Table of contents for The Princeton companion to mathematics / Timothy Gowers, editor ; June Barrow-Green, Imre Leader, associate editors.

Bibliographic record and links to related information available from the Library of Congress catalog.

Note: Contents data are machine generated based on pre-publication provided by the publisher. Contents may have variations from the printed book or be incomplete or contain other coding.


Counter
Contents
Preface ix
List of Contributors xvii
Part I Introduction
I.1 What Is Mathematics About? 1
I.2 The Language and Grammar of Mathematics 8
I.3 Some Fundamental Mathematical Definitions 16
I.4 The General Goals of Mathematical Research 46
Part II The Origins of Modern
Mathematics
II.1 From Numbers to Number Systems 77
II.2 Geometry 83
II.3 The Development of Abstract Algebra 96
II.4 Algorithms 107
II.5 The Development of Rigor in
Mathematical Analysis 118
II.6 The Development of the Idea of Proof 130
II.7 The Crisis in the Foundations of Mathematics 143
Part III Mathematical Concepts
III.1 The Axiom of Choice 159
III.2 The Axiom of Determinacy 161
III.3 Bayesian Analysis 161
III.4 Braid Groups 162
III.5 Buildings 163
III.6 Calabi-Yau Manifolds 164
III.7 Cardinals 167
III.8 Categories 167
III.9 Compactness and Compactification 169
III.10 Computational Complexity Classes 171
III.11 Countable and Uncountable Sets 172
III.12 C?-Algebras 173
III.13 Curvature 174
III.14 Designs 174
III.15 Determinants 175
III.16 Differential Forms and Integration 176
III.17 Dimension 182
III.18 Distributions 186
III.19 Duality 189
III.20 Dynamical Systems and Chaos 191
III.21 Elliptic Curves 192
III.22 The Euclidean Algorithm and
Continued Fractions 193
III.23 The Euler and Navier-Stokes Equations 195
III.24 The Exponential and Logarithmic Functions 198
III.25 Expanders 201
III.26 The Fast Fourier Transform 204
III.27 The Fourier Transform 206
III.28 Fuchsian Groups 209
III.29 Function Spaces 211
III.30 Galois Groups 214
III.31 The Gamma Function 215
III.32 Generating Functions 216
III.33 Genus 216
III.34 Graphs 217
III.35 Hamiltonians 217
III.36 The Heat Equation 218
III.37 Hilbert Spaces 221
III.38 Holomorphic Functions 222
III.39 Homology and Cohomology 222
III.40 Homotopy Groups 223
III.41 The Hyperbolic Plane 223
III.42 The Ideal Class Group 223
III.43 Irrational and Transcendental Numbers 224
III.44 The Ising Model 225
III.45 Jordan Normal Form 225
III.46 Knot Polynomials 227
III.47 K-Theory 229
III.48 The Leech Lattice 229
III.49 L-Functions 230
III.50 Lie Theory 231
III.51 Linear and Nonlinear Waves and Solitons 236
III.52 Linear Operators and Their Properties 241
III.53 Local and Global in Number Theory 243
III.54 Optimization and Lagrange Multipliers 246
_
vi Contents
III.55 The Mandelbrot Set 248
III.56 Manifolds 248
III.57 Matroids 249
III.58 Measures 250
III.59 Metric Spaces 252
III.60 Models of Set Theory 253
III.61 Modular Arithmetic 253
III.62 Modular Forms 255
III.63 Moduli Spaces 256
III.64 The Monster Group 256
III.65 Normed Spaces and Banach Spaces 257
III.66 Number Fields 258
III.67 Orbifolds 259
III.68 Ordinals 260
III.69 The Peano Axioms 260
III.70 Permutation Groups 261
III.71 Phase Transitions 263
III.72 ? 263
III.73 Probability Distributions 264
III.74 Projective Space 269
III.75 Quadratic Forms 269
III.76 Quantum Computation 270
III.77 Quantum Groups 273
III.78 Quaternions, Octonions, and Normed
Division Algebras 277
