Del i serien Graduate Studies in Mathematics

Differential Galois Theory Through Riemann-Hilbert Correspondence

An Elementary Introduction

Inbunden, Engelska, 2017

1 999 kr

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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.

Produktinformation

Utgivningsdatum2017-03-30
Mått178 x 254 x undefined mm
Vikt688 g
FormatInbunden
SpråkEngelska
SerieGraduate Studies in Mathematics
Antal sidor275
FörlagAmerican Mathematical Society
ISBN9781470430955

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Part 1. A quick introduction to complex analytic functions: The complex exponential functionPower seriesAnalytic functionsThe complex logarithmFrom the local to the globalPart 2. Complex linear differential equations and their monodromy: Two basic equations and their monodromyLinear complex analytic differential equationsA functorial point of view on analytic continuation: Local systemsPart 3. The Riemann-Hilbert correspondence: Regular singular points and the local Riemann-Hilbert correspondenceLocal Riemann-Hilbert correspondence as an equivalence of categoriesHypergeometric series and equationsThe global Riemann-Hilbert correspondencePart 4. Differential Galois theory: Local differential Galois theoryThe local Schlesinger density theoremThe universal (Fuchsian local) Galois groupThe universal group as proalgebraic hull of the fundamental groupBeyond local Fuchsian differential Galois theoryAppendix 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 matrixAppendix C. Jordan decomposition in a linear algebraic groupAppendix D. Tannaka duality without schemesAppendix E. Duality for diagonalizable algebraic groupsAppendix F. Revision problemsBibliographyIndex.

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