Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

D. V. Anosov^a, V. P. Leksin<sup>b

a</sup> Steklov Mathematical Institute, Russian Academy of Sciences
^b Moscow State Regional Socio-Humanitarian Institute

Abstract: This paper contains an account of A. A. Bolibrukh's results obtained in the new directions of research that arose in the analytic theory of differential equations as a consequence of his sensational counterexample to the Riemann–Hilbert problem. A survey of results of his students in developing topics first considered by Bolibrukh is also presented. The main focus is on the role of the reducibility/irreducibility of systems of linear differential equations and their monodromy representations. A brief synopsis of results on the multidimensional Riemann–Hilbert problem and on isomonodromic deformations of Fuchsian systems is presented, and the main methods in the modern analytic theory of differential equations are sketched.
Bibliography: 69 titles.

Keywords: regular and Fuchsian systems of linear differential equations, monodromy representations of meromorphic systems of differential equations, Riemann–Hilbert problem, reducible and irreducible monodromy representations and systems of differential equations, isomonodromic deformations.

UDC: 517.925

MSC: 01A70, 34-03, 34M35, 34M50, 34M56

Received: 02.12.2010

DOI: 10.4213/rm9401

 English version: Russian Mathematical Surveys, 2011, 66:1, 1–33

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