V.1 The ABC Conjecture

Chapters in this book

1. Frontmatter i
2. Contents v
3. Preface ix
4. Contributors xvii
5. Part I. Introduction
6. I.1 What Is Mathematics About? 1
7. I.2 The Language and Grammar of Mathematics 8
8. I.3 Some Fundamental Mathematical Definitions 16
9. I.4 The General Goals of Mathematical Research 47
10. Part II. The Origins of Modern Mathematics
11. II.1 From Numbers to Number Systems 77
12. II.2 Geometry 83
13. II.3 The Development of Abstract Algebra 95
14. II.4 Algorithms 106
15. II.5 The Development of Rigor in Mathematical Analysis 117
16. II.6 The Development of the Idea of Proof 129
17. II.7 The Crisis in the Foundations of Mathematics 142
18. Part III. Mathematical Concepts
19. III.1 The Axiom of Choice 157
20. III.2 The Axiom of Determinacy 159
21. III.3 Bayesian Analysis 159
22. III.4 Braid Groups 160
23. III.5 Buildings 161
24. III.6 Calabi–Yau Manifolds 163
25. III.7 Cardinals 165
26. III.8 Categories 165
27. III.9 Compactness and Compactification 167
28. III.10 Computational Complexity Classes 169
29. III.11 Countable and Uncountable Sets 170
30. III.12 C*-Algebras 172
31. III.13 Curvature 172
32. III.14 Designs 172
33. III.15 Determinants 174
34. III.16 Differential Forms and Integration 175
35. III.17 Dimension 180
36. III.18 Distributions 184
37. III.19 Duality 187
38. III.20 Dynamical Systems and Chaos 190
39. III.21 Elliptic Curves 190
40. III.22 The Euclidean Algorithm and Continued Fractions 191
41. III.23 The Euler and Navier–Stokes Equations 193
42. III.24 Expanders 196
43. III.25 The Exponential and Logarithmic Functions 199
44. III.26 The Fast Fourier Transform 202
45. III.27 The Fourier Transform 204
46. III.28 Fuchsian Groups 208
47. III.29 Function Spaces 210
48. III.30 Galois Groups 213
49. III.31 The Gamma Function 213
50. III.32 Generating Functions 214
51. III.33 Genus 215
52. III.34 Graphs 215
53. III.35 Hamiltonians 215
54. III.36 The Heat Equation 216
55. III.37 Hilbert Spaces 219
56. III.38 Homology and Cohomology 221
57. III.39 Homotopy Groups 221
58. III.40 The Ideal Class Group 221
59. III.41 Irrational and Transcendental Numbers 222
60. III.42 The Ising Model 223
61. III.43 Jordan Normal Form 223
62. III.44 Knot Polynomials 225
63. III.45 K-Theory 227
64. III.46 The Leech Lattice 227
65. III.47 L-Functions 228
66. III.48 Lie Theory 229
67. III.49 Linear and Nonlinear Waves and Solitons 234
68. III.50 Linear Operators and Their Properties 239
69. III.51 Local and Global in Number Theory 241
70. III.52 The Mandelbrot Set 244
71. III.53 Manifolds 244
72. III.54 Matroids 244
73. III.55 Measures 246
74. III.56 Metric Spaces 247
75. III.57 Models of Set Theory 248
76. III.58 Modular Arithmetic 249
77. III.59 Modular Forms 250
78. III.60 Moduli Spaces 252
79. III.61 The Monster Group 252
80. III.62 Normed Spaces and Banach Spaces 252
81. III.63 Number Fields 254
82. III.64 Optimization and Lagrange Multipliers 255
83. III.65 Orbifolds 257
84. III.66 Ordinals 258
85. III.67 The Peano Axioms 258
86. III.68 Permutation Groups 259
87. III.69 Phase Transitions 261
88. III.70 π 261
89. III.71 Probability Distributions 263
90. III.72 Projective Space 267
91. III.73 Quadratic Forms 267
92. III.74 Quantum Computation 269
93. III.75 Quantum Groups 272
94. III.76 Quaternions, Octonions, and Normed Division Algebras 275
95. III.77 Representations 279
96. III.78 Ricci Flow 279
97. III.79 Riemann Surfaces 282
98. III.80 The Riemann Zeta Function 283
99. III.81 Rings, Ideals, and Modules 284
100. III.82 Schemes 285
101. III.83 The Schrödinger Equation 285
102. III.84 The Simplex Algorithm 288
103. III.85 Special Functions 290
104. III.86 The Spectrum 294
105. III.87 Spherical Harmonics 295
