van der Put / Singer Galois Theory of Linear Differential Equations

Erscheinungsjahr 2012
ISBN: 978-3-642-55750-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 328, 438 Seiten, eBook

Reihe: Grundlehren der mathematischen Wissenschaften

Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used. This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

Zielgruppe

Research

Autoren/Hrsg.

van der Put, Marius

Singer, Michael F.

Inhaltsverzeichnis

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.