# The Riemann-Hilbert Problem

## A Publication from the Steklov Institute of Mathematics Adviser: Armen Sergeev

Part of the Aspects of Mathematics book series (ASMA, volume 22)

Part of the Aspects of Mathematics book series (ASMA, volume 22)

This book is devoted to Hilbert's 21st problem (the Riemann-Hilbert problem) which belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concems the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this tumed out to be a rare case of a wrong forecast made by hirn. In 1989 the second author (A.B.) discovered a counterexample, thus 1 obtaining a negative solution to Hilbert's 21st problem. After we recognized that some "data" (singularities and monodromy) can be obtai ned from a Fuchsian system and some others cannot, we are enforced to change our point of view. To make the terminology more precise, we shaII caII the foIIowing problem the Riemann-Hilbert problem for such and such data: does there exist a Fuchsian system having these singularities and monodromy? The contemporary version of the 21 st Hilbert problem is to find conditions implying a positive or negative solution to the Riemann-Hilbert problem.

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- DOI https://doi.org/10.1007/978-3-322-92909-9
- Copyright Information Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH, Wiesbaden 1994
- Publisher Name Vieweg+Teubner Verlag, Wiesbaden
- eBook Packages Springer Book Archive
- Print ISBN 978-3-322-92911-2
- Online ISBN 978-3-322-92909-9
- Series Print ISSN 0179-2156
- About this book