Preface Introduction Part 1． Algebraic Geometry Chapter 1．Affine and Projective Varieties 1．1．Affine varieties 1．2．Abstract affine varieties 1．3．Projective varieties Exercises Chapter 2．Algebraic Varieties 2．1．Prevarieties 2．2．Varieties Exercises Part 2． Algebraic Groups Chapter 3．Basic Notions 3．1．The notion of Mgebraic group 3．2．Connected algebraic groups 3．3．Subgroups and morphisms 3．4．Linearization of affine algebraic groups 3．5．Homogeneous spaces 3．6．Characters and semi-invariants 3．7．Quotients Exercises Chapter 4．Lie Algebras and Algebraic Groups 4．1．Lie algebras 4．2．The Lie algebra of a linear algebraic group 4．3．Decomposition of algebraic groups 4．4．Solvable algebraic groups 4．5．Correspondence between algebraic groups and Lie algebras 4．6．Subgroups of SL（2， C） Exercises Part 3． Differential Galois Theory Chapter 5．Picard-Vessiot Extensions 5．1．Derivations 5．2．Differential rings 5．3．Differential extensions 5．4．The ring of differential operators 5．5．Homogeneous linear differential equations 5．6．The Picard-Vessiot extension Exercises Chapter 6．The Galois Correspondence 6．1．Differential Galois group 6．2．The differential Galois group as a linear algebraic group 6．3．The fundamental theorem of differential Galois theory 6．4．Liouville extensions 6．5．Generalized Liouville extensions Exercises Chapter 7．Differential Equations over C（z） 7．1．Fuchsian differential equations 7．2．Monodromy group 7．3．Kovacic's algorithmExercises Chapter 8．Suggestions for Further Reading Bibliography Index
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