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

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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Content:
Front Matter....Pages I-IX
Introduction....Pages 1-13
Counterexample to Hilbert’s 21st problem....Pages 14-50
The Plemelj theorem....Pages 51-76
Irreducible representations....Pages 77-88
Miscellaneous topics....Pages 89-132
The case p = 3....Pages 133-157
Fuchsian equations....Pages 158-184
Back Matter....Pages 185-193

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Content:
Front Matter....Pages I-IX
Introduction....Pages 1-13
Counterexample to Hilbert’s 21st problem....Pages 14-50
The Plemelj theorem....Pages 51-76
Irreducible representations....Pages 77-88
Miscellaneous topics....Pages 89-132
The case p = 3....Pages 133-157
Fuchsian equations....Pages 158-184
Back Matter....Pages 185-193
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