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# Differential Galois Theory through Riemann-Hilbert Correspondence

### An Elementary Introduction

ISBN: 9781470430955

Publication Date: Mar 2017

Format: Hardback

Offers a hands-on transcendental approach to differential Galois theory, based on the Riemann-Hilbert correspondence. Along the way, it provides a smooth, down-to-earth introduction to algebraic geometry, category theory and tannakian duality. A large variety of examples, exercises, and theoretical constructions offers an accessible entry into this exciting area.
£88.95

Temporarily out of stock: usually despatched in 10-14 days

• Full Description
• Author Biography
• Customer Reviews
Differential Galois theory is an important, fast developing area which appears more and more in graduate courses since it mixes fundamental objects from many different areas of mathematics in a stimulating context. For a long time, the dominant approach, usually called Picard-Vessiot Theory, was purely algebraic. This approach has been extensively developed and is well covered in the literature. An alternative approach consists in tagging algebraic objects with transcendental information which enriches the understanding and brings not only new points of view but also new solutions. It is very powerful and can be applied in situations where the Picard-Vessiot approach is not easily extended. This book offers a hands-on transcendental approach to differential Galois theory, based on the Riemann-Hilbert correspondence. Along the way, it provides a smooth, down-to-earth introduction to algebraic geometry, category theory and tannakian duality.

Since the book studies only complex analytic linear differential equations, the main prerequisites are complex function theory, linear algebra, and an elementary knowledge of groups and of polynomials in many variables. A large variety of examples, exercises, and theoretical constructions, often via explicit computations, offers first-year graduate students an accessible entry into this exciting area.
Pages 279 254 x 178 30 Mar 2017 American Mathematical Society Graduate Studies in Mathematics 177 Number theory   Differential calculus & equations
• Part 1. A quick introduction to complex analytic functions: The complex exponential function
• Power series
• Analytic functions
• The complex logarithm
• From the local to the global
• Part 2. Complex linear differential equations and their monodromy: Two basic equations and their monodromy
• Linear complex analytic differential equations
• A functorial point of view on analytic continuation: Local systems
• Part 3. The Riemann-Hilbert correspondence: Regular singular points and the local Riemann-Hilbert correspondence
• Local Riemann-Hilbert correspondence as an equivalence of categories
• Hypergeometric series and equations
• The global Riemann-Hilbert correspondence
• Part 4. Differential Galois theory: Local differential Galois theory
• The local Schlesinger density theorem
• The universal (Fuchsian local) Galois group
• The universal group as proalgebraic hull of the fundamental group
• Beyond local Fuchsian differential Galois theory
• Appendix A. Another proof of the surjectivity of $\mathrm{exp}:\mathrm{Mat}_n(\mathbf{C})\rightarrow \mathrm{GL}_n(\mathbf{C})$
• Appendix B. Another construction of the logarithm of a matrix
• Appendix C. Jordan decomposition in a linear algebraic group
• Appendix D. Tannaka duality without schemes
• Appendix E. Duality for diagonalizable algebraic groups
• Appendix F. Revision problems
• Bibliography
• Index.
Jacques Sauloy, Institut de Mathematiques de Toulouse, France.