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 algorithm
