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The Princeton companion to mathematics

Author: Gowers, Timothy ; Barrow-Green, June ; Leader, ImrePublisher: Princeton University Press, 2008.Language: EnglishDescription: 1034 p. : Graphs ; 26 cm.ISBN: 9780691118802Type of document: BookBibliography/Index: Includes bibliographical references and index
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Book Europe Campus
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Print QA40 .P75 2008
(Browse shelf)
001194913
Available 001194913
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Includes bibliographical references and index

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The Princeton Companion to Mathematics Contents Preface Contributors ix xvii III.15 Determinants III.16 Differential Forms and Integration III.17 Dimension III.18 Distributions III.19 Duality 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 Part I Introduction I.1 What Is Mathematics About? I.2 The Language and Grammar of Mathematics I.3 Some Fundamental Mathematical Definitions I.4 The General Goals of Mathematical Research 1 8 16 48 III.20 Dynamical Systems and Chaos III.21 Elliptic Curves III.22 The Euclidean Algorithm and Continued Fractions III.23 The Euler and Navier-Stokes Equations III.24 Expanders III.25 The Exponential and Logarithmic Functions III.26 The Fast Fourier Transform III.27 The Fourier Transform III.28 Fuchsian Groups III.29 Function Spaces 77 83 95 106 117 129 142 III.30 Galois Groups III.31 The Gamma Function III.32 Generating Functions III.33 Genus III.34 Graphs III.35 Hamiltonians III.36 The Heat Equation III.37 Hilbert Spaces III.38 Homology and Cohomology III.39 Homotopy Groups III.40 The Ideal Class Group 157 159 159 160 161 163 165 165 167 169 170 172 172 172 III.41 Irrational and Transcendental Numbers III.42 The Ising Model III.43 Jordan Normal Form III.44 Knot Polynomials III.45 K-Theory III.46 The Leech Lattice III.47 L-Functions III.48 Lie Theory III.49 Linear and Nonlinear Waves and Solitons III.50 Linear Operators and Their Properties III.51 Local and Global in Number Theory III.52 The Mandelbrot Set III.53 Manifolds III.54 Matroids III.55 Measures Part II The Origins of Modern Mathematics II.1 From Numbers to Number Systems II.2 Geometry II.3 The Development of Abstract Algebra II.4 Algorithms II.5 The Development of Rigor in Mathematical Analysis II.6 The Development of the Idea of Proof II.7 The Crisis in the Foundations of Mathematics Part III Mathematical Concepts III.1 The Axiom of Choice III.2 The Axiom of Determinacy III.3 Bayesian Analysis III.4 Braid Groups III.5 Buildings III.6 Calabi-Yau Manifolds III.7 Cardinals III.8 Categories III.9 Compactness and Compactification III.10 Computational Complexity Classes III.11 Countable and Uncountable Sets III.12 C*-Algebras III.13 Curvature III.14 Designs III.56 Metric Spaces III.57 Models of Set Theory III.58 Modular Arithmetic III.59 Modular Forms III.60 Moduli Spaces III.61 The Monster Group III.62 Normed Spaces and Banach Spaces III.63 Number Fields III.64 Optimization and Lagrange Multipliers III.65 Orbifolds III.66 Ordinals III.67 The Peano Axioms III.68 Permutation Groups III.69 Phase Transitions III.70 n III.71 Probability Distributions III.72 Projective Space III.73 Quadratic Forms III.74 Quantum Computation III.75 Quantum Groups III.76 Quaternions, Octonions, and Normed Division Algebras III.77 Representations III.78 Ricci Flow III.79 Riemann Surfaces III.80 The Riemann Zeta Function III.81 Rings, Ideals, and Modules III.82 Schemes III.83 The Schrödinger