**Graduate Studies in Mathematics**

Volume: 177;
2016;
275 pp;
Hardcover

MSC: Primary 12; 30; 34;

Print ISBN: 978-1-4704-3095-5

Product Code: GSM/177

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-3593-6
Product Code: GSM/177.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Differential Galois Theory through Riemann-Hilbert Correspondence: An Elementary Introduction

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*Jacques Sauloy*

Differential Galois theory is an important,
fast developing area which appears more and more in graduate courses
since it mixes fundamental objects from many different areas of
mathematics in a stimulating context. For a long time, the dominant
approach, usually called Picard-Vessiot Theory, was purely
algebraic. This approach has been extensively developed and is well
covered in the literature. An alternative approach consists in tagging
algebraic objects with transcendental information which enriches the
understanding and brings not only new points of view but also new
solutions. It is very powerful and can be applied in situations where
the Picard-Vessiot approach is not easily extended. This book offers a
hands-on transcendental approach to differential Galois theory, based
on the Riemann-Hilbert correspondence. Along the way, it provides a
smooth, down-to-earth introduction to algebraic geometry, category
theory and tannakian duality.

Since the book studies only complex analytic linear differential
equations, the main prerequisites are complex function theory, linear
algebra, and an elementary knowledge of groups and of polynomials in
many variables. A large variety of examples, exercises, and
theoretical constructions, often via explicit computations, offers
first-year graduate students an accessible entry into this exciting
area.

#### Readership

Graduate students and researchers interested in differential Galois theory.

#### Reviews & Endorsements

Jacques Sauloy's book is an introduction to differential Galois theory, an important area of mathematics having different powerful applications (for example, to the classical problem of integrability of dynamical systems in mechanics and physics)...Sauloy offers an alternative approach to the subject which is based on the monodromy representation...Enriching the understanding of differential Galois theory, this point of view also brings new solutions, which makes the book especially valuable...There are a lot of nice exercises, both inside and at the end of each chapter.

-- Renat R. Gontsov, Mathematical Reviews

The book is an elementary introduction to the differential Galois theory and is intended for undergraduate students of mathematical departments. It is not overloaded with redundant definitions, constructs and results. Everything that is minimally necessary for understanding the whole presentation is given in full. The reader can find the rest [of the] details from a well-designed references system. And at the same time, the book contains quite a lot of carefully selected examples and exercises.

-- Mykola Grygorenko, Zentralblatt MATH

It's an excellent book about a beautiful and deep subject...There are loads of exercises, and I think the book is very well-paced, as well as very clearly written. It's a fabulous entry in the AMS GSM series.

-- Michael Berg, MAA Reviews

#### Table of Contents

# Table of Contents

## Differential Galois Theory through Riemann-Hilbert Correspondence: An Elementary Introduction

- Cover Cover11
- Title page iii4
- Contents vii8
- Foreword xiii14
- Preface xv16
- Introduction xvii18
- Part 1 . A Quick Introduction to Complex Analytic Functions 126
- Part 2 . Complex Linear Differential Equations and their Monodromy 5580
- Part 3 . The Riemann-Hilbert Correspondence 115140
- Chapter 9. Regular singular points and the local Riemann-Hilbert correspondence 117142
- 9.1. Introduction and motivation 118143
- 9.2. The condition of moderate growth in sectors 120145
- 9.3. Moderate growth condition for solutions of a system 123148
- 9.4. Resolution of systems of the first kind and monodromy of regular singular systems 124149
- 9.5. Moderate growth condition for solutions of an equation 128153
- 9.6. Resolution and monodromy of regular singular equations 132157
- Exercises 135160

- Chapter 10. Local Riemann-Hilbert correspondence as an equivalence of categories 137162
- Chapter 11. Hypergeometric series and equations 145170
- Chapter 12. The global Riemann-Hilbert correspondence 161186

- Part 4 . Differential Galois Theory 167192
- Chapter 13. Local differential Galois theory 169194
- Chapter 14. The local Schlesinger density theorem 181206
- 14.1. Calculation of the differential Galois group in the semi-simple case 182207
- 14.2. Calculation of the differential Galois group in the general case 186211
- 14.3. The density theorem of Schlesinger in the local setting 188213
- 14.4. Why is Schlesinger’s theorem called a “density theorem”? 191216
- Exercises 192217

- Chapter 15. The universal (fuchsian local) Galois group 193218
- Chapter 16. The universal group as proalgebraic hull of the fundamental group 201226
- Chapter 17. Beyond local fuchsian differential Galois theory 219244
- Appendix A. Another proof of the surjectivity of exp:𝑀𝑎𝑡_{𝑛}(\C)→𝐺𝐿_{𝑛}(\C) 229254
- Appendix B. Another construction of the logarithm of a matrix 233258
- Appendix C. Jordan decomposition in a linear algebraic group 237262
- Appendix D. Tannaka duality without schemes 243268
- Appendix E. Duality for diagonalizable algebraic groups 255280
- Appendix F. Revision problems 259284
- Bibliography 267292
- Index 271296

- Back Cover Back Cover1303