Differential Galois theory through Riemann-Hilbert correspondence : an elementary introduction
著者
書誌事項
Differential Galois theory through Riemann-Hilbert correspondence : an elementary introduction
(Graduate studies in mathematics, v. 177)
American Mathematical Society, c2016
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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.
目次
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.
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