Equation III.84 The Simplex Algorithm III.85 Special Functions III.86 The Spectrum III.87 Spherical Harmonics III.88 Symplectic Manifolds III.89 Tensor Products III.90 Topological Spaces III.91 Transforms III.92 Trigonometric Functions III.93 Universal Covers III.94 Variational Methods III.95 Varieties III.96 Vector Bundles III.97 Von Neumann Algebras III.98 Wavelets III.99 The Zermelo-Fraenkel Axioms 247 248 249 250 252 252 252 254 255 257 258 258 259 261 261 263 267 267 269 272 275 279 279 282 283 284 285 285 288 290 294 295 297 301 301 303 307 309 310 313 313 313 313 314 IV.5 Arithmetic Geometry IV.6 Algebraic Topology IV.7 Differential Topology IV.8 Moduli Spaces IV.9 Representation Theory IV.10 Geometric and Combinatorial Group Theory IV.11 Harmonic Analysis IV.12 Partial Differential Equations IV.13 General Relativity and the Einstein Equations IV.14 Dynamics IV.15 Operator Algebras IV.16 Mirror Symmetry IV.17 Vertex Operator Algebras IV.18 Enumerative and Algebraic Combinatorics IV.19 Extremal and Probabilistic Combinatorics IV.20 Computational Complexity IV.21 Numerical Analysis IV.22 Set Theory IV.23 Logic and Model Theory IV.24 Stochastic Processes IV.25 Probabilistic Models of Critical Phenomena IV.26 High-Dimensional Geometry and Its Probabilistic Analogues 372 383 396 408 419 431 448 455 483 493 510 523 539 550 562 575 604 615 635 647 657 670 Part V Theorems and Problems V.1 The ABC Conjecture V.2 The Atiyah-Singer Index Theorem V.3 The Banach-Tarski Paradox V.4 The Birch-Swinnerton-Dyer Conjecture V.5 Carleson's Theorem V.6 The Central Limit Theorem V.7 The Classification of Finite Simple Groups V.8 Dirichlet's Theorem V.9 Ergodic Theorems V.10 Fermat's Last Theorem V.11 Fixed Point Theorems V.12 The Four-Color Theorem V.13 The Fundamental Theorem of Algebra V.14 The Fundamental Theorem of Arithmetic V.15 Gödel's Theorem V.16 Gromov's Polynomial-Growth Theorem V.17 Hilbert's Nullstellensatz V.18 The Independence of the Continuum Hypothesis V.19 Inequalities V.20 The Insolubility of the Halting Problem V.21 The Insolubility of the Quintic V.22 Liouville's Theorem and Roth's Theorem V.23 Mostow's Strong Rigidity Theorem V.24 The P versus NT Problem V.25 The Poincaré Conjecture 681 681 684 685 686 687 687 689 689 691 693 696 698 699 700 702 703 703 703 706 708 710 711 713 714 Part IV Branches of Mathematics IV.1 Algebraic Numbers IV.2 Analytic Number Theory IV.3 Computational Number Theory IV.4 Algebraic Geometry 315 332 348 363 V.26 The Prime Number Theorem and the Riemann Hypothesis V.27 Problems arid 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 Riemarm-Roch Theorem V.32 The Robertson-Seymour Theorem V.33 The Three-Body Problem V.34 The Uniformization Theorem V.35 The Weil Conjectures 714 715 718 720 722 723 725 726 728 729 Part VI Mathematicians VI.1 Pythagoras (ca. 569 B.c.E.-ca. 494 B.c.E.) VI.2 Euclid (ca. 325 B.c.E.-ca. 265 B.c.E.) VI.3 Archimedes (ca. 287 B.c.E.-212 B.c.E.) W.4 Apollonius (ca. 262 B.c.E.-ca.190 B.c.E.) VI.5 Abu Ja'far Muhammad ibn Musa al-Khwarizmi (800-847) W.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 François Viète (1540-1603) VI.10 Simon Stevin (1548-1620) VI.11 René Descartes (1596-1650) VI.12 Pierre Fermat (160?-1665) VI.13 Blaise Pascal (1623-1662) V1.14 Isaac Newton (1642-1727) W.15 Gottfried Wilhelm Leibniz (1646-1716) VI.16 Brook Taylor (1685-1731) VI.17 Christian Goldbach (1690-1764) VI.18 The Bernoullis (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) W.