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WorldCat.org

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Presents 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, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible 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

Preface ix   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 48      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 95  II.4 Algorithms 106  II.5 The Development of Rigor in Mathematical Analysis 117  II.6 The Development of the Idea of Proof 129  II.7 The Crisis in the Foundations of Mathematics 142     Part III Mathematical Concepts  III.1 The Axiom of Choice 157  III.2 The Axiom of Determinacy 159  III.3 Bayesian Analysis 159  III.4 Braid Groups 160  III.5 Buildings 161  III.6 Calabi-Yau Manifolds 163  III.7 Cardinals 165  III.8 Categories 165  III.9 Compactness and Compactification 167  III.10 Computational Complexity Classes 169  III.11 Countable and Uncountable Sets 170  III.12 C*-Algebras 172  III.13 Curvature 172  III.14 Designs 172  III.15 Determinants 174  III.16 Differential Forms and Integration 175  III.17 Dimension 180  III.18 Distributions 184  III.19 Duality 187  III.20 Dynamical Systems and Chaos 190  III.21 Elliptic Curves 190  III.22 The Euclidean Algorithm and Continued Fractions 191  III.23 The Euler and Navier-Stokes Equations 193  III.24 Expanders 196  III.25 The Exponential and Logarithmic Functions 199  III.26 The Fast Fourier Transform 202  III.27 The Fourier Transform 204  III.28 Fuchsian Groups 208  III.29 Function Spaces 210  III.30 Galois Groups 213  III.31 The Gamma Function 213  III.32 Generating Functions 214  III.33 Genus 215  III.34 Graphs 215  III.35 Hamiltonians 215  III.36 The Heat Equation 216  III.37 Hilbert Spaces 219  III.38 Homology and Cohomology 221  III.39 Homotopy Groups 221  III.40 The Ideal Class Group 221  III.41 Irrational and Transcendental Numbers 222  III.42 The Ising Model 223  III.43 Jordan Normal Form 223  III.44 Knot Polynomials 225  III.45 K-Theory 227  III.46 The Leech Lattice 227  III.47 L-Functions 228  III.48 Lie Theory 229  III.49 Linear and Nonlinear Waves and Solitons 234  III.50 Linear Operators and Their Properties 239  III.51 Local and Global in Number Theory 241  III.52 The Mandelbrot Set 244  III.53 Manifolds 244  III.54 Matroids 244  III.55 Measures 246  III.56 Metric Spaces 247  III.57 Models of Set Theory 248  III.58 Modular Arithmetic 249  III.59 Modular Forms 250  III.60 Moduli Spaces 252  III.61 The Monster Group 252  III.62 Normed Spaces and Banach Spaces 252  III.63 Number Fields 254  III.64 Optimization and Lagrange Multipliers 255  III.65 Orbifolds 257  III.66 Ordinals 258  III.67 The Peano Axioms 258  III.68 Permutation Groups 259  III.69 Phase Transitions 261  III.70 p 261  III.71 Probability Distributions 263  III.72 Projective Space 267  III.73 Quadratic Forms 267  III.74 Quantum Computation 269  III.75 Quantum Groups 272  III.76 Quaternions, Octonions, and Normed Division Algebras 275  III.77 Representations 279  III.78 Ricci Flow 279  III.79 Riemann Surfaces 282  III.80 The Riemann Zeta Function 283  III.81 Rings, Ideals, and Modules 284  III.82 Schemes 285  III.83 The Schrodinger Equation 285  III.84 The Simplex Algorithm 288  III.85 Special Functions 290  III.86 The Spectrum 294  III.87 Spherical Harmonics 295  III.88 Symplectic Manifolds 297  III.89 Tensor Products 301  III.90 Topological Spaces 301  III.91 Transforms 303  III.92 Trigonometric Functions 307  III.93 Universal Covers 309  III.94 Variational Methods 310  III.95 Varieties 313  III.96 Vector Bundles 313  III.97 Von Neumann Algebras 313  III.98 Wavelets 313  III.99 The Zermelo-Fraenkel Axioms 314     Part IV Branches of Mathematics  IV.1 Algebraic Numbers 315  IV.2 Analytic Number Theory 332  IV.3 Computational Number Theory 348  IV.4 Algebraic Geometry 363  IV.5  Arithmetic Geometry 372  IV.6 Algebraic Topology 383  IV.7 Differential Topology 396  IV.