III.79 Representations 280
III.80 Ricci Flow 281
III.81 Riemannian Metrics 283
III.82 Riemann Surfaces 284
III.83 Rings, Ideals, and Modules 286
III.84 Schemes 287
III.85 The Schrodinger Equation 288
III.86 The Simplex Algorithm 290
III.87 Special Functions 292
III.88 The Spectrum 296
III.89 Spherical Harmonics 297
III.90 Symplectic Manifolds 299
III.91 Tensor Products 303
III.92 Topological Spaces 303
III.93 Transforms 305
III.94 Trigonometric Functions 309
III.95 Variational Methods 311
III.96 Varieties 313
III.97 Vector Bundles 314
III.98 Von Neumann Algebras 314
III.99 Wavelets 314
III.100 Zeta Functions 315
III.101 The Zermelo-Fraenkel Axioms 315
Part IV Branches of Mathematics
IV.1 Set Theory 317
IV.2 Logic and Model Theory 336
IV.3 Algebraic Numbers 348
IV.4 Analytic Number Theory 365
IV.5 Computational Number Theory 382
IV.6 Arithmetic Geometry 396
IV.7 Algebraic Geometry 407
IV.8 Moduli Spaces 416
IV.9 Differential Topology 428
IV.10 Algebraic Topology 440
IV.11 Geometric and Combinatorial Group Theory 453
IV.12 Representation Theory 470
IV.13 Vertex Operator Algebras 482
IV.14 Mirror Symmetry 493
IV.15 Dynamics 509
IV.16 Partial Differential Equations 526
IV.17 General Relativity and the Einstein Equations 553
IV.18 Harmonic Analysis 553
IV.19 Operator Algebras 560
IV.20 Numerical Analysis 573
IV.21 Computational Complexity 585
IV.22 Enumerative and Algebraic Combinatorics 613
IV.23 Extremal and Probabilistic Combinatorics 625
IV.24 High-Dimensional Geometry and Its
Probabilistic Analogues 639
IV.25 Stochastic Processes 649
IV.26 Probabilistic Models of Critical Phenomena 660
Part V Theorems and Problems
V.1 The ABC Conjecture 673
V.2 The Atiyah-Singer Index Theorem 673
V.3 The Banach-Tarski Paradox 676
V.4 The Birch-Swinnerton-Dyer Conjecture 677
V.5 Carleson's Theorem 678
V.6 Cauchy's Theorem 679
V.7 The Central Limit Theorem 679
V.8 The Classification of Finite Simple Groups 679
V.9 Dirichlet's Theorem 681
V.10 Dvoretzky's Theorem 681
V.11 Ergodic Theorems 682
V.12 Fermat's Last Theorem 684
V.13 Fixed-Point Theorems 686
V.14 The Four-Color Theorem 689
V.15 The Fundamental Theorem of Algebra 691
V.16 The Fundamental Theorem of Arithmetic 692
V.17 The Fundamental Theorem of Calculus 693
V.18 Godel's Theorem 693
V.19 Gromov's Polynomial-Growth Theorem 695
V.20 Hilbert's Nullstellensatz 696
_
Contents vii
V.21 The Independence of the
Continuum Hypothesis 696
V.22 Inequalities 696
V.23 The Insolubility of the Halting Problem 699
V.24 The Insolubility of the Quintic 701
V.25 Liouville's Theorem and Roth's Theorem 703
V.26 Rational Points on Curves and
the Mordell Conjecture 704
V.27 Mostow's Strong Rigidity Theorem 706
V.28 The P = NP Problem 708
V.29 The Poincare Conjecture 709
V.30 Problems and Results in
Additive Number Theory 709
V.31 From Quadratic Reciprocity to
Class Field Theory 712
V.32 The Resolution of Singularities 715
V.33 The Riemann Hypothesis 715
V.34 The Riemann-Roch Theorem 716
V.35 The Robertson-Seymour Theorem 718
V.36 The Three-Body Problem 719
V.37 The Uniformization Theorem 721
V.38 The Weil Conjectures 722
Part VI Mathematicians
VI.1 Pythagoras 727
VI.2 Euclid 728
VI.3 Archimedes 728
VI.4 Apollonius 729
VI.5 Leonardo of Pisa (known as Fibonacci) 730
VI.6 Girolamo Cardano 730
VI.7 Rafael Bombelli 730
VI.8 Francois Viete 731
VI.9 Simon Stevin 732
VI.10 Rene Descartes 732
VI.11 Pierre Fermat 734
VI.12 Blaise Pascal 735
VI.13 Isaac Newton 735
