# 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)

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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
- Buy this book on publisher's site