Description
Book SynopsisOffers 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.
Trade ReviewJacques 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 ReviewsTable 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.
