Algebraic Theory 1 Picard-Vessiot Theory 1.1 Differential Rings and Fields 1.2 Linear Differential Equations 1.3 Picard-Vessiot Extensions 1.4 The Differential Galois Group 1.5 Liouvillian Extensions 2 Differential Operators and Differential Modules 2.1 The Ring D=k[e] of Differential Operators 2.2 Constructions with Differential Modules 2.3 Constructions with Differential Operators 2.4 Differential Modules and Representations 3 Formal Local Theory 3.1 Formal Classification of Differential Equations 3.1.1 Regular Singular Equations 3.1.2 Irregular Singular Equations 3.2 The Universal Picard-Vessiot Ring of K 3.3 Newton Polygons 4 Algorithmic Considerations 4.1 Rational and Exponential Solutions 4.2 Factoring Linear Operators 4.2.1 Beke's Algorithm 4.2.2 Eigenring and Factorizations 4.3 Liouvillian Solutions 4.3.1 Group Theory 4.3.2 Liouvillian Solutions for a Differential Module 4.3.3 Liouvillian Solutions for a Differential Operator 4.3.4 Second Order Equations 4.3.5 Third Order Equations 4.4 Finite Differential Galois groups 4.4.1 Generalities on Scalar Fuchsian Equations 4.4.2 Restrictions on the Exponents 4.4.3 Representations of Finite Groups 4.4.4 A Calculation of the Accessory Parameter 4.4.5 Examples Analytic Theory 5 Monodromy, the Riemann-Hilbert Problem,and the Differential Galois Group 5.1 Monodromy of a Differential Equation 5.1.1 Local Theory of Regular Singular Equations 5.1.2 Regular Singular Equations on p1 5.2 A Solution of the Inverse Problem 5.3 The Riemann-Hilbert Problem 6 Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem 6.1 Differentials and Connections 6.2 Vector Bundles and Connections 6.3 Fuchsian Equations 6.3.1 From Scalar Fuchsian to Matrix Fuchsian 6.3.2 A Criterion for a Scalar Fuchsian Equation 6.4 The Riemann-Hilbert Problem, Weak Form 6.5 Irreducible Connections 6.6 Counting Fuchsian Equations 7 Exact Asymptotics 7.1 Introduction and Notation 7.2 The Main Asymptotic Existence Theorem 7.3 The Inhomogeneous Equation of Order One 7.4 The Sheaves A,A0,A1/k, A01/k 7.5 The Equation (δ - q)f = g Revisited 7.6 The Laplace and Borel Transforms 7.7 The k-Summation Theorem 7.8 The Multisummation Theorem 8 Stokes Phenomenon and Differential Galois Groups 8.1 Introduction 8.2 The Additive Stokes Phenomenon 8.3 Construction of the Stokes Matrices 9 Stokes Matrices and Meromorphic Classification 9.1 Introduction 9.2 The Category Gr2 9.3 The Cohomology Set HI(s1, STS) 9.4 Explicit 1-cocycles for HI(s1, STS) 9.4.1 One Level k 9.4.2 Two Levels k19.4.3 The General Case 9.5 H1 (S1, STS) as an Algebraic Variety 10 Universal Picard-Vessiot Rings and Galois Groups 10.1 Introduction 10.2 Regular Singular Differential Equations 10.3 Formal Differential Equations 10.4 Meromorphic Differential Equations 11 Inverse Problems 11.1 Introduction 11.2 The Inverse Problem for C((z)) 11.3 Some Topics on Linear Algebraic Groups 11.4 The Local Theorem 11.5 The Global Theorem 11.6 More on Abhyankar's Conjecture 11.7 The Constructive Inverse Problem 12 Moduli for Singular Differential Equations 12.1 Introduction 12.2 The Moduli Functor 12.3 An Example 12.3.1 Construction of the Moduli Space 12.3.2 Comparison with the Meromorphic Classification 12.3.3 Invariant Line Bundles 12.3.4 The Differential Galois Group 12.4 Unramified Irregular Singularities 12.5 The Ramified Case 12.6 The Meromorphic Classification 13 Positive Characteristic 13.1 Classification of Differential Modules 13.2 Algorithmic Aspects 13.2.1 The Equation b(p-1) q- bp = a 13.2.2 The p-Curvature and Its Minimal Polynomial 13.2.3 Example: Operators of Order Two 13.3 Iterative Differential Modules 13.3.1 Picard-Vessiot Theory and Some Examples 13.3.2 Global Iterative Differential Equations 13.3.3 p-Adic Differential Equations Appendices A Algebraic Geometry A.1 Affine Varieties A.1.1 Basic Definitions and Results A. 1.2 Products of Affine Varieties over k A. 1.3 Dimension of an Affine Variety A.1.4 Tangent Spaces, Smooth Points, and Singular Points A.2 Linear Algebraic Groups A.2.1 Basic Definitions and Results A.2.2 The Lie Algebra of a Linear Algebraic Group A.2.3 Torsors B Tannakian Categories B. 1 Galois Categories B.2 Affine Group Schemes B.3 Tannakian Categories C Sheavesand Cohomology C.1 Sheaves: Definition and Examples C.I.1 Germs and Stalks C. 1.2 Sheaves of Groups and Rings C.1.3 From Presheaf to Sheaf C.1.4 Moving Sheaves C.1.5 Complexes and Exact Sequences C.2 Cohomology of Sheaves C.2.1 The Idea and the Formalism C.2.2 Construction of the Cohomology Groups C.2.3 More Results and Examples D Partial Differential Equations D. 1 The Ring of Partial Differential Operators D.2 Picard-Vessiot Theory and Some Remarks Bibliography List of Notation Index
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