The Princeton Companion to Mathematics

EDITOR

Timothy Gowers University of Cambridge

ASSOCIATE EDITORS

June Barrow-Green The Open University

Imre Leader University of Cambridge

Princeton University Press Princeton and Oxford

Contents

Preface Contributors

Parti Introduction LI 1.2 1.3 1.4

What Is Mathematics About? The Language and Grammar of Mathematics Some Fundamental Mathematical Definitions The General Goals of Mathematical Research

Part II The Origins of Modern

П.1 П.2 П.З П.4 П.5 .

П.6 П.7

Mathematics From Numbers to Number Systems Geometry The Development of Abstract Algebra Algorithms The Development of Rigor in Mathematical Analysis The Development of the Idea of Proof The Crisis in the Foundations of Mathematics

Part III Mathematical Concepts Ш.1 Ш.2 Ш.З Ш.4 Ш.5 Ш.6 Ш.7 Ш.8 Ш.9 Ш.10 Ш.11 Ш.12 Ш.13 Ш.14

The Axiom of Choice The Axiom of Determinacy Bayesian Analysis Braid Groups Buildings Calabi-Yau Manifolds Cardinals Categories Compactness and Compactfflcation Computational Complexity Classes Countable and Uncountable Sets С*-Algebras Curvature Designs

ix xvii

1 8

16 47

77 83 95

106

117 129 142

157 159 159 160 161 163 165 165 167 169 170 172 172 172

Ш.15 Ш.16 Ш.17 Ш.18 III. 19 Ш.20 Ш.21 Ш.22

Ш.23 Ш.24 Ш.25 Ш.26 Ш.27 Ш.28 Ш.29 Ш.30 Ш.31 Ш.32 Ш.ЗЗ Ш.34 Ш.35 Ш.36 Ш.37 Ш.38 Ш.39 Ш.40 Ш.41 Ш.42 Ш.43 Ш.44 Ш.45 Ш.46 Ш.47 Ш.48 Ш.49 Ш.50 Ш.51 Ш.52 Ш.53 Ш.54 Ш.55

Determinants Differential Forms and Integration Dimension Distributions Duality Dynamical Systems and Chaos Elliptic Curves The Euclidean Algorithm and Continued Fractions The Euler and Navier-Stokes Equations Expanders The Exponential and Logarithmic Functions The Fast Fourier Transform The Fourier Transform Fuchsian Groups Function Spaces Galois Groups The Gamma Function Generating Functions Genus Graphs Hamiltonians The Heat Equation Hubert Spaces Homology and Cohomology Homotopy Groups The Ideal Class Group Irrational and Transcendental Numbers The Ising Model Jordan Normal Form Knot Polynomials ^-Theory The Leech Lattice I-Functions Lie Theory Linear and Nonlinear Waves and Solitons Linear Operators and Their Properties Local and Global in Number Theory The Mandelbrot Set Manifolds Matroids Measures

174 175 180 184 187 190 190

191 193 196 199 202 204 208 210 213 213 214 215 215 215 216 219 221 221 221 222 223 223 225 227 227 228 229 234 239 241 244 244 244 246

VI Contents

Ш.56 Metrie Spaces 247 Ш.57 Models of Set Theory 248 Ш.58 Modular Arithmetic 249 Ш.59 Modular Forms 250 Ш.60 Moduli Spaces 252 Ш.61 The Monster Group 252 Ш.62 Normed Spaces and Banach Spaces 252 Ш.63 Number Fields 254 Ш.64 Optimization and Lagrange Multipliers 255 Ш.65 Orbifolds 257 Ш.66 Ordinals 258 Ш.67 The Peano Axioms 258 Ш.68 Permutation Groups 259 Ш.69 Phase Transitions 261 Ш.70 n 261 Ш.71 Probability Distributions 263 Ш.72 Projective Space 267 Ш.73 Quadratic Forms 267 Ш.74 Quantum Computation 269 Ш.75 Quantum Groups 272 Ш.76 Quaternions, Octonions, and Normed

Division Algebras 275 Ш.77 Representations 279 Ш.78 Ricci Flow 279 Ш.79 Riemann Surfaces 282 Ш.80 The Riemann Zeta Function 283 Ш.81 Rings, Ideals, and Modules 284 Ш.82 Schemes 285 Ш.83 The Schrödinger Equation 285 Ш.84 The Simplex Algorithm 288 Ш.85 Special Functions 290 Ш.86 The Spectrum 294 Ш.87 Spherical Harmonics 295 Ш.88 Symplectic Manifolds 297 Ш.89 Tensor Products 301 Ш.90 Topological Spaces 301 Ш.91 Transforms 303 Ш.92 Trigonometric Functions 307 Ш.93 Universal Covers 309 Ш.94 Variational Methods 310 Ш.95 Varieties 313 Ш.96 Vector Bundles 313 Ш.97 Von Neumann Algebras 313 Ш.98 Wavelets 313 Ш.99 The Zermelo-Fraenkel Axioms 314

Part IV Branches of Mathematics IV. 1 Algebraic Numbers IV.2 Analytic Number Theory IV. 3 Computational Number Theory rv.4 Algebraic Geometry