106. III.88 Symplectic Manifolds 297
107. III.89 Tensor Products 301
108. III.90 Topological Spaces 301
109. III.91 Transforms 303
110. III.92 Trigonometric Functions 307
111. III.93 Universal Covers 309
112. III.94 Variational Methods 310
113. III.95 Varieties 313
114. III.96 Vector Bundles 313
115. III.97 Von Neumann Algebras 313
116. III.98 Wavelets 313
117. III.99 The Zermelo–Fraenkel Axioms 314
118. Part IV. Branches of Mathematics
119. IV.1 Algebraic Numbers 315
120. IV.2 Analytic Number Theory 332
121. IV.3 Computational Number Theory 348
122. IV.4 Algebraic Geometry 363
123. IV.5 Arithmetic Geometry 372
124. IV.6 Algebraic Topology 383
125. IV.7 Differential Topology 396
126. IV.8 Moduli Spaces 408
127. IV.9 Representation Theory 419
128. IV.10 Geometric and Combinatorial Group Theory 431
129. IV.11 Harmonic Analysis 448
130. IV.12 Partial Differential Equations 455
131. IV.13 General Relativity and the Einstein Equations 483
132. IV.14 Dynamics 493
133. IV.15 Operator Algebras 510
134. IV.16 Mirror Symmetry 523
135. IV.17 Vertex Operator Algebras 539
136. IV.18 Enumerative and Algebraic Combinatorics 550
137. IV.19 Extremal and Probabilistic Combinatorics 562
138. IV.20 Computational Complexity 575
139. IV.21 Numerical Analysis 604
140. IV.22 Set Theory 615
141. IV.23 Logic and Model Theory 635
142. IV.24 Stochastic Processes 647
143. IV.25 Probabilistic Models of Critical Phenomena 657
144. IV.26 High-Dimensional Geometry and Its Probabilistic Analogues 670
145. Part V. Theorems and Problems
146. V.1 The ABC Conjecture 681
147. V.2 The Atiyah–Singer Index Theorem 681
148. V.3 The Banach–Tarski Paradox 684
149. V.4 The Birch–Swinnerton-Dyer Conjecture 685
150. V.5 Carleson’s Theorem 686
151. V.6 The Central Limit Theorem 687
152. V.7 The Classification of Finite Simple Groups 687
153. V.8 Dirichlet’s Theorem 689
154. V.9 Ergodic Theorems 689
155. V.10 Fermat’s Last Theorem 691
156. V.11 Fixed Point Theorems 693
157. V.12 The Four-Color Theorem 696
158. V.13 The Fundamental Theorem of Algebra 698
159. V.14 The Fundamental Theorem of Arithmetic 699
160. V.15 Gödel’s Theorem 700
161. V.16 Gromov’s Polynomial-Growth Theorem 702
162. V.17 Hilbert’s Nullstellensatz 703
163. V.18 The Independence of the Continuum Hypothesis 703
164. V.19 Inequalities 703
165. V.20 The Insolubility of the Halting Problem 706
166. V.21 The Insolubility of the Quintic 708
167. V.22 Liouville’s Theorem and Roth’s Theorem 710
168. V.23 Mostow’s Strong Rigidity Theorem 711
169. V.24 The P versus NP Problem 713
170. V.25 The Poincaré Conjecture 714
171. V.26 The Prime Number Theorem and the Riemann Hypothesis 714
172. V.27 Problems and Results in Additive Number Theory 715
173. V.28 From Quadratic Reciprocity to Class Field Theory 718
174. V.29 Rational Points on Curves and the Mordell Conjecture 720
175. V.30 The Resolution of Singularities 722
176. V.31 The Riemann–Roch Theorem 723
177. V.32 The Robertson–Seymour Theorem 725
178. V.33 The Three-Body Problem 726
179. V.34 The Uniformization Theorem 728
180. V.35 The Weil Conjectures 729
181. Part VI. Mathematicians
182. VI.1 Pythagoras 733
183. VI.2 Euclid 734
184. VI.3 Archimedes 734
185. VI.4 Apollonius 735
186. VI.5 Abu Ja’far Muhammad ibn Mūsā al-Khwārizmī 736
187. VI.6 Leonardo of Pisa (known as Fibonacci) 737
188. VI.7 Girolamo Cardano 737
189. VI.8 Rafael Bombelli 737
190. VI.9 François Viète 737
191. VI.10 Simon Stevin 738
192. VI.11 René Descartes 739
193. VI.12 Pierre Fermat 740
194. VI.13 Blaise Pascal 741
195. VI.14 Isaac Newton 742
196. VI.15 Gottfried Wilhelm Leibniz 743
197. VI.16 Brook Taylor 745
198. VI.17 Christian Goldbach 745
199. VI.18 The Bernoullis 745
200. VI.19 Leonhard Euler 747