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) W.26 Carl Friedrich Gauss (1777-1855) VI.27 Siméon-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 Lobachevskii (1792-1856) VI.32 George Green (1793-1841) VI.33 Niels Henrik Abel (1802-1829) 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 Janos Bolyai (1802-1860) VI.35 Carl Gustav Jacob Jacobi (1804-1851) VI.36 Peter Gustav Lejeune Dirichlet (1805-1859) VI.37 William Rowan Hamilton (1805-1865) VI.38 Augustus De Morgan (1806-1871) VI.39 Joseph Liouville (1809-1882) VI.40 Eduard Kummer (1810-1893) VI.41 Evariste Galois (1811-1832) VI.42 James Joseph Sylvester (1814-1897) W.43 George Boole (1815-1864) VI.44 Karl Weierstrass (1815-1897) VI.45 Pafnuty Chebyshev (1821-1894) VI.46 Arthur Cayley (1821-1895) VI.47 Charles Hermite (1822-1901) VI.48 Leopold Kronecker (1823-1891) VI.49 Georg Friedrich Bernhard Riemann (1826-1866) VI.50 Julius Wilhelm Richard Dedekind (1831-1916) VI.51 Emile Leonard Mathieu (1835-1890) VI.52 Camille Jordan (1838-1922) VL53 Sophus Lie (1842-1899) VI.54 Georg Cantor (1845-1918) VI. 55 William Kingdon Clifford (1845-1879) VI.56 Gottlob Frege (1848-1925) VI.57 Christian Felix Klein (1849-1925) VI.58 Ferdinand Georg Frobenius (1849-1917) VI.59 Sofya (Sonya) Kovalevskaya (1850-1891) VI.60 William Burnside (1852-1927) VI.61 Jules Henri Poincarê (1854-1912) VI.62 Giuseppe Peano (1858-1932) VI.63 David Hilbert (1862-1943) VI.64 Hermann Minkowski (1864-1909) VI.65 Jacques Hadamard (1865-1963) VI.66 Ivar Fredholm (1866-1927) VI.67 Charles-Jean de la Vallee Poussin (1866-1962) 792 VI.68 Felix Hausdorff (1868-1942) VI.69 Elie Joseph Cartan (1869-1951) VI.70 Emile Borel (1871-1956) VI.71 Bertrand Arthur William Russell (1872-1970) 795 VI.72 Henri Lebesgue (1875-1941) VI.73 Godfrey Harold Hardy (1877-1947) VI.74 Frigyes (Frédéric) Riesz (1880-1956) VI.75 Luitzen Egbertus Jan Brouwer (1881-1966) VI.76 Emmy Noether (1882-1935) VI.77 Waclaw Sierpinski (1882-1969) VI.78 George Birkhoff (1884-1944) VI.79 John Edensor Littlewood (1885-1977) VI.80 Hermann Weyl (1885-1955) VI.81 Thoralf Skolem (1887-1963) VI.82 Srinivasa Ramanujan (1887-1920) VI.83 Richard Courant (1888-1972) VI.84 Stefan Banach (1892-1945) VI.85 Norbert Wiener (1894-1964) 762 762 764 765 765 766 767 767 768 769 770 771 772 773 773 774 776 776 777 777 778 780 780 782 783 784 785 785 787 788 789 790 791 792 794 795 796 797 798 799 800 801 802 803 805 806 807 808 809 811 VL86 Emil Artin (1898-1962) VI.87 Alfred Tarski (1901-1983) VI.88 Andrei Nikolaevich Kolmogorov (1903-1987) 814 VL89 Alonzo Church (1903-1995) VL90 William Valiance Douglas Hodge (1903-1975) 816 VI.91 John von Neumann (1903-1957) VI.92 Kurt GOdel (1906-1978) VI.93 Andre Weil (1906-1998) VI.94 Alan Turing (1912-1954) VI.95 Abraham Robinson (1918-1974) VI.96 Nicolas Bourbaki (1935-) 812 813 816 817 819 819 821 822 823 VII.8 Mathematics and Economic Reasoning VII.9 The Mathematics of Money VII.10 Mathematical Statistics VII.11 Mathematics and Medical Statistics VII.12 Analysis, Mathematical and Philosophical VII.13 Mathematics and Music VII.14 Mathematics and Art 895 910 916 921 928 935 944 Part VIII Final Perspectives VIII.1 The Art of Problem Solving VIII.2 "Why Mathematics?" You Might Ask 955 966 977 983 991 1000 1010 Part VII The Influence of Mathematics VII.1 Mathematics and Chemistry VII.2 Mathematical Biology VII.3 Wavelets and Applications VII.4 The Mathematics of Traffic in Networks VII.5 The Mathematics of Algorithm Design VII.6 Reliable Transmission of Information VII.7 Mathematics and Cryptography 827 837 848 862 871 878 887 VIII.3 The Ubiquity of Mathematics VIII.4 Numeracy Mathematics: An Experimental Science VIII.6 Advice to a Young Mathematician VIII.7 A Chