Differential Galois Theory Through Riemann-hilbert Correspondence: An Elementary Introduction (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 177)

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.Part 1. A quick introduction to complex analytic functionsChapter 1. The complex exponential functionChapter 2. Power seriesChapter 3. Analytic functionsChapter 4. The complex logarithmChapter 5. From the local to the globalPart 2. Complex linear differential equations and their monodromyChapter 6. Two basic equations and their monodromyChapter 7. Linear complex analytic differential equationsChapter 8. A functorial point of view on analytic continuation: Local systemsPart 3. The Riemann-Hilbert correspondenceChapter 9. Regular singular points and the local Riemann-Hilbert correspondenceChapter 10. Local Riemann-Hilbert correspondence 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” z^{α} 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 n and systems of rank n 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 ρ_{A} of π1 16.2. Essential image of this functor 16.3. The structure of the semi-simple component of π1 16.4. Rational representations of π1 16.5. Galois correspondence and the proalgebraic hull of π1 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:Mat_{n}(\C)→GL_{n}(\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