Author | Put, Marius van der. author |
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Title | Galois Theory of Linear Differential Equations [electronic resource] / by Marius van der Put, Michael F. Singer |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2003 |
Connect to | http://dx.doi.org/10.1007/978-3-642-55750-7 |
Descript | XVII, 438 p. online resource |
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