VI.14 Gottfried Wilhelm Leibniz 737
VI.15 The Bernoullis 738
VI.16 Brooke Taylor 740
VI.17 Christian Goldbach 740
VI.18 Leonhard Euler 740
VI.19 Jean Le Rond d'Alembert 742
VI.20 Edward Waring 744
VI.21 Joseph Louis Lagrange 744
VI.22 Pierre-Simon Laplace 745
VI.23 Adrien-Marie Legendre 747
VI.24 Jean-Baptiste Joseph Fourier 748
VI.25 Carl Friedrich Gauss 749
VI.26 Simeon-Denis Poisson 750
VI.27 Bernard Bolzano 750
VI.28 Augustin-Louis Cauchy 751
VI.29 August Ferdinand Mobius 752
VI.30 Nicolai Ivanovich Lobachevskii 752
VI.31 George Green 753
VI.32 Niels Henrik Abel 753
VI.33 Janos Bolyai 755
VI.34 Carl Gustav Jacob Jacobi 755
VI.35 Peter Gustav Lejeune Dirichlet 757
VI.36 William Rowan Hamilton 758
VI.37 Augustus De Morgan 759
VI.38 Joseph Liouville 759
VI.39 Eduard Kummer 760
VI.40 Evariste Galois 760
VI.41 James Joseph Sylvester 761
VI.42 George Boole 762
VI.43 Karl Weierstrass 763
VI.44 Pafnuty Chebyshev 764
VI.45 Arthur Cayley 765
VI.46 Charles Hermite 766
VI.47 Leopold Kronecker 766
VI.48 Georg Bernhard Friedrich Riemann 767
VI.49 Julius Wilhelm Richard Dedekind 769
VI.50 Emile Leonard Mathieu 769
VI.51 Camille Jordan 770
VI.52 Sophus Lie 770
VI.53 Georg Cantor 771
VI.54 William Kingdon Clifford 773
VI.55 Gottlob Frege 773
VI.56 Christian Felix Klein 775
VI.57 Ferdinand Georg Frobenius 776
VI.58 Sonya Kovalevskaya 777
VI.59 William Burnside 778
VI.60 Jules Henri Poincare 778
VI.61 Giuseppe Peano 780
VI.62 David Hilbert 781
VI.63 Hermann Minkowski 782
VI.64 Jacques Hadamard 783
VI.65 Ivar Fredholm 784
VI.66 Charles-Jean de la Vallee Poussin 784
VI.67 Felix Hausdorff 785
VI.68 Elie Joseph Cartan 786
VI.69 Emile Borel 787
VI.70 Bertrand Arthur William Russell 788
VI.71 Henri Lebesgue 788
VI.72 Godfrey Harold Hardy 790
VI.73 Frigyes (Frederic) Riesz 791
VI.74 Luitzen Egbertus Jan Brouwer 792
VI.75 Emmy Noether 793
VI.76 Waclaw Sierpi?nski 794
VI.77 George Birkhoff 795
VI.78 John Edensor Littlewood 796
VI.79 Hermann Weyl 798
VI.80 Thoralf Skolem 799
_
viii Contents
VI.81 Srinivasa Ramanujan 800
VI.82 Richard Courant 801
VI.83 Stefan Banach 802
VI.84 Norbert Wiener 804
VI.85 Emil Artin 805
VI.86 Alfred Tarski 806
VI.87 Andrei Nikolaevich Kolmogorov 807
VI.88 William Vallance Douglas Hodge 809
VI.89 John von Neumann 810
VI.90 Kurt Godel 811
VI.91 Andre Weil 812
VI.92 Alan Turing 813
VI.93 Abraham Robinson 814
VI.94 Nicolas Bourbaki 816
Part VII The Influence of Mathematics
VII.1 Mathematics and Chemistry 819
VII.2 Mathematical Biology 830
VII.3 Wavelets and Applications 840
VII.4 The Mathematics of Traffic in Networks 855
VII.5 The Mathematics of Algorithm Design 863
VII.6 Reliable Transmission of Information 871
VII.7 Mathematics and Cryptography 879
VII.8 Mathematics and Economic Reasoning 888
VII.9 The Mathematics of Money 903
VII.10 Mathematical Statistics 909
VII.11 Mathematics and Medical Statistics 914
VII.12 Analysis, Mathematical and Philosophical 921
VII.13 Mathematics and Music 928
VII.14 Mathematics and Art 937
Part VIII Final Perspectives
VIII.1 The Art of Problem Solving 949
VIII.2 "Why Mathematics?" You Might Ask 960
VIII.3 The Ubiquity of Mathematics 971
VIII.4 Numeracy 977
VIII.5 Mathematics: An Experimental Science 985
VIII.6 Advice to a Young Mathematician 994
VIII.7 A Chronology of Major Mathematical Events 1003
Index

Library of Congress Subject Headings for this publication:

Mathematics -- Study and teaching (Higher).
Princeton University.