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  IV.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.23 Logic and Model Theory 635  IV.24 Stochastic Processes 647  IV.25 Probabilistic Models of Critical Phenomena 657  IV.26 High-Dimensional Geometry and Its Probabilistic Analogues 670     Part V Theorems and Problems  V.1 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.11 Fixed Point Theorems 693  V.12 The Four-Color Theorem 696  V.13 The Fundamental Theorem of Algebra 698  V.14 The Fundamental Theorem of Arithmetic 699  V.15 Godel's Theorem 700  V.16 Gromov's Polynomial-Growth Theorem 702  V.17 Hilbert's Nullstellensatz 703  V.18 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 P versus NP Problem 713  V.25 The Poincare Conjecture 714  V.26 The Prime Number Theorem and the Riemann Hypothesis 714  V.27 Problems and Results in Additive Number Theory 715  V.28 From Quadratic Reciprocity to Class Field Theory 718  V.29 Rational Points on Curves and the Mordell Conjecture 720  V.30 The Resolution of Singularities 722  V.31 The Riemann-Roch Theorem 723  V.32 The Robertson-Seymour Theorem 725  V.33 The Three-Body Problem 726  V.34 The Uniformization Theorem 728  V.35 The Weil Conjectures 729     Part VI Mathematicians  VI.1 Pythagoras (ca. 569 B.C.E.-ca. 494 B.C.E.) 733  VI.2 Euclid (ca. 325 B.C.E.-ca. 265 B.C.E.) 734  VI.3 Archimedes (ca. 287 B.C.E.-212 B.C.E.) 734  VI.4 Apollonius (ca. 262 B.C.E.-ca. 190 B.C.E.) 735  VI.5 Abu Ja'far Muhammad ibn Musa al-Khwarizmi (800-847) 736  VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170-ca. 1250) 737  VI.7 Girolamo Cardano (1501-1576) 737  VI.8 Rafael Bombelli (1526-after 1572) 737  VI.9 Francois Viete (1540-1603) 737  VI.10 Simon Stevin (1548-1620) 738  VI.11 Rene Descartes (1596-1650) 739  VI.12 Pierre Fermat (160?-1665) 740  VI.13 Blaise Pascal (1623-1662) 741  VI.14 Isaac Newton (1642-1727) 742  VI.15 Gottfried Wilhelm Leibniz (1646-1716) 743  VI.16 Brook Taylor (1685-1731) 745  VI.17 Christian Goldbach (1690-1764) 745  VI.18 The Bernoullis (fl. 18th century) 745  VI.19 Leonhard Euler (1707-1783) 747  VI.20 Jean Le Rond d'Alembert (1717-1783) 749  VI.21 Edward Waring (ca. 1735-1798) 750  VI.22 Joseph Louis Lagrange (1736-1813) 751  VI.23 Pierre-Simon Laplace (1749-1827) 752  VI.24 Adrien-Marie Legendre (1752-1833) 754  VI.25 Jean-Baptiste Joseph Fourier (1768-1830) 755  VI.26 Carl Friedrich Gauss (1777-1855) 755  VI.27 Simeon-Denis Poisson (1781-1840) 757  VI.28 Bernard Bolzano (1781-1848) 757  VI.29 Augustin-Louis Cauchy (1789-1857) 758  VI.30 August Ferdinand Mobius (1790-1868) 759  VI.31 Nicolai Ivanovich Lobachevskii (1792-1856) 759  VI.32 George Green (1793-1841) 760  VI.33 Niels Henrik Abel (1802-1829) 760  VI.34 Janos Bolyai (1802-1860) 762  VI.35 Carl Gustav Jacob Jacobi (1804-1851) 762  VI.36 Peter Gustav Lejeune Dirichlet (1805-1859) 764  VI.37 William Rowan Hamilton (1805-1865) 765  VI.38 Augustus De Morgan (1806-1871) 765  VI.39 Joseph Liouville (1809-1882) 766  VI.40 Eduard Kummer (1810-1893) 767  VI.41 Evariste Galois (1811-1832) 767  VI.42 James Joseph Sylvester (1814-1897) 768  VI.43 George Boole (1815-1864) 769  VI.44 Karl Weierstrass (1815-1897) 770  VI.45 Pafnuty