Hardcover ISBN:  9781470430955 
Product Code:  GSM/177 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470435936 
Product Code:  GSM/177.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470430955 
eBook: ISBN:  9781470435936 
Product Code:  GSM/177.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 
Hardcover ISBN:  9781470430955 
Product Code:  GSM/177 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470435936 
Product Code:  GSM/177.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470430955 
eBook ISBN:  9781470435936 
Product Code:  GSM/177.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 

Book DetailsGraduate Studies in MathematicsVolume: 177; 2016; 275 ppMSC: Primary 12; 30; 34
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 PicardVessiot 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 PicardVessiot approach is not easily extended. This book offers a handson transcendental approach to differential Galois theory, based on the RiemannHilbert correspondence. Along the way, it provides a smooth, downtoearth 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 firstyear graduate students an accessible entry into this exciting area.
ReadershipGraduate students and researchers interested in differential Galois theory.

Table of Contents

Part 1. A quick introduction to complex analytic functions

Chapter 1. The complex exponential function

Chapter 2. Power series

Chapter 3. Analytic functions

Chapter 4. The complex logarithm

Chapter 5. From the local to the global

Part 2. Complex linear differential equations and their monodromy

Chapter 6. Two basic equations and their monodromy

Chapter 7. Linear complex analytic differential equations

Chapter 8. A functorial point of view on analytic continuation: Local systems

Part 3. The RiemannHilbert correspondence

Chapter 9. Regular singular points and the local RiemannHilbert correspondence

Chapter 10. Local RiemannHilbert correspondence as an equivalence of categories

Chapter 11. Hypergeometric series and equations

Chapter 12. The global RiemannHilbert correspondence

Part 4. Differential Galois theory

Chapter 13. Local differential Galois theory

Chapter 14. The local Schlesinger density theorem

Chapter 15. The universal (Fuchsian local) Galois group

Chapter 16. The universal group as proalgebraic hull of the fundamental group

Chapter 17. Beyond local Fuchsian differential Galois theory

Appendix A. Another proof of the surjectivity of $\mathrm {exp}:\mathrm {Mat}_n(\mathbf {C})\rightarrow \mathrm {GL}_n(\mathbf {C})$

Appendix B. Another construction of the logarithm of a matrix

Appendix C. Jordan decomposition in a linear algebraic group

Appendix D. Tannaka duality without schemes

Appendix E. Duality for diagonalizable algebraic groups

Appendix F. Revision problems


Additional Material

Reviews

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 welldesigned 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 wellpaced, as well as very clearly written. It's a fabulous entry in the AMS GSM series.
Michael Berg, MAA Reviews


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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 PicardVessiot 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 PicardVessiot approach is not easily extended. This book offers a handson transcendental approach to differential Galois theory, based on the RiemannHilbert correspondence. Along the way, it provides a smooth, downtoearth 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 firstyear graduate students an accessible entry into this exciting area.
Graduate students and researchers interested in differential Galois theory.

Part 1. A quick introduction to complex analytic functions

Chapter 1. The complex exponential function

Chapter 2. Power series

Chapter 3. Analytic functions

Chapter 4. The complex logarithm

Chapter 5. From the local to the global

Part 2. Complex linear differential equations and their monodromy

Chapter 6. Two basic equations and their monodromy

Chapter 7. Linear complex analytic differential equations

Chapter 8. A functorial point of view on analytic continuation: Local systems

Part 3. The RiemannHilbert correspondence

Chapter 9. Regular singular points and the local RiemannHilbert correspondence

Chapter 10. Local RiemannHilbert correspondence as an equivalence of categories

Chapter 11. Hypergeometric series and equations

Chapter 12. The global RiemannHilbert correspondence

Part 4. Differential Galois theory

Chapter 13. Local differential Galois theory

Chapter 14. The local Schlesinger density theorem

Chapter 15. The universal (Fuchsian local) Galois group

Chapter 16. The universal group as proalgebraic hull of the fundamental group

Chapter 17. Beyond local Fuchsian differential Galois theory

Appendix A. Another proof of the surjectivity of $\mathrm {exp}:\mathrm {Mat}_n(\mathbf {C})\rightarrow \mathrm {GL}_n(\mathbf {C})$

Appendix B. Another construction of the logarithm of a matrix

Appendix C. Jordan decomposition in a linear algebraic group

Appendix D. Tannaka duality without schemes

Appendix E. Duality for diagonalizable algebraic groups

Appendix F. Revision problems

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 welldesigned 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 wellpaced, as well as very clearly written. It's a fabulous entry in the AMS GSM series.
Michael Berg, MAA Reviews