Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used. This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

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From the reviews: "This is a great book, which will hopefully become a classic in the subject of differential Galois theory. [...] the specialist, as well as the novice, have long been missing an introductory book covering also specific and advanced research topics. This gap is filled by the volume under review, and more than satisfactorily." Mathematical Reviews

Inhalt

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.