Galois Theory of Linear Differential Equations

Algebraic Theory.- 1 Picard-Vessiot Theory.- 2 Differential Operators and Differential Modules.- 3 Formal Local Theory.- 4 Algorithmic Considerations.- Analytic Theory.- 5 Monodromy, the Riemann-Hilbert Problem, and the Differential Galois Group.- 6 Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem.- 7 Exact Asymptotics.- 8 Stokes Phenomenon and Differential Galois Groups.- 9 Stokes Matrices and Meromorphic Classification.- 10 Universal Picard-Vessiot Rings and Galois Groups.- 11 Inverse Problems.- 12 Moduli for Singular Differential Equations.- 13 Positive Characteristic.- 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.