Chebyshev (1821-1894) 771  VI.46 Arthur Cayley (1821-1895) 772  VI.47 Charles Hermite (1822-1901) 773  VI.48 Leopold Kronecker (1823-1891) 773  VI.49 Georg Friedrich Bernhard Riemann (1826-1866) 774  VI.50 Julius Wilhelm Richard Dedekind (1831-1916) 776  VI.51 Emile Leonard Mathieu (1835-1890) 776  VI.52 Camille Jordan (1838-1922) 777  VI.53 Sophus Lie (1842-1899) 777  VI.54 Georg Cantor (1845-1918) 778  VI.55 William Kingdon Clifford (1845-1879) 780  VI.56 Gottlob Frege (1848-1925) 780  VI.57 Christian Felix Klein (1849-1925) 782  VI.58 Ferdinand Georg Frobenius (1849-1917) 783  VI.59 Sofya (Sonya) Kovalevskaya (1850-1891) 784  VI.60 William Burnside (1852-1927) 785  VI.61  Jules Henri Poincare (1854-1912) 785 [Illustration credit: Portrait courtesy of Henri Poincare Archives (CNRS,UMR 7117, Nancy)]  VI.62 Giuseppe Peano (1858-1932) 787  VI.63 David Hilbert (1862-1943) 788  VI.64 Hermann Minkowski (1864-1909) 789  VI.65 Jacques Hadamard (1865-1963) 790  VI.66 Ivar Fredholm (1866-1927) 791  VI.67 Charles-Jean de la Vallee Poussin (1866-1962) 792  VI.68 Felix Hausdorff (1868-1942) 792  VI.69 Elie Joseph Cartan (1869-1951) 794  VI.70 Emile Borel (1871-1956) 795  VI.71 Bertrand Arthur William Russell (1872-1970) 795  VI.72 Henri Lebesgue (1875-1941) 796  VI.73 Godfrey Harold Hardy (1877-1947) 797  VI.74 Frigyes (Frederic) Riesz (1880-1956) 798  VI.75 Luitzen Egbertus Jan Brouwer (1881-1966) 799  VI.76 Emmy Noether (1882-1935) 800  VI.77 Wac?aw Sierpinski (1882-1969) 801  VI.78 George Birkhoff (1884-1944) 802  VI.79 John Edensor Littlewood (1885-1977) 803  VI.80 Hermann Weyl (1885-1955) 805  VI.81 Thoralf Skolem (1887-1963) 806  VI.82 Srinivasa Ramanujan (1887-1920) 807  VI.83 Richard Courant (1888-1972) 808  VI.84 Stefan Banach (1892-1945) 809  VI.85 Norbert Wiener (1894-1964) 811  VI.86 Emil Artin (1898-1962) 812  VI.87 Alfred Tarski (1901-1983) 813  VI.88 Andrei Nikolaevich Kolmogorov (1903-1987) 814  VI.89 Alonzo Church (1903-1995) 816  VI.90 William Vallance Douglas Hodge (1903-1975) 816  VI.91 John von Neumann (1903-1957) 817  VI.92 Kurt Godel (1906-1978) 819  VI.93 Andre Weil (1906-1998) 819  VI.94 Alan Turing (1912-1954) 821  VI.95 Abraham Robinson (1918-1974) 822  VI.96 Nicolas Bourbaki (1935-) 823     Part VII The Influence of Mathematics  VII.1 Mathematics and Chemistry 827  VII.2  Mathematical Biology 837  VII.3 Wavelets and Applications 848  VII.4 The Mathematics of Traffic in Networks 862  VII.5 The Mathematics of Algorithm Design 871  VII.6 Reliable Transmission of Information 878  VII.7 Mathematics and Cryptography 887  VII.8 Mathematics and Economic Reasoning 895  VII.9 The Mathematics of Money 910  VII.10 Mathematical Statistics 916  VII.11 Mathematics and Medical Statistics 921  VII.12 Analysis, Mathematical and Philosophical 928  VII.13 Mathematics and Music 935  VII.14 Mathematics and Art 944     Part VIII Final Perspectives  VIII.1 The Art of Problem Solving 955  VIII.2 "Why Mathematics?" You Might Ask 966  VIII.3 The Ubiquity of Mathematics 977  VIII.4 Numeracy 983  VIII.5 Mathematics: An Experimental Science 991  VIII.6  Advice to a Young Mathematician 1000  VIII.7 A Chronology of Mathematical Events 1010    Index 1015

1 online resource (292 entries) : 5346 images, digital files

Presents two hundred entries that introduce basic mathematical tools and vocabulary. This title traces the development of modern mathematics. It explains essential terms and concepts; examines core ideas in major areas of mathematics; and describes the achievements of scores of famous mathematicians.

The Princeton companion to mathematics