Abstract

In my opinion the (general) Riemann-Hilbert problem (RHP) is THE fundamental boundary value problem in complex analysis. It comprises the well-known linear boundary and transmission problems of Riemann-Hilbert type, but also conformal mapping and conformal welding and has interrelations with singular integral equations, Toeplitz operators, Wiener-Hopf factorization, and extremal problems in geometric function theory. Though many aspects of the problem are by now fairly well understood, the problem has not yet received the attention it deserves. A main aim of the talk is to stimulate the interest of the young generation in this fascinating topic and to encourage further research in theory and applications of RHPs.This includes the development of universal numerical methods and a direct proof of the existence of solutions via an extremal principle that generalizes Caratheodory’s principle for conformal mappings.