Constructive Solvability Conditions for the Riemann-Hilbert Problem

Sufficient and necessary conditions for the solvability of the Riemann-Hilbert problem are studied. These conditions consist in the possibility of constructing stable and semistable pairs (of bundles and connections) for a given monodromy. The obtained results make it possible to develop algorithms for testing the solvability conditions for the Riemann-Hilbert problem.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 643–655.Original Russian Text Copyright ©2005 by I. V. V’yugin.I dedicate this work to the blessed memory of my teacher Andrei Andreevich Bolibrukh