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Mostly Surfaces [Minkštas viršelis]

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  • Formatas: Paperback / softback, 312 pages, weight: 396 g
  • Serija: Student Mathematical Library
  • Išleidimo metai: 30-Jul-2011
  • Leidėjas: American Mathematical Society
  • ISBN-10: 0821853686
  • ISBN-13: 9780821853689
Kitos knygos pagal šią temą:
Mostly SurfacesDidesnis paveikslėlis
  • Formatas: Paperback / softback, 312 pages, weight: 396 g
  • Serija: Student Mathematical Library
  • Išleidimo metai: 30-Jul-2011
  • Leidėjas: American Mathematical Society
  • ISBN-10: 0821853686
  • ISBN-13: 9780821853689
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780821853689.html
Raktažodžiai:
This book presents a number of topics related to surfaces, such as Euclidean, spherical and hyperbolic geometry, the fundamental group, universal covering surfaces, Riemannian manifolds, the Gauss-Bonnet Theorem, and the Riemann mapping theorem. The main idea is to get to some interesting mathematics without too much formality. The book also includes some material only tangentially related to surfaces, such as the Cauchy Rigidity Theorem, the Dehn Dissection Theorem, and the Banach-Tarski Theorem.

The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigourous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis.
Preface xi
Chapter 1 Book Overview
1(20)
§1.1 Behold, the Torus!
1(2)
§1.2 Gluing Polygons
3(2)
§1.3 Drawing on a Surface
5(3)
§1.4 Covering Spaces
8(1)
§1.5 Hyperbolic Geometry and the Octagon
9(2)
§1.6 Complex Analysis and Riemann Surfaces
11(2)
§1.7 Cone Surfaces and Translation Surfaces
13(1)
§1.8 The Modular Group and the Veech Group
14(2)
§1.9 Moduli Space
16(1)
§1.10 Dessert
17(4)
Part 1 Surfaces and Topology
Chapter 2 Definition of a Surface
21(10)
§2.1 A Word about Sets
21(1)
§2.2 Metric Spaces
22(1)
§2.3 Open and Closed Sets
23(1)
§2.4 Continuous Maps
24(1)
§2.5 Homeomorphisms
25(1)
§2.6 Compactness
26(1)
§2.7 Surfaces
27(1)
§2.8 Manifolds
28(3)
Chapter 3 The Gluing Construction
31(12)
§3.1 Gluing Spaces Together
31(2)
§3.2 The Gluing Construction in Action
33(3)
§3.3 The Classification of Surfaces
36(1)
§3.4 The Euler Characteristic
37(6)
Chapter 4 The Fundamental Group
43(10)
§4.1 A Primer on Groups
43(2)
§4.2 Homotopy Equivalence
45(1)
§4.3 The Fundamental Group
46(2)
§4.4 Changing the Basepoint
48(1)
§4.5 Functoriality
49(2)
§4.6 Some First Steps
51(2)
Chapter 5 Examples of Fundamental Groups
53(12)
§5.1 The Winding Number
53(3)
§5.2 The Circle
56(1)
§5.3 The Fundamental Theorem of Algebra
57(1)
§5.4 The Torus
58(1)
§5.5 The 2-Sphere
58(1)
§5.6 The Projective Plane
59(1)
§5.7 A Lens Space
59(3)
§5.8 The Poincare Homology Sphere
62(3)
Chapter 6 Covering Spaces and the Deck Group
65(14)
§6.1 Covering Spaces
65(1)
§6.2 The Deck Group
66(1)
§6.3 A Flat Torus
67(2)
§6.4 More Examples
69(1)
§6.5 Simply Connected Spaces
70(1)
§6.6 The Isomorphism Theorem
71(1)
§6.7 The Bolzano---Weierstrass Theorem
72(1)
§6.8 The Lifting Property
73(1)
§6.9 Proof of the Isomorphism Theorem
74(5)
Chapter 7 Existence of Universal Covers
79(8)
§7.1 The Main Result
80(2)
§7.2 The Covering Property
82(2)
§7.3 Simple Connectivity
84(3)
Part 2 Surfaces and Geometry
Chapter 8 Euclidean Geometry
87(16)
§8.1 Euclidean Space
87(3)
§8.2 The Pythagorean Theorem
90(1)
§8.3 The X Theorem
91(1)
§8.4 Pick's Theorem
92(4)
§8.5 The Polygon Dissection Theorem
96(2)
§8.6 Line Integrals
98(2)
§8.7 Green's Theorem for Polygons
100(3)
Chapter 9 Spherical Geometry
103(12)
§9.1 Metrics, Tangent Planes, and Isometries
103(2)
§9.2 Geodesics
105(2)
§9.3 Geodesic Triangles
107(3)
§9.4 Convexity
110(1)
§9.5 Stereographic Projection
111(2)
§9.6 The Hairy Ball Theorem
113(2)
Chapter 10 Hyperbolic Geometry
115(18)
§10.1 Linear Fractional Transformations
115(1)
§10.2 Circle Preserving Property
116(2)
§10.3 The Upper Half-Plane Model
118(3)
§10.4 Another Point of View
121(1)
§10.5 Symmetries
121(2)
§10.6 Geodesics
123(2)
§10.7 The Disk Model
125(2)
