverse scattering transform, to name a few.
To motivate the discussion, I will begin the talk by illustrating some well known applications of Riemann-Hilbert problems. Then, I will discuss some basic properties of Cauchy type integrals and why they are pivotal in the study of Riemann-Hilbert problems. I will continue by showing how certain singular integral equations can be recast as Riemann-Hilbert problems. I will finish the talk by showing how the theory of Riemann-Hilbert problems can be applied to solve a simple boundary value problem, recover the inverse Fourier transform in a simple example, and recover a basic form of the Schwarz reflection principle. The talk is intended to be intuitive and I will only assume some background in complex variables.
Speaker: Alfredo Wetzel