315 332 348 363

IV.5 Arithmetic Geometry 372 rv.6 Algebraic Topology 383 rv.7 Differential Topology 396 rv.8 Moduli Spaces 408 IV.9 Representation Theory 419 IV. 10 Geometric and Combinatorial Group Theory 431 IV. 11 Harmonic Analysis 448 IV. 12 Partial Differential Equations 455 IV. 13 General Relativity and the Einstein Equations 483 rv.14 Dynamics 493 IV.15 Operator Algebras 510 IV.16 Mirror Symmetry 523 IV. 17 Vertex Operator Algebras 539 IV. 18 Enumerative and Algebraic Combinatorics 550 IV.19 Extremal and Probabilistic Combinatorics 562 IV.20 Computational Complexity 575 IV. 21 Numerical Analysis 604 IV.22 Set Theory 615 IV.2 3 Logic and Model Theory 63 5 IV. 2 4 Stochastic Processes 647 IV.25 ProbablUstic Models of Critical Phenomena 657 IV.26 High-Dimensional Geometry and Its

Probabilistic Analogues 670

Part V Theorems and Problems V.l The ABC Conjecture 681 V.2 The Atiyah-Singer Index Theorem 681 V.3 The Banach-Tarski Paradox 684 V.4 The Birch-Swinnerton-Dyer Conjecture 685 V.5 Carleson's Theorem 686 V.6 The Central Limit Theorem 687 V.7 The Classification of Finite Simple Groups 687 V.8 Dirichlet's Theorem 689 V.9 Ergodic Theorems 689 V.10 Fermat's Last Theorem 691 V.ll Fixed Point Theorems 693 V.l2 The Four-Color Theorem 696 V. 13 The Fundamental Theorem of Algebra 698 V.l4 The Fundamental Theorem of Arithmetic 699 V.l 5 Gödel's Theorem 700 V.16 Gromov's Polynomial-Growth Theorem 702 V.17 Hubert's Nullstellensatz 703 V.l8 The Independence of the

Continuum Hypothesis 703 V.19 Inequalities 703 V.20 The Insolubility of the Halting Problem 706 V.21 The Insolubility of the Quintic 708 V.22 Liouville's Theorem and Roth's Theorem 710 V.23 Mostow's Strong Rigidity Theorem 711 V.24 The У versus WP Problem 713 V.25 The Poincare Conjecture 714

Contents vu

V.26 The Prime Number Theorem and the Riemann Hypothesis

V.27 Problems and Results in Additive Number Theory

V.28 From Quadratic Reciprocity to Class Field Theory

V.29 Rational Points on Curves and the Mordell Conjecture

V.30 The Resolution of Singularities V.31 The Riemann-Roch Theorem V.32 The Robertson-Seymour Theorem V.33 The Three-Body Problem V.34 The Uniformization Theorem V.35 The Weil Conjectures

Part VI Mathematicians VI. 1 Pythagoras (ca. 569 в.С.Е.-са. 494 B.C.E.) VI.2 Euclid (ca. 325 B.c.E.-ca. 265 B.CE.) VI.3 Archimedes (ca. 287 B.C.E.-212 B.C.E.) VI.4 Apollonius (ca. 262 B.c.E.-ca. 190 B.C.E.) VI. 5 Abu Ja'far Muhammad ibn Müsä al-Khwärizml

(800-847)

VI. 6 Leonardo of Pisa (known as Fibonacci) (ca.1170-ca.1250)

VI.7 Girolamo Cardano (1501-1576) VI.8 Rafael Bombelli (1526-after 1572) VI.9 Francois Viete (1540-1603) VI.10 Simon Stevin (1548-1620) VI.11 Rene Descartes (1596-1650) VI.12 Pierre Fermat (1607-1665) VI.13 Blaise Pascal (1623-1662) VI.14 Isaac Newton (1642-1727) VI.15 Gottfried Wilhelm Leibniz (1646-1716) VI.16 Brook Taylor (1685-1731) VI.17 Christian Goldbach (1690-1764) VI. 18 The BernoulUs (fl. 18th century) VI.19 Leonhard Euler (1707-1783) VI.20 Jean Le Rond d'Alembert (1717-1783) VI.21 Edward Waring (ca. 1735-1798) VI.22 Joseph Louis Lagrange (1736-1813) VI.23 Pierre-Simon Laplace (1749-1827) VI.24 Adrien-Marie Legendre (1752-1833) VI.25 Jean-Baptiste Joseph Fourier (1768-1830) VI.26 Carl Friedrich Gauss (1777-1855) VI.27 Simeon-Denis Poisson (1781-1840) VI.28 Bernard Bolzano (1781-1848) VI.29 Augustin-Louis Cauchy (1789-1857) VI.30 August Ferdinand Möbius (1790-1868) VI.31 Nicolai Ivanovich Lobachevskü (1792-1856) VI.32 George Green (1793-1841) VI.33 Niels Henrik Abel (1802-1829)

714

715

718

720 722 723 725 726 728 729

733 734 734 735

736

737 737 737 737 738 739 740 741 742 743 745 745 745 747 749 750 751 752 754 755 755 757 757 758 759 759 760 760

