Princeton Companion to Mathematics

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.

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Part I IntroductionI.1 What Is Mathematics About? 1I.2 I.3 Some Fundamental Mathematical Definitions 16I.4 The General Goals of Mathematical Research 48

Part II The Origins of Modern MathematicsII.1 From Numbers to Number Systems 77II.2 II.3 The Development of Abstract Algebra 95II.4 Algorithms 106II.5 The Development of Rigor in Mathematical Analysis 117II.6 The Development of the Idea of Proof 129II.7 The Crisis in the Foundations of Mathematics 142

Part III Mathematical ConceptsIII.1 The Axiom of Choice 157III.2 The Axiom of Determinacy 159III.3 Bayesian Analysis 159III.4 Braid Groups 160III.5 Buildings 161III.6 Calabi-Yau Manifolds 163III.7 Cardinals 165III.8 Categories 165III.9 Compactness and Compactification 167III.10 Computational Complexity Classes 169III.11 Countable and Uncountable Sets 170III.12 C*-Algebras 172III.13 Curvature 172III.14 Designs 172III.15 Determinants 174III.16 Differential Forms and Integration 175III.17 Dimension 180III.18 Distributions 184III.19 Duality 187III.20 Dynamical Systems and Chaos 190III.21 Elliptic Curves 190III.22 The Euclidean Algorithm and Continued Fractions 191III.23 The Euler and Navier-Stokes Equations 193III.24 Expanders 196III.25 The Exponential and Logarithmic Functions 199III.26 The Fast Fourier Transform 202III.27 The Fourier Transform 204III.28 Fuchsian Groups 208III.29 Function Spaces 210III.30 Galois Groups 213III.31 The Gamma Function 213III.32 Generating Functions 214III.33 Genus 215III.34 Graphs 215III.35 Hamiltonians 215III.36 The Heat Equation 216III.37 Hilbert Spaces 219III.38 Homology and Cohomology 221III.39 Homotopy Groups 221III.40 The Ideal Class Group 221III.41 Irrational and Transcendental Numbers 222III.42 The Ising Model 223III.43 Jordan Normal Form 223III.44 Knot Polynomials 225III.45 K-Theory 227III.46 The Leech Lattice 227III.47 L-Functions 228III.48 Lie Theory 229III.49 Linear and Nonlinear Waves and Solitons 234III.50 Linear Operators and Their Properties 239III.51 Local and Global in Number Theory 241III.52 The Mandelbrot Set 244III.53 Manifolds 244III.54 Matroids 244III.55 Measures 246III.56 Metric Spaces 247III.57 Models of Set Theory 248III.58 Modular Arithmetic 249III.59 Modular Forms 250III.60 Moduli Spaces 252III.61 The Monster Group 252III.62 Normed Spaces and Banach Spaces 252III.63 Number Fields 254III.64 Optimization and Lagrange Multipliers 255III.65 Orbifolds 257III.66 Ordinals 258III.67 The Peano Axioms 258III.68 Permutation Groups 259III.69 Phase Transitions 261III.70 p 261III.71 Probability Distributions 263III.72 Projective Space 267III.73 Quadratic Forms 267III.74 Quantum Computation 269III.75 Quantum Groups 272III.76 Quaternions, Octonions, and Normed Division Algebras 275III.77 Representations 279III.78 Ricci Flow 279III.79 Riemann Surfaces 282III.80 The Riemann Zeta Function 283III.81 Rings, Ideals, and Modules 284III.82 Schemes 285III.83 The Schrödinger Equation 285III.84 The Simplex Algorithm 288III.85 Special Functions 290III.86 The Spectrum 294III.87 Spherical Harmonics 295III.88 Symplectic Manifolds 297III.89 Tensor Products 301III.90 Topological Spaces 301III.91 Transforms 303III.92 Trigonometric Functions 307III.93 Universal Covers 309III.94 Variational Methods 310III.95 Varieties 313III.96 Vector Bundles 313III.97 Von Neumann Algebras 313III.98 Wavelets 313III.99 The Zermelo-Fraenkel Axioms 314

Part IV Branches of MathematicsIV.1 Algebraic Numbers 315IV.2 Analytic Number Theory 332IV.3 Computational Number Theory 348IV.4 Algebraic Geometry 363IV.5 IV.6 Algebraic Topology 383IV.7 Differential Topology 396IV.8 Moduli Spaces 408IV.9 Representation Theory 419IV.10 Geometric and Combinatorial Group Theory 431IV.11 Harmonic Analysis 448IV.12 Partial Differential Equations 455IV.13 General Relativity and the Einstein Equations 483IV.14 Dynamics 493IV.15 Operator Algebras 510IV.16 Mirror Symmetry 523IV.17 Vertex Operator Algebras 539IV.18 Enumerative and Algebraic Combinatorics 550IV.19 Extremal and Probabilistic Combinatorics 562IV.20 Computational Complexity 575IV.21 IV.22 Set Theory 615IV.23 Logic and Model Theory 635IV.24 Stochastic Processes 647IV.25 Probabilistic Models of Critical Phenomena 657IV.26 High-Dimensional Geometry and Its Probabilistic Analogues 670

