Differential Galois Theory Through Riemannhilbert Correspondence Jacques Sauloy by Jacques Sauloy 9781470430955, 1470430959 instant download after payment.

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.

Part 1. A quick introduction to complex analytic functions

Chapter 1. The complex exponential function

Chapter 2. Power series

Chapter 3. Analytic functions

Chapter 4. The complex logarithm

Chapter 5. From the local to the global

Part 2. Complex linear differential equations and their monodromy

Chapter 6. Two basic equations and their monodromy

Chapter 7. Linear complex analytic differential equations

Chapter 8. A functorial point of view on analytic continuation: Local systems

Part 3. The Riemann-Hilbert correspondence

Chapter 9. Regular singular points and the local Riemann-Hilbert correspondence

Chapter 10. Local Riemann-Hilbert correspondence