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E-raamat: Differential Galois Theory through Riemann-Hilbert Correspondence

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Differential Galois Theory through Riemann-Hilbert CorrespondenceSuurem pilt
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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.

Arvustused

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

Foreword xiii
Preface xv
Introduction xvii
Index of notation xix
Part 1: A Quick Introduction to Complex Analytic Functions
Chapter 1 The complex exponential function
3(12)
1.1 The series
3(1)
1.2 The function exp is C-derivable
4(3)
1.3 The exponential function as a covering map
7(1)
1.4 The exponential of a matrix
8(2)
1.5 Application to differential equations
10(2)
Exercises
12(3)
Chapter 2 Power series
15(14)
2.1 Formal power series
15(5)
2.2 Convergent power series
20(2)
2.3 The ring of power series
22(1)
2.4 C-derivability of power series
23(2)
2.5 Expansion of a power series at a point not equal to 0
25(1)
2.6 Power series with values in a linear space
26(1)
Exercises
27(2)
Chapter 3 Analytic functions
29(10)
3.1 Analytic and holomorphic functions
29(3)
3.2 Singularities
32(1)
3.3 Cauchy theory
33(3)
3.4 Our first differential algebras
36(1)
Exercises
37(2)
Chapter 4 The complex logarithm
39(6)
4.1 Can one invert the complex exponential function?
39(1)
4.2 The complex logarithm via trigonometry
40(1)
4.3 The complex logarithm as an analytic function
41(1)
4.4 The logarithm of an invertible matrix
42(2)
Exercises
44(1)
Chapter 5 From the local to the global
45(12)
5.1 Analytic continuation
45(2)
5.2 Monodromy
47(3)
5.3 A first look at differential equations with a singularity
50(2)
Exercises
52(5)
Part 2: Complex Linear Differential Equations and their Monodromy
Chapter 6 Two basic equations and their monodromy
57(20)
6.1 The "characters" zalpha
57(13)
6.2 A new look at the complex logarithm
70(4)
6.3 Back again to the first example
74(1)
Exercises
75(2)
Chapter 7 Linear complex analytic differential equations
77(26)
7.1 The Riemann sphere
77(4)
7.2 Equations of order n and systems of rank n
81(6)
7.3 The existence theorem of Cauchy
87(2)
7.4 The sheaf of solutions
89(2)
7.5 The monodromy representation
91(4)
7.6 Holomorphic and meromorphic equivalences of systems
95(6)
Exercises
101(2)
Chapter 8 A functorial point of view on analytic continuation: Local systems
103(14)
8.1 The category of differential systems on Omega
103(2)
8.2 The category Ls of local systems on Omega
105(2)
8.3 A functor from differential systems to local systems
107(2)
8.4 From local systems to representations of the fundamental group
109(4)
Exercises
113(4)
Part 3: The Riemann-Hilbert Correspondence
Chapter 9 Regular singular points and the local Riemann-Hilbert correspondence
117(20)
9.1 Introduction and motivation
118(2)
9.2 The condition of moderate growth in sectors
120(3)
9.3 Moderate growth condition for solutions of a system
123(1)
9.4 Resolution of systems of the first kind and monodromy of regular singular systems
124(4)
9.5 Moderate growth condition for solutions of an equation
128(4)
9.6 Resolution and monodromy of regular singular equations
132(3)
Exercises
135(2)
Chapter 10 Local Riemann-Hilbert correspondence as an equivalence of categories
137(8)
10.1 The category of singular regular differential systems at 0
138(1)
10.2 About equivalences and isomorphisms of categories
139(2)
10.3 Equivalence with the category of representations of the local fundamental group
141(1)
10.4 Matricial representation
142(2)
Exercises
144(1)
Chapter 11 Hypergeometric series and equations
145(16)
11.1 Fuchsian equations and systems
145(4)
11.2 The hypergeometric series
149(1)
11.3 The hypergeometric equation
150(3)
11.4 Global monodromy according to Riemann
153(4)
11.5 Global monodromy using Barnes' connection formulas
157(2)
Exercises
159(2)
Chapter 12 The global Riemann-Hilbert correspondence
161(8)
12.1 The correspondence
161(1)
12.2 The twenty-first problem of Hilbert
162(4)
Exercises
166(3)
Part 4: Differential Galois Theory
Chapter 13 Local differential Galois theory
169(12)
13.1 The differential algebra generated by the solutions
170(2)
13.2 The differential Galois group
172(3)
13.3 The Galois group as a linear algebraic group
175(4)
Exercises
179(2)
Chapter 14 The local Schlesinger density theorem
181(12)
14.1 Calculation of the differential Galois group in the semi-simple case
182(4)
14.2 Calculation of the differential Galois group in the general case
186(2)
14.3 The density theorem of Schlesinger in the local setting
188(3)
14.4 Why is Schlesinger's theorem called a "density theorem"?
191(1)
Exercises
192(1)
Chapter 15 The universal (fuchsian local) Galois group
193(8)
15.1 Some algebra, with replicas
194(2)
15.2 Algebraic groups and replicas of matrices
196(3)
15.3 The universal group
199(1)
Exercises
200(1)
Chapter 16 The universal group as proalgebraic hull of the fundamental group
201(18)
16.1 Functoriality of the representation rhoA of pi1
201(2)
16.2 Essential image of this functor
203(4)
16.3 The structure of the semi-simple component of pi1
207(6)
16.4 Rational representations of pi1
213(1)
16.5 Galois correspondence and the proalgebraic hull of pi1
214(2)
Exercises
216(3)
Chapter 17 Beyond local fuchsian differential Galois theory
219(18)
17.1 The global Schlesinger density theorem
220(1)
17.2 Irregular equations and the Stokes phenomenon
221(5)
17.3 The inverse problem in differential Galois theory
226(1)
17.4 Galois theory of nonlinear differential equations
227(2)
Appendix A. Another proof of the surjectivity of exp : Matn(C)—>GLn(C) 229(4)
Appendix B. Another construction of the logarithm of a matrix 233(4)
Appendix C. Jordan decomposition in a linear algebraic group 237(6)
C.1 Dunford-Jordan decomposition of matrices
237(4)
C.2 Jordan decomposition in an algebraic group
241(2)
Appendix D. Tannaka duality without schemes 243(12)
D.1 One weak form of Tannaka duality
245(1)
D.2 The strongest form of Tannaka duality
246(2)
D.3 The proalgebraic hull of Z
248(3)
D.4 How to use tannakian duality in differential Galois theory
251(4)
Appendix E. Duality for diagonalizable algebraic groups 255(4)
E.1 Rational functions and characters
255(2)
E.2 Diagonalizable groups and duality
257(2)
Appendix F. Revision problems 259(8)
F.1 2012 exam (Wuhan)
259(1)
F.2 2013 exam (Toulouse)
260(3)
F.3 Some more revision problems
263(4)
Bibliography 267(4)
Index 271
Jacques Sauloy, Institut de Mathematiques de Toulouse, France.
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