Algebraic Theory 1 Picard Vessiot Theory 1.1 Differential Rings and Fidlds 1.2 Linear Differential Eqations 1.3 Picard-Vessiot Extensions 1.4 The Differential Galois Group 1.5 Liouvillian Extensions 2 Differential Operatiors and Differential Modules 2.1 The Ring＝k of Differential Operatiors 2.2 Constuctions with Differential Modules 2.3 Constuctions with Differential Operatiors 2.4 Differential Modules and Representations 3 Formal Local Theory 3.1 Formal Classification of Differential Equations 3.2 The Universal Picalr-Vessiot Ring of K 3.3 Newton Polygons 4 Algorithimc Considerations 4.1 Rational and Exponential Solutions 4.2 Factoring Linear Operatiors 4.3 Liouvillinan Solutions 4.4 Finnite Differential Galois Groups Analytic Theory 5 Monodromy,the Riemann-Hilbert Problem,and the Differential Galois Group 6 Differential Equations on the Complex Sphere and the Rimann-Hillbert Problem 7 Exact Asymptotics 8 Stokes Phenmenon and Differential Galois Groups 9 Stookes Matrices and Meromorphic Classification 10 Universal Picard-Vessiot Rings and Galois Groups 11 Inverse Problems 12 Modeli for Singular Differential Equations 13 Positive Characteristic Appendices A Algebraic Geometry B Tannakian Categories C Sheaves and Cohmology D Partial Differential Equations Bibliography List of Notiation Index
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