VI.34 VI.3 5 VI.36 VI.37 VI.38 VI.39 VI.40 VI.41 VI.42 VI.43 VI.44 VI.45 VI.46 VI.47

VI.48 VI.49

VI.50 VI.51 VI.52 VI.53 VI.54 VI.55 VI.56 VI.57 VI.58 VI.59 VI.60 VI.61 VI.62 VI.63 VI.64 VI.65 VI.66 VI.67 VI.68 VI.69 VI. 70 VI.71 VI.72 VI.73 VI.74 VI.75 VI. 76 VI. 77 VI. 78 VI.79 VI.80 VI.81 VI.82 VI.83 VI.84 VI.85

Jänos Bolyai (1802-1860) 762 Carl Gustav Jacob Jacobi (1804-1851) 762 Peter Gustav Lejeune Dirichlet (1805-1859) 764 William Rowan Hamilton (1805-1865) 765 Augustus De Morgan (1806-1871) 765 Joseph Liouville (1809-1882) 766 Ernst Eduard Kummer (1810-1893) 767 Evariste Galois (1811-1832) 767 James Joseph Sylvester (1814-1897) 768 George Boole (1815-1864) 769 Karl Weierstrass (1815-1897) 770 Pafnuty Chebyshev (1821-1894) 771 Arthur Cayley (1821-1895) 772 Charles Hermite (1822-1901) 773 Leopold Kronecker (1823-1891) 773 Georg Friedrich Bernhard Riemann (1826-1866) 774 Julius Wilhelm Richard Dedekind (1831-1916) 776 Emile Leonard Mathieu (1835-1890) 776 Camille Jordan (1838-1922) 777 SophusLie (1842-1899) 777 Georg Cantor (1845-1918) 778 William Kingdon Clifford (1845-1879) 780 Gottlob Frege (1848-1925) 780 Christian Felix Klein (1849-1925) 782 Ferdinand Georg Frobenius (1849-1917) 783 Sofya (Sonya) Kovalevskaya (1850-1891) 784 William Burnside (1852-1927) 785 Jules Henri Poincare (1854-1912) 785 Giuseppe Peano (1858-1932) 787 David Hubert (1862-1943) 788 Hermann Minkowski (1864-1909) 789 Jacques Hadamard (1865-1963) 790 Ivar Fredholm (1866-1927) 791 Charles-Jean de la Vallee Poussin (1866-1962) 792 Felix Hausdorff (1868-1942) 792 Ehe Joseph Cartan (1869-1951) 794 Emile Borel (1871-1956) 795 Bertrand Arthur William Russell (1872-1970) 795 Henri Lebesgue (1875-1941) 796 Godfrey Harold Hardy (1877-1947) 797 Frigyes (Frederic) Riesz (1880-1956) 798 Luitzen Egbertus Jan Brouwer (1881-1966) 799 Emmy Noether (1882-1935) 800 Waclaw Sierpinski (1882-1969) 801 George Birkhoff (1884-1944) 802 John Edensor Littlewood (1885-1977) 803 Hermann Weyl (1885-1955) 805 Thoralf Skolem (1887-1963) 806 Srinivasa Ramanujan (1887-1920) 807 Richard Courant (1888-1972) 808 Stefan Banach (1892-1945) 809 Norbert Wiener (1894-1964) 811

VUl Contents

VI.86 VI.87 VI.88 VI.89 VI.90 VI.91 VI.92 VI.93 VI.94 VI.95 VI.96

Emil Artin (1898-1962) Alfred Tarsia (1901-1983) Andrei Nikolaevich Kolmogorov (1903-1987) Alonzo Church (1903-1995) William Vallance Douglas Hodge (1903-1975) John von Neumann (1903-1957) Kurt Gödel (1906-1978) Andre Weil (1906-1998) Alan Turing (1912-1954) Abraham Robinson (1918-1974) Nicolas Bourbaki (1935-)

812 813 814 816 816 817 819 819 821 822 823

Part VII The Influence of Mathematics vn.l Mathematics and Chemistry 827 VII.2 Mathematical Biology 837 VII. 3 Wavelets and Applications 848 VII.4 The Mathematics of Traffic in Networks 862 Vn.5 The Mathematics of Algorithm Design 871 Vn.6 Reliable Transmission of Information 878 VII. 7 Mathematics and Cryptography 887

Vn.8 Vn.9 VII. 10 m i l Vn.12 vn.i3 VII.14

Mathematics and Economic Reasoning The Mathematics of Money Mathematical Statistics Mathematics and Medical Statistics Analysis, Mathematical and Philosophical Mathematics and Music Mathematics and Art

895 910 916 921 928 935 944

PartVIII Final Perspectives

vmu vni.2 vni.3 vm.4 vni.5 vrn.6 VIII.7

The Art of Problem Solving "Why Mathematics?" You Might Ask The Ubiquity of Mathematics Numeracy Mathematics: An Experimental Science Advice to a Young Mathematician A Chronology of Mathematical Events

955 966 977 983 991

1000 1010

Index .1015