The Riemann-hilbert Problem: A Publication From The Steklov Institute Of Mathematics Adviser: Armen Sergeev (aspects Of Mathematics)

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. 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 obtaiƯ 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 The Riemann-hilbert Problem (hilbert's 21st Problem) Belongs To The Theory Of Linear Systems Of Ordinary Differential Equations In The Complex Domain. The Problem Concerns The Existence Of A Fuchsian System With Prescribed Singularities And Monodromy. Hilbert Was Convinced That Such A System Always Exists. However, This Turned Out To Be A Rare Case Of A Wrong Forecast Made By Him. In 1989 The Second Author (a. B.) Discovered A Counterexample, Thus Obtaining A Negative Solution To Hilbert's 21st Problem In Its Original Form. 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