Differential Equations and Quantum Groups

This special volume is dedicated to the memory of Andrey A. Bolibrukh. It contains two expository articles devoted to some aspects of Bolibrukh's work, followed by ten refereed research articles.

Topics cover complex linear and nonlinear differential equations and quantum groups: monodromy, Fuchsian linear systems, Riemann-Hilbert problem, differential Galois theory, differential algebraic groups, multisummability, isomonodromy, Painlevé equations, Schlesinger equations, integrable systems, KZ equations, complex reflection groups, root systems.

Contents:

Yulij Ilyashenko:

Realization of irreducible monodromy by Fuchsian systems and
reduction to the Birkhoff standard form (by Andrey Bolibrukh);

Claude Sabbah:

The work of Andrey Bolibrukh on isomonodromic deformations;

Michèle Audin:

Two notions of integrability;

Werner Balser:

Formal power series solutions of the heat equation in one spatial variable;

Prakash Belkale, Evgeny Mukhin and Alexander Varchenko:

Multiplicity of critical points of master functions and Schubert calculus;

Philip Boalch:

Some explicit solutions to the Riemann-Hilbert problem;

Phyllis J. Cassidy and Michael F. Singer:

Galois theory of parameterized differential equations and linear differential algebraic groups;

Boris Dubrovin and Marta Mazzocco:

On the reductions and classical solutions of the Schlesinger equations;

Valentina A. Glubeva:

On the Riemann-Hilbert correspondence for generalized Knizhnik-Zamolodchikov equations for different root systems;

Vladimir Petrov Kostov:

Monodromy groups of regular systems on the Riemann sphere;

Vladimir P. Leksin:

Monodromy of Cherednik-Kohno-Veselov connections;

Hiroshi Umemura:

Invitation to Galois theory