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E-grāmata: Riemann Surfaces [Oxford Scholarship Online E-books]

(Royal Society Research Professor, Imperial College, London)
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Riemann SurfacesPalielināt
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The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics.

This graduate text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind.

Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved. Following this section, the remainder of the text illustrates various facets of the more advanced theory.
PART I PRELIMINARIES
1 Holomorphic functions
3(7)
1.1 Simple examples: algebraic functions
3(2)
1.2 Analytic continuation: differential equations
5(5)
Exercises
8(2)
2 Surface topology
10(19)
2.1 Classification of surfaces
10(11)
2.2 Discussion: the mapping class group
21(8)
Exercises
24(5)
PART II BASIC THEORY
3 Basic definitions
29(13)
3.1 Riemann surfaces and holomorphic maps
29(2)
3.2 Examples
31(11)
3.2.1 First examples
31(1)
3.2.2 Algebraic curves
32(4)
3.2.3 Quotients
36(4)
Exercises
40(2)
4 Maps between Riemann surfaces
42(15)
4.1 General properties
42(3)
4.2 Monodromy and the Riemann Existence Theorem
45(12)
4.2.1 Digression into algebraic topology
45(3)
4.2.2 Monodromy of covering maps
48(2)
4.2.3 Compactifying algebraic curves
50(4)
4.2.4 The Riemann surface of a holomorphic function
54(1)
4.2.5 Quotients
55(1)
Exercises
56(1)
5 Calculus on surfaces
57(25)
5.1 Smooth surfaces
57(10)
5.1.1 Cotangent spaces and 1-forms
57(4)
5.1.2 2-forms and integration
61(6)
5.2 de Rham cohomology
67(6)
5.2.1 Definition and examples
67(3)
5.2.2 Cohomology with compact support, and Poincare duality
70(3)
5.3 Calculus on Riemann surfaces
73(9)
5.3.1 Decomposition of the 1-forms
73(4)
5.3.2 The Laplace operator and harmonic functions
77(1)
5.3.3 The Dirichlet norm
78(3)
Exercises
81(1)
6 Elliptic functions and integrals
82(15)
6.1 Elliptic integrals
82(4)
6.2 The Weierstrass ℘ function
86(2)
6.3 Further topics
88(9)
6.3.1 Theta functions
88(3)
6.3.2 Classification
91(5)
Exercises
96(1)
7 Applications of the Euler characteristic
97(14)
7.1 The Euler characteristic and meromorphic forms
97(3)
7.1.1 Topology
97(2)
7.1.2 Meromorphic forms
99(1)
7.2 Applications
100(11)
7.2.1 The Riemann-Hurwitz formula
100(2)
7.2.2 The degree-genus formula
102(1)
7.2.3 Real structures and Harnack's bound
103(2)
7.2.4 Modular curves
105(2)
Exercises
107(4)
PART III DEEPER THEORY
8 Meromorphic functions and the Main Theorem for compact Riemann surfaces
111(7)
8.1 Consequences of the Main Theorem
113(2)
8.2 The Riemann-Roch formula
115(3)
Exercises
117(1)
9 Proof of the Main Theorem
118(13)
9.1 Discussion and motivation
118(2)
9.2 The Riesz Representation Theorem
120(2)
9.3 The heart of the proof
122(4)
9.4 Weyl's Lemma
126(5)
Exercises
129(2)
10 The Uniformisation Theorem
131(16)
10.1 Statement
131(2)
10.2 Proof of the analogue of the Main Theorem
133(14)
10.2.1 Set-up
133(2)
10.2.2 Classification of behaviour at infinity
135(3)
10.2.3 The main argument
138(3)
10.2.4 Proof of Proposition 30
141(1)
Exercises
142(5)
PART IV FURTHER DEVELOPMENTS
11 Contrasts in Riemann surface theory
147(31)
11.1 Algebraic aspects
147(12)
11.1.1 Fields of meromorphic functions
147(4)
11.1.2 Valuations
151(5)
11.1.3 Connections with algebraic number theory
156(3)
11.2 Hyperbolic surfaces
159(19)
11.2.1 Definitions
159(2)
11.2.2 Models of the hyperbolic plane
161(2)
11.2.3 Self-isometries
163(1)
11.2.4 Hyperbolic surfaces
164(1)
11.2.5 Geodesics
164(3)
11.2.6 Discussion
167(2)
11.2.7 The Gauss-Bonnet Theorem
169(2)
11.2.8 Right-angled hexagons
171(2)
11.2.9 Closed geodesics
173(4)
Exercises
177(1)
12 Divisors, line bundles and Jacobians
178(32)
12.1 Cohomology and line bundles
178(18)
12.1.1 Sheaves and cohomology
178(4)
12.1.2 Line bundles
182(6)
12.1.3 Line bundles and projective embeddings
188(6)
12.1.4 Divisors and unique factorisation
194(2)
12.2 Jacobians of Riemann surfaces
196(14)
12.2.1 The Abel-Jacobi Theorem
196(2)
12.2.2 Abstract theory
198(1)
12.2.3 Geometry of symmetric products
199(2)
12.2.4 Remarks in the direction of algebraic topology
201(2)
12.2.5 Digression into projective geometry
203(5)
Exercises
208(2)
13 Moduli and deformations
210(17)
13.1 Almost-complex structures, Beltrami differentials and the integrability theorem
210(4)
13.2 Deformations and cohomology
214(6)
13.3 Appendix
220(7)
Exercises
226(1)
14 Mappings and moduli
227(33)
14.1 Diffeomorphisms of the plane
227(2)
14.2 Braids, Dehn twists and quadratic singularities
229(14)
14.2.1 Classification of branched covers
229(7)
14.2.2 Monodromy and Dehn twists
236(4)
14.2.3 Plane curves
240(3)
14.3 Hyperbolic geometry
243(6)
14.4 Compactification of the moduli space
249(11)
14.4.1 Collars and cusps
252(7)
Exercises
259(1)
15 Ordinary differential equations
260(22)
15.1 Conformal mapping
260(7)
15.2 Periods of holomorphic forms and ordinary differential equations
267(15)
15.2.1 The hypergeometric equation
267(2)
15.2.2 The Gauss-Manin connection
269(3)
15.2.3 Singular points
272(9)
Exercises
281(1)
References 282(3)
Index 285
Simon Donaldson gained a BA from Cambridge in 1979. In 1980 he began graduate work in Oxford, supervised by Nigel Hitchin and Sir Michael Atiyah. His PhD thesis studied mathematical aspects of Yang-Mills theory. In 1986, aged 29, he was awarded a Fields Medal and was elected to the Royal Society. He was Wallis Professor of Mathematics in Oxford between 1985 and 1998 when he moved to Imperial College London. Most of his work since has been on the interface between differential geometry and complex algebraic geometry. The recipient of numerous awards, including the Shaw Prize in 2009 with Clifford Taubes, he is also a Foreign Member of the US, French & Swedish academies. Donaldson has supervised more than 40 doctoral students, many of whom have gone on to become leading figures in research.
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