Divergent Series, Summability and Resurgence I
Monodromy and Resurgence

Authors: Claude Mitschi, David Sauzin



Features an elementary, self-contained introduction to analytic differential Galois theory and the Riemann-Hilbert problem

Provides a foundation that will allow the reader to independently explore and understand the specialized literature in the field

Focuses on 1-summability and resurgence, with a view to nonlinear problems

For the first time, resurgence theory is introduced with great pedagogical care and the main proofs are given in full detail

All notions are motivated and illustrated by numerous examples and applications

Providing an elementary introduction to analytic continuation and monodromy, the first part of this volume applies these notions to the local and global study of complex linear differential equations, their formal solutions at singular points, their monodromy and their differential Galois groups. The Riemann-Hilbert problem is discussed from Bolibrukh’s point of view.
The second part expounds 1-summability and Ecalle’s theory of resurgence under fairly general conditions. It contains numerous examples and presents an analysis of the singularities in the Borel plane via “alien calculus”, which provides a full description of the Stokes phenomenon for linear or non-linear differential or difference equations.

The first of a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists interested in geometric, algebraic or local analytic properties of dynamical systems. It includes useful exercises with solutions. The prerequisites are a working knowledge of elementary complex analysis and differential algebra.

Table of contents

Front Matter
Pages i-xxi

Monodromy in Linear Differential Equations
Front Matter
Pages 1-2
Analytic continuation and monodromy
Pages 3-23
Differential Galois Theory
Pages 25-73
Inverse Problems
Pages 75-86
The Riemann-Hilbert problem
Pages 87-119

Introduction to 1-Summability and Resurgence
Front Matter
Pages 121-122
Borel-Laplace Summation
Pages 123-171
Resurgent Functions and Alien Calculus
Pages 173-271
The Resurgent Viewpoint on Holomorphic Tangent-to-Identity Germs
Pages 273-293

Back Matter
Pages 295-298