§10.8 Geodesic Polygons
127(3)
§10.9 Classification of Isometries
130(3)
Chapter 11 Riemannian Metrics on Surfaces
133(10)
§11.1 Curves in the Plane
133(1)
§11.2 Riemannian Metrics on the Plane
134(1)
§11.3 Diffeomorphisms and Isometries
135(1)
§11.4 Atlases and Smooth Surfaces
136(1)
§11.5 Smooth Curves and the Tangent Plane
137(2)
§11.6 Riemannian Surfaces
139(4)
Chapter 12 Hyperbolic Surfaces
143(20)
§12.1 Definition
143(2)
§12.2 Gluing Recipes
145(2)
§12.3 Gluing Recipes Lead to Surfaces
147(2)
§12.4 Some Examples
149(1)
§12.5 Geodesic Triangulations
150(2)
§12.6 Riemannian Covers
152(2)
§12.7 Hadamard's Theorem
154(2)
§12.8 The Hyperbolic Cover
156(7)
Part 3 Surfaces and Complex Analysis
Chapter 13 A Primer on Complex Analysis
163(14)
§13.1 Basic Definitions
163(2)
§13.2 Cauchy's Theorem
165(2)
§13.3 The Cauchy Integral Formula
167(1)
§13.4 Differentiability
168(2)
§13.5 The Maximum Principle
170(1)
§13.6 Removable Singularities
171(1)
§13.7 Power Series
172(2)
§13.8 Taylor Series
174(3)
Chapter 14 Disk and Plane Rigidity
177(6)
§14.1 Disk Rigidity
177(2)
§14.2 Liouville's Theorem
179(2)
§14.3 Stereographic Projection Revisited
181(2)
Chapter 15 The Schwarz---Christoffel Transformation
183(12)
§15.1 The Basic Construction
184(1)
§15.2 The Inverse Function Theorem
185(1)
§15.3 Proof of Theorem 15.1
186(2)
§15.4 The Range of Possibilities
188(1)
§15.5 Invariance of Domain
189(1)
§15.6 The Existence Proof
190(5)
Chapter 16 Riemann Surfaces and Uniformization
195(12)
§16.1 Riemann Surfaces
195(2)
§16.2 Maps Between Riemann Surfaces
197(2)
§16.3 The Riemann Mapping Theorem
199(2)
§16.4 The Uniformization Theorem
201(1)
§16.5 The Small Picard Theorem
202(1)
§16.6 Implications for Compact Surfaces
203(4)
Part 4 Flat Cone Surfaces
Chapter 17 Flat Cone Surfaces
207(14)
§17.1 Sectors and Euclidean Cones
207(1)
§17.2 Euclidean Cone Surfaces
208(1)
§17.3 The Gauss---Bonnet Theorem
209(2)
§17.4 Translation Surfaces
211(2)
§17.5 Billiards and Translation Surfaces
213(4)
§17.6 Special Maps on a Translation Surface
217(2)
§17.7 Existence of Periodic Billiard Paths
219(2)
Chapter 18 Translation Surfaces and the Veech Group
221(18)
§18.1 Affine Automorphisms
221(2)
§18.2 The Diffential Representation
223(1)
§18.3 Hyperbolic Group Actions
224(2)
§18.4 Proof of Theorem 18.1
226(2)
§18.5 Triangle Groups
228(1)
§18.6 Linear and Hyperbolic Reflections
229(3)
§18.7 Behold, The Double Octagon!
232(7)
Part 5 The Totality of Surfaces
Chapter 19 Continued Fractions
239(12)
§19.1 The Gauss Map
239(2)
§19.2 Continued Fractions
241(1)
§19.3 The Farey Graph
242(2)
§19.4 Structure of the Modular Group
244(1)
§19.5 Continued Fractions and the Farey Graph
245(2)
§19.6 The Irrational Case
247(4)
Chapter 20 Teichmuller Space and Moduli Space
251(12)
§20.1 Parallelograms
251(1)
§20.2 Flat Tori
252(2)
§20.3 The Modular Group Again
254(2)
§20.4 Moduli Space
256(2)
§20.5 Teichmuller Space
258(2)
§20.6 The Mapping Class Group
260(3)
Chapter 21 Topology of Teichmuller Space
263(12)
§21.1 Pairs of Pants
263(2)
§21.2 Pants Decompositions
265(2)
§21.3 Special Maps and Triples
267(2)
§21.4 The End of the Proof
269(6)
Part 6 Dessert
Chapter 22 The Banach---Tarski Theorem
275(12)
§22.1 The Result
275(1)
§22.2 The Schroeder---Bernstein Theorem
276(2)
§22.3 The Doubling Theorem
278(1)
§22.4 Depleted Balls
279(1)
§22.5 The Depleted Ball Theorem
280(2)
§22.6 The Injective Homomorphism
282(5)
Chapter 23 Dehn's Dissection Theorem
287(8)
§23.1 The Result
287(1)
§23.2 Dihedral Angles
288(1)
§23.3 Irrationality Proof
289(1)
§23.4 Rational Vector Spaces
290(1)
§23.5 Dehn's Invariant
291(1)
§23.6 Clean Dissections
292(2)
§23.7 The Proof
294(1)
Chapter 24 The Cauchy Rigidity Theorem
295(14)
§24.1 The Main Result
295(1)
§24.2 The Dual Graph
296(1)
§24.3 Outline of the Proof
297(1)
§24.4 Proof of Lemma 24.3
298(3)
§24.5 Proof of Lemma 24.2
301(2)
§24.6 Euclidean Intuition Does Not Work
303(1)
§24.7 Proof of Cauchy's Arm Lemma
304(5)
Bibliography 309(2)
Index 311
Richard Evan Schwartz is at Brown University, Providence, RI, USA.