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

### About this Title

**Jacques Sauloy**, *Institut de Mathématiques de Toulouse, Toulouse, France*

Publication: Graduate Studies in Mathematics

Publication Year:
2016; Volume 177

ISBNs: 978-1-4704-3095-5 (print); 978-1-4704-3593-6 (online)

DOI: https://doi.org/10.1090/gsm/177

MathSciNet review: MR3585800

MSC: Primary 34Mxx; Secondary 12H05, 30B10, 30Fxx

### Table of Contents

**Front/Back Matter**

**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 Riemann-Hilbert correspondence **

- Chapter 9. Regular singular points and the local Riemann-Hilbert correspondence
- Chapter 10. Local Riemann-Hilbert correspondence as an equivalence of categories
- Chapter 11. Hypergeometric series and equations
- Chapter 12. The global Riemann-Hilbert 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