Galois Theory of Linear Differential Equations

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[?] 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, $$ A_{{1/k}}^0 $$.- 7.5 The Equation $$ (\delta - q)\hat{f} = g $$ 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 H1(S1STS).- 9.4 Explicit 1-cocycles for H1(S1, STS).- 9.4.1 One Level k.- 9.4.2 Two Levels k1 Meromorphic Classification.- 13 Positive Characteristic.- 13.1 Classification of Differential Modules.- 13.2 Algorithmic Aspects.- 13.2.1 The Equation b(p-1)+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 Sheaves and Cohomology.- C.l Sheaves: Definition and Examples.- C.1.1 Germs and Stalks.- C.1.2 Sheaves of Groups and Rings.- C. 1.3 From Presheaf to Sheaf.- C. 1.4 Moving Sheaves.- C.l.5 Complexes and Exact Sequences.- C.2 Cohomology of Sheaves.- C.2.1 The Idea and the Formahsm.- 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.- List of Notation.