REVIEW OF THE BOOK “DIFFERENTIAL GALOISTHEORY THROUGH RIEMANN-HILBERTCORRESPONDENCE: AN ELEMENTARY APPROACH” BY
JACQUES SAULOYJULIEN ROQUESThe linear differential equations are at the crossroads between several
areas of mathematics. The book under review starts with an explorationof the analytic theory of (regular singular) linear differential equations andconcludes with an invitation to their differential Galois theory. Let me stressthat Sauloy’s intention is not to expose the differential Galois theory in its
full generality, for linear differential equations with coefficients in an arbitrary differential field K, but rather to focus on the case K = C(z), taking
advantage of the analytic tools at our disposal in this context, and making apparent some beautiful interplay between analysis and algebra. The
intention of Sauloy is clearly to explain the meaning of the adage “differential Galois theory is what the algebra can say about the dynamics” andto convince the reader that the modern differential Galois theory takes its
roots in the analytic theory of differential equations (and, more precisely, inRiemann’s monodromy representations). And it is a success.
I shall now give an idea of the content of this book.
Part I is an introduction to complex analysis, and a prelude to the analytic
study of linear differential equations undertaken in Parts II and III.
Parts II and III are mainly concerned with the analytic theory of theregular singular differential equations on P1 (C), say of the form
(1)

an (z)y (n) (z) + · · · + a1 (z)y 0 (z) + a0 (z)y(z) = 0,

with a0 (z), . . . , an (z) ∈ C(z), an (z) 6= 0, and more specifically with theirmonodromy representation. Before describing in more details what can befound in Parts II and III, I would like to say a few words about the concept ofmonodromy, which is the hero of the book under review. The staring pointis the so-called Cauchy theorem. Denoting by S the set of singularities inP1 (C) of (1) and by U its complement in P1 (C), Cauchy theorem reads: forany z0 ∈ U , the complex vector space Vz0 of germs of analytic functions atz0 solutions of (1) has dimension n. These solutions possess a marvelous
property : any f ∈ Vz0 can be continued analytically along any loop γ in U
based at z0 and the germ of analytic function f γ resulting from this process
is again a solution of (1), i.e., f γ ∈ Vz0 . In this way, we get a C-linear actionof the fundamental group π1 (U, z0 ) on Vz0 given, for all [γ] ∈ π1 (U, z0 ) andf ∈ Vz0 , by [γ]f := f γ . In other words, we get a finite dimensional linear
representationρ : π1 (U, z0 ) → GL(Vz0 ).
Date: April 19, 2018.
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JULIEN ROQUESThis is the monodromy representation attached to (1). The following twoquestions are natural:— is the sole knowledge of the monodromy representation sufficient toreconstruct the differential equation we started with ?
— is any finite dimensional linear representation of π1 (U, z0 ) the monodromy representation of some differential equation ?
In whole generality the answer to the first question is negative but becomes
positive if we restrict our attention to a special class of differential equations, namely to those having only regular singularities. The answer to thesecond question is positive, even if we restrict our attention to differential
equations having only regular singularities. These facts are usually formalized as a certain equivalence of categories, known as the Riemann-Hilbertcorrespondance. I am now in a position to describe the content of Parts
II and III. Part II starts with Cauchy theorem, continues with the proof of
the analytic continuation property of the solutions mentioned above, andintroduces the monodromy representation. Part III starts with the notionof regular singularity and with a detailed study of this type of singularity.
A local version of the Riemann-Hilbert correspondance is then stated andproved. The statement and a sketch of proof of the (global) Riemann-Hilbertcorrespondance (over P1 (C)) are then given. It is worth noticing that Part
III also contains the determination of the monodromy representation of theso-called hypergeometric differential equations (following Riemann’s originalmethod).
With Part IV, the reader enters in the word of differential Galois theory.
The first few pages of this chapter contain the definition of the local differential Galois groups and a proof of the fact that they are complex linear
algebraic groups. The transition from Parts II and III to Part IV is done in avery natural and smooth way via the so-called Schlesinger density theorem:
the monodromy is galoisian and gives rise to Zariski-dense subgroups of theabove mentionned local differential Galois groups in the regular singularcase. The rest of Part IV is essentially concerned with the regular singular
local universal Galois groups; roughly speaking, this universal Galois groupis the algebraic hull of the local topological fundamental group and it canbe used to formulate an “algebraic Riemann-Hilbert correspondance”.
The book ends with a selection of further developments and readings.
This is an accessible book, well-suited for students. It is mainly selfcontained. Some “advanced concepts” such as sheaves, categories, linear
algebraic groups, etc., are used. These concepts are introduced progressively all along the book, when needed, without any attempt of systematicpresentation or maximal generality. This is a nice and efficient choice. Several exercises are given along the book.
I warmly recommand this book for those looking for an accessible text
about the analytic theory of regular singular differential equations and theirdifferential Galois theory.

0.1. About the author. Julien Roques is maı̂tre de conférences at theInstitut Fourier of the Université Grenoble Alpes. He is notably working on

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algebra, number theory and functional equations, on their interactions andapplications.
Université Grenoble Alpes, Institut Fourier, CNRS UMR 5582, CS 40700,38058 Grenoble Cedex 09
E-mail address: Julien.Roques@univ-grenoble-alpes.fr