201. VI.20 Jean Le Rond d’Alembert 749
202. VI.21 Edward Waring 750
203. VI.22 Joseph Louis Lagrange 751
204. VI.23 Pierre-Simon Laplace 752
205. VI.24 Adrien-Marie Legendre 754
206. VI.25 Jean-Baptiste Joseph Fourier 755
207. VI.26 Carl Friedrich Gauss 755
208. VI.27 Siméon-Denis Poisson 757
209. VI.28 Bernard Bolzano 757
210. VI.29 Augustin-Louis Cauchy 758
211. VI.30 August Ferdinand Möbius 759
212. VI.31 Nicolai Ivanovich Lobachevskii 759
213. VI.32 George Green 760
214. VI.33 Niels Henrik Abel 760
215. VI.34 János Bolyai 762
216. VI.35 Carl Gustav Jacob Jacobi 762
217. VI.36 Peter Gustav Lejeune Dirichlet 764
218. VI.37 William Rowan Hamilton 765
219. VI.38 Augustus De Morgan 765
220. VI.39 Joseph Liouville 766
221. VI.40 Ernst Eduard Kummer 767
222. VI.41 Évariste Galois 767
223. VI.42 James Joseph Sylvester 768
224. VI.43 George Boole 769
225. VI.44 Karl Weierstrass 770
226. VI.45 Pafnuty Chebyshev 771
227. VI.46 Arthur Cayley 772
228. VI.47 Charles Hermite 773
229. VI.48 Leopold Kronecker 773
230. VI.49 Georg Friedrich Bernhard Riemann 774
231. VI.50 Julius Wilhelm Richard Dedekind 776
232. VI.51 Émile Léonard Mathieu 776
233. VI.52 Camille Jordan 777
234. VI.53 Sophus Lie 777
235. VI.54 Georg Cantor 778
236. VI.55 William Kingdon Clifford 780
237. VI.56 Gottlob Frege 780
238. VI.57 Christian Felix Klein 782
239. VI.58 Ferdinand Georg Frobenius 783
240. VI.59 Sofya (Sonya) Kovalevskaya 784
241. VI.60 William Burnside 785
242. VI.61 Jules Henri Poincaré 785
243. VI.62 Giuseppe Peano 787
244. VI.63 David Hilbert 788
245. VI.64 Hermann Minkowski 789
246. VI.65 Jacques Hadamard 790
247. VI.66 Ivar Fredholm 791
248. VI.67 Charles-Jean de la Vallée Poussin 792
249. VI.68 Felix Hausdorff 792
250. VI.69 Élie Joseph Cartan 794
251. VI.70 Emile Borel 795
252. VI.71 Bertrand Arthur William Russell 795
253. VI.72 Henri Lebesgue 796
254. VI.73 Godfrey Harold Hardy 797
255. VI.74 Frigyes (Frédéric) Riesz 798
256. VI.75 Luitzen Egbertus Jan Brouwer 799
257. VI.76 Emmy Noether 800
258. VI.77 Wacław Sierpiński 801
259. VI.78 George Birkhoff 802
260. VI.79 John Edensor Littlewood 803
261. VI.80 Hermann Weyl 805
262. VI.81 Thoralf Skolem 806
263. VI.82 Srinivasa Ramanujan 807
264. VI.83 Richard Courant 808
265. VI.84 Stefan Banach 809
266. VI.85 Norbert Wiener 811
267. VI.86 Emil Artin 812
268. VI.87 Alfred Tarski 813
269. VI.88 Andrei Nikolaevich Kolmogorov 814
270. VI.89 Alonzo Church 816
271. VI.90 William Vallance Douglas Hodge 816
272. VI.91 John von Neumann 817
273. VI.92 Kurt Gödel 819
274. VI.93 André Weil 819
275. VI.94 Alan Turing 821
276. VI.95 Abraham Robinson 822
277. VI.96 Nicolas Bourbaki 823
278. Part VII. The Influence of Mathematics
279. VII.1 Mathematics and Chemistry 827
280. VII.2 Mathematical Biology 837
281. VII.3 Wavelets and Applications 848
282. VII.4 The Mathematics of Traffic in Networks 862
283. VII.5 The Mathematics of Algorithm Design 871
284. VII.6 Reliable Transmission of Information 878
285. VII.7 Mathematics and Cryptography 887
286. VII.8 Mathematics and Economic Reasoning 895
287. VII.9 The Mathematics of Money 910
288. VII.10 Mathematical Statistics 916
289. VII.11 Mathematics and Medical Statistics 921
290. VII.12 Analysis, Mathematical and Philosophical 928
291. VII.13 Mathematics and Music 935
292. VII.14 Mathematics and Art 944
293. Part VIII. Final Perspectives
294. VIII.1 The Art of Problem Solving 955
295. VIII.2 “Why Mathematics?” You Might Ask 966
296. VIII.3 The Ubiquity of Mathematics 977
297. VIII.4 Numeracy 983
298. VIII.5 Mathematics: An Experimental Science 991
299. VIII.6 Advice to a Young Mathematician 1000
300. VIII.7 A Chronology of Mathematical Events 1010
301. Index 1015