The Riemann-Hilbert Problem

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

  • D. V. Anosov
  • A. A. Bolibruch

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

Table of contents

  1. Front Matter
    Pages I-IX
  2. D. V. Anosov, A. A. Bolibruch
    Pages 1-13
  3. D. V. Anosov, A. A. Bolibruch
    Pages 14-50
  4. D. V. Anosov, A. A. Bolibruch
    Pages 51-76
  5. D. V. Anosov, A. A. Bolibruch
    Pages 77-88
  6. D. V. Anosov, A. A. Bolibruch
    Pages 89-132
  7. D. V. Anosov, A. A. Bolibruch
    Pages 133-157
  8. D. V. Anosov, A. A. Bolibruch
    Pages 158-184
  9. Back Matter
    Pages 185-193

About this book

Introduction

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.

Keywords

Monodromy differential equation eXist equation form mathematics ordinary differential equation presentation system

Authors and affiliations

  • D. V. Anosov
    • 1
  • A. A. Bolibruch
    • 1
  1. 1.Steklov Institute of MathematicsMoscow/CISRussia

Bibliographic information

  • 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