The Riemann-Hilbert Problem: A Publication from the Steklov Institute of Mathematics Adviser: Armen Sergeev

The Riemann-Hilbert problem (Hilberts 21st problem) belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concerns the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this turned out to be a rare case of a wrong forecast made by him. In 1989 the second author (A. B.) discovered a counterexample, thus obtaining a negative solution to Hilberts 21st problem in its original form.