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

Cover
Title page
Contents
Foreword
Preface
Introduction
Index of notation
Part 1 . A Quick Introduction to Complex Analytic Functions
Chapter 1. The complex exponential function
1.1. The series
1.2. The function exp is \C-derivable
1.3. The exponential function as a covering map
1.4. The exponential of a matrix
1.5. Application to differential equations
Exercises
Chapter 2. Power series
2.1. Formal power series
2.2. Convergent power series
2.3. The ring of power series
2.4. \C-derivability of power series
2.5. Expansion of a power series at a point ≠0
2.6. Power series with values in a linear space
Exercises
Chapter 3. Analytic functions
3.1. Analytic and holomorphic functions
3.2. Singularities
3.3. Cauchy theory
3.4. Our first differential algebras
Exercises
Chapter 4. The complex logarithm
4.1. Can one invert the complex exponential function?
4.2. The complex logarithm via trigonometry
4.3. The complex logarithm as an analytic function
4.4. The logarithm of an invertible matrix
Exercises
Chapter 5. From the local to the global
5.1. Analytic continuation
5.2. Monodromy
5.3. A first look at differential equations with a singularity
Exercises
Part 2 . Complex Linear Differential Equations and their Monodromy
Chapter 6. Two basic equations and their monodromy
6.1. The “characters” ?^{?}
6.2. A new look at the complex logarithm
6.3. Back again to the first example
Exercises
Chapter 7. Linear complex analytic differential equations
7.1. The Riemann sphere
7.2. Equations of order ? and systems of rank ?
7.3. The existence theorem of Cauchy
7.4. The sheaf of solutions
7.5. The monodromy representation
7.6. Holomorphic and meromorphic equivalences of systems
Exercises
Chapter 8. A functorial point of view on analytic continuation: Local systems
8.1. The category of differential systems on Ω
8.2. The category \Ls of local systems on Ω
8.3. A functor from differential systems to local systems
8.4. From local systems to representations of the fundamental group
Exercises
Part 3 . The Riemann-Hilbert Correspondence
Chapter 9. Regular singular points and the local Riemann-Hilbert correspondence
9.1. Introduction and motivation
9.2. The condition of moderate growth in sectors
9.3. Moderate growth condition for solutions of a system
9.4. Resolution of systems of the first kind and monodromy of regular singular systems
9.5. Moderate growth condition for solutions of an equation
9.6. Resolution and monodromy of regular singular equations
Exercises
Chapter 10. Local Riemann-Hilbert correspondence as an equivalence of categories
10.1. The category of singular regular differential systems at 0
10.2. About equivalences and isomorphisms of categories
10.3. Equivalence with the category of representations of the local fundamental group
10.4. Matricial representation
Exercises
Chapter 11. Hypergeometric series and equations
11.1. Fuchsian equations and systems
11.2. The hypergeometric series
11.3. The hypergeometric equation
11.4. Global monodromy according to Riemann
11.5. Global monodromy using Barnes’ connection formulas
Exercises
Chapter 12. The global Riemann-Hilbert correspondence
12.1. The correspondence
12.2. The twenty-first problem of Hilbert
Exercises
Part 4 . Differential Galois Theory
Chapter 13. Local differential Galois theory
13.1. The differential algebra generated by the solutions
13.2. The differential Galois group
13.3. The Galois group as a linear algebraic group
Exercises
Chapter 14. The local Schlesinger density theorem
14.1. Calculation of the differential Galois group in the semi-simple case
14.2. Calculation of the differential Galois group in the general case
14.3. The density theorem of Schlesinger in the local setting
14.4. Why is Schlesinger’s theorem called a “density theorem”?
Exercises
Chapter 15. The universal (fuchsian local) Galois group
15.1. Some algebra, with replicas
15.2. Algebraic groups and replicas of matrices
15.3. The universal group
Exercises
Chapter 16. The universal group as proalgebraic hull of the fundamental group
16.1. Functoriality of the representation ?_{?} of ?₁
16.2. Essential image of this functor
16.3. The structure of the semi-simple component of ?₁
16.4. Rational representations of ?₁
16.5. Galois correspondence and the proalgebraic hull of ?₁
Exercises
Chapter 17. Beyond local fuchsian differential Galois theory
17.1. The global Schlesinger density theorem
17.2. Irregular equations and the Stokes phenomenon
17.3. The inverse problem in differential Galois theory
17.4. Galois theory of nonlinear differential equations
Appendix A. Another proof of the surjectivity of exp:???_{?}(\C)→??_{?}(\C)
Appendix B. Another construction of the logarithm of a matrix
Appendix C. Jordan decomposition in a linear algebraic group
C.1. Dunford-Jordan decomposition of matrices
C.2. Jordan decomposition in an algebraic group
Appendix D. Tannaka duality without schemes
D.1. One weak form of Tannaka duality
D.2. The strongest form of Tannaka duality
D.3. The proalgebraic hull of \Z
D.4. How to use tannakian duality in differential Galois theory
Appendix E. Duality for diagonalizable algebraic groups
E.1. Rational functions and characters
E.2. Diagonalizable groups and duality
Appendix F. Revision problems
F.1. 2012 exam (Wuhan)
F.2. 2013 exam (Toulouse)
F.3. Some more revision problems
Bibliography
Index
Back Cover