Differential Galois Theory Through Riemann-Hilbert Correspondence: An Elementary Introduction(Graduate Studies in Mathematics)

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.

About the Author :
Jacques Sauloy, Institut de Mathematiques de Toulouse, France.

Review :
Jacques Sauloy's book is an introduction to differential Galois theory, an important area of mathematics having different powerful applications (for example, to the classical problem of integrability of dynamical systems in mechanics and physics)...Sauloy offers an alternative approach to the subject which is based on the monodromy representation...Enriching the understanding of differential Galois theory, this point of view also brings new solutions, which makes the book especially valuable...There are a lot of nice exercises, both inside and at the end of each chapter." — Renat R. Gontsov, Mathematical Reviews "The book is an elementary introduction to the differential Galois theory and is intended for undergraduate students of mathematical departments. It is not overloaded with redundant definitions, constructs and results. Everything that is minimally necessary for understanding the whole presentation is given in full. The reader can find the rest [of the] details from a well-designed references system. And at the same time, the book contains quite a lot of carefully selected examples and exercises." — Mykola Grygorenko, Zentralblatt MATH "It's an excellent book about a beautiful and deep subject...There are loads of exercises, and I think the book is very well-paced, as well as very clearly written. It's a fabulous entry in the AMS GSM series." — Michael Berg, MAA Reviews