Part V Theorems and ProblemsV.1 The ABC Conjecture 681V.2 The Atiyah-Singer Index Theorem 681V.3 The Banach-Tarski Paradox 684V.4 The Birch-Swinnerton-Dyer Conjecture 685V.5 Carleson's Theorem 686V.6 The Central Limit Theorem 687V.7 The Classification of Finite Simple Groups 687V.8 Dirichlet's Theorem 689V.9 Ergodic Theorems 689V.10 V.11 Fixed Point Theorems 693V.12 The Four-Color Theorem 696V.13 The Fundamental Theorem of Algebra 698V.14 The Fundamental Theorem of Arithmetic 699V.15 Gödel's Theorem 700V.16 Gromov's Polynomial-Growth Theorem 702V.17 Hilbert's Nullstellensatz 703V.18 The Independence of the Continuum Hypothesis 703V.19 Inequalities 703V.20 The Insolubility of the Halting Problem 706V.21 The Insolubility of the Quintic 708V.22 Liouville's Theorem and Roth's Theorem 710V.23 Mostow's Strong Rigidity Theorem 711V.24 The P versus NP Problem 713V.25 The Poincaré Conjecture 714V.26 The Prime Number Theorem and the Riemann Hypothesis 714V.27 Problems and Results in Additive Number Theory 715V.28 From Quadratic Reciprocity to Class Field Theory 718V.29 Rational Points on Curves and the Mordell Conjecture 720V.30 The Resolution of Singularities 722V.31 The Riemann-Roch Theorem 723V.32 The Robertson-Seymour Theorem 725V.33 The Three-Body Problem 726V.34 The Uniformization Theorem 728V.35 The Weil Conjectures 729

