Differential Galois theory through Riemann-Hilbert correspondence : an elementary introduction

Author(s)

• Sauloy, Jacques

Bibliographic Information

Differential Galois theory through Riemann-Hilbert correspondence : an elementary introduction

Jacques Sauloy

（Graduate studies in mathematics, v. 177）

American Mathematical Society, c2016

Note

Includes bibliographical references and index

Description and Table of Contents

Description

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.

Table of Contents

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.

by "Nielsen BookData"

Details

• NCID: BB22707083
• ISBN:
  • 9781470430955
• LCCN: 2016033241
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Providence, R.I.
• Pages/Volumes: xxiii, 275 p.
• Size: 27 cm
• Classification:
  • LCC : QA214
  • DC23 : 512/.32
• Subject Headings:
  • LCSH : Galois theory
  • LCSH : Riemann-Hilbert problems
• Parent Bibliography ID:
  • BA20542214