The Riemann-Hilbert Problem: A Publication from the Steklov Institute of Mathematics Adviser: Armen Sergeev(22 Aspects of Mathematics)

The Riemann-Hilbert problem deals with linear systems of ordinary differential euqations in the complex domain. Namely, the question is whether there is a Fuchsian system with prescribed singularities and monodromy. Hilbert was, in fact, convinced that such a system does always exist. However, as it turned out only recently this is one of the very rare cases of a wrong prediction by him. In 1989, the second author of this book, A. Bolibruch, discovered a counterexample, thus giving a negative answer to Hilbert's famous 21st problem in its original form. This fact has immediately changed the point of view. Now, one has to ask for conditions on singularities and monodromy implying existence and respectively nonexistence of corresponding Fuchsian systems. This book treats all known results on the problem, both positive and negative. Besides this, it also contains other related results on scalar linear ordinary differential equations in the complex domain. Many examples are given. In the book, it is only assumed that the reader is acquainted with the basics from linear algebra, ordinary differential equations and functions of one complex variable. The more complicated material needed for the treatment of the Riemann-Hilbert problem is presented in the first chapters. They contain: the local theory, including its new version due to A.H.M. Levelt; Birkhoff-Grothendieck's theorem on vector bundles over the Riemann sphere or, equivalently, Birkhoff's factorization result for certain matrix functions; Plemelj's theorem providing a positive answer to a question similar to Hilbert's 21st problem concerning so-called regular systems instead of Fuchsian ones. The exposition of this "preliminary" material turns the book also into a useful introduction to several important chapters of the contemporary theory of ordinary differential equations in the complex domain.

Table of Contents:
Introduction - Counterexample to Hilbert's 21st problem - Irreducible representations - Miscellaneous topics - The case p 3 - Fuchsian equations.

About the Author :
Prof. Anosov und Prof. Bolibrukh sind beide am Steklov Institut in Moskau tatig.