Matrix-valued Riemann-Hilbert problems (RHPs) are emerging as a kind of common language for representing many apparently different sorts of problems from pure and applied mathematics. After describing what is a RHP, we will give examples of their application ranging from the theory of orthogonal polynomials and combinatorics to inverse scattering theory and the solution of integrable nonlinear partial differential equations. Then, we will specialize to a special sort of RHP for complicated contours with self-intersections. By an appropriately defined equivalence with a singular integral equation, and by establishing the right amount of compactness we will prove an alternative theorem giving conditions under which the RHP can be solved. Or, at the very least, we will state the theorem and have an exciting discussion!
 

Convenors:Klaus Ecker, Alan Pryde