Part VI MathematiciansVI.1 Pythagoras (ca. 569 B.C.E.-ca. 494 B.C.E.) 733VI.2 Euclid (ca. 325 B.C.E.-ca. 265 B.C.E.) 734VI.3 Archimedes (ca. 287 B.C.E.-212 B.C.E.) 734VI.4 Apollonius (ca. 262 B.C.E.-ca. 190 B.C.E.) 735VI.5 Abu Ja'far Muhammad ibn Musa al-Khwarizmi (800-847) 736VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170-ca. 1250) 737VI.7 Girolamo Cardano (1501-1576) 737VI.8 Rafael Bombelli (1526-after 1572) 737VI.9 François Viète (1540-1603) 737VI.10 Simon Stevin (1548-1620) 738VI.11 René Descartes (1596-1650) 739VI.12 Pierre Fermat (160?-1665) 740VI.13 Blaise Pascal (1623-1662) 741VI.14 Isaac Newton (1642-1727) 742VI.15 Gottfried Wilhelm Leibniz (1646-1716) 743VI.16 Brook Taylor (1685-1731) 745VI.17 Christian Goldbach (1690-1764) 745VI.18 The Bernoullis (fl. 18th century) 745VI.19 Leonhard Euler (1707-1783) 747VI.20 Jean Le Rond d'Alembert (1717-1783) 749VI.21 Edward Waring (ca. 1735-1798) 750VI.22 Joseph Louis Lagrange (1736-1813) 751VI.23 Pierre-Simon Laplace (1749-1827) 752VI.24 Adrien-Marie Legendre (1752-1833) 754VI.25 Jean-Baptiste Joseph Fourier (1768-1830) 755VI.26 Carl Friedrich Gauss (1777-1855) 755VI.27 Siméon-Denis Poisson (1781-1840) 757VI.28 Bernard Bolzano (1781-1848) 757VI.29 Augustin-Louis Cauchy (1789-1857) 758VI.30 August Ferdinand Möbius (1790-1868) 759VI.31 Nicolai Ivanovich Lobachevskii (1792-1856) 759VI.32 George Green (1793-1841) 760VI.33 Niels Henrik Abel (1802-1829) 760VI.34 János Bolyai (1802-1860) 762VI.35 Carl Gustav Jacob Jacobi (1804-1851) 762VI.36 Peter Gustav Lejeune Dirichlet (1805-1859) 764VI.37 William Rowan Hamilton (1805-1865) 765VI.38 Augustus De Morgan (1806-1871) 765VI.39 Joseph Liouville (1809-1882) 766VI.40 Eduard Kummer (1810-1893) 767VI.41 Évariste Galois (1811-1832) 767VI.42 James Joseph Sylvester (1814-1897) 768VI.43 George Boole (1815-1864) 769VI.44 Karl Weierstrass (1815-1897) 770VI.45 Pafnuty Chebyshev (1821-1894) 771VI.46 Arthur Cayley (1821-1895) 772VI.47 Charles Hermite (1822-1901) 773VI.48 Leopold Kronecker (1823-1891) 773VI.49 Georg Friedrich Bernhard Riemann (1826-1866) 774VI.50 Julius Wilhelm Richard Dedekind (1831-1916) 776VI.51 Émile Léonard Mathieu (1835-1890) 776VI.52 Camille Jordan (1838-1922) 777VI.53 Sophus Lie (1842-1899) 777VI.54 Georg Cantor (1845-1918) 778VI.55 William Kingdon Clifford (1845-1879) 780VI.56 Gottlob Frege (1848-1925) 780VI.57 Christian Felix Klein (1849-1925) 782VI.58 Ferdinand Georg Frobenius (1849-1917) 783VI.59 Sofya (Sonya) Kovalevskaya (1850-1891) 784VI.60 William Burnside (1852-1927) 785VI.61 [Illustration credit: Portrait courtesy of Henri Poincaré Archives (CNRS,UMR 7117, Nancy)]VI.62 Giuseppe Peano (1858-1932) 787VI.63 David Hilbert (1862-1943) 788VI.64 Hermann Minkowski (1864-1909) 789VI.65 Jacques Hadamard (1865-1963) 790VI.66 Ivar Fredholm (1866-1927) 791VI.67 Charles-Jean de la Vallée Poussin (1866-1962) 792VI.68 Felix Hausdorff (1868-1942) 792VI.69 Élie Joseph Cartan (1869-1951) 794VI.70 Emile Borel (1871-1956) 795VI.71 Bertrand Arthur William Russell (1872-1970) 795VI.72 Henri Lebesgue (1875-1941) 796VI.73 Godfrey Harold Hardy (1877-1947) 797VI.74 Frigyes (Frédéric) Riesz (1880-1956) 798VI.75 Luitzen Egbertus Jan Brouwer (1881-1966) 799VI.76 Emmy Noether (1882-1935) 800VI.77 Wac?aw Sierpinski (1882-1969) 801VI.78 George Birkhoff (1884-1944) 802VI.79 John Edensor Littlewood (1885-1977) 803VI.80 Hermann Weyl (1885-1955) 805VI.81 Thoralf Skolem (1887-1963) 806VI.82 Srinivasa Ramanujan (1887-1920) 807VI.83 Richard Courant (1888-1972) 808VI.84 Stefan Banach (1892-1945) 809VI.85 Norbert Wiener (1894-1964) 811VI.86 Emil Artin (1898-1962) 812VI.87 Alfred Tarski (1901-1983) 813VI.88 Andrei Nikolaevich Kolmogorov (1903-1987) 814VI.89 Alonzo Church (1903-1995) 816VI.90 William Vallance Douglas Hodge (1903-1975) 816VI.91 John von Neumann (1903-1957) 817VI.92 Kurt Gödel (1906-1978) 819VI.93 André Weil (1906-1998) 819VI.94 Alan Turing (1912-1954) 821VI.95 Abraham Robinson (1918-1974) 822VI.96 Nicolas Bourbaki (1935-) 823

Part VII The Influence of MathematicsVII.1 Mathematics and Chemistry 827VII.2 VII.3 Wavelets and Applications 848VII.4 The Mathematics of Traffic in Networks 862VII.5 The Mathematics of Algorithm Design 871VII.6 Reliable Transmission of Information 878VII.7 Mathematics and Cryptography 887VII.8 Mathematics and Economic Reasoning 895VII.9 The Mathematics of Money 910VII.10 Mathematical Statistics 916VII.11 Mathematics and Medical Statistics 921VII.12 Analysis, Mathematical and Philosophical 928VII.13 Mathematics and Music 935VII.14 Mathematics and Art 944

Part VIII Final PerspectivesVIII.1 The Art of Problem Solving 955VIII.2 "Why Mathematics?" You Might Ask 966VIII.3 The Ubiquity of Mathematics 977VIII.4 Numeracy 983VIII.5 Mathematics: An Experimental Science 991VIII.6 VIII.7 A Chronology of Mathematical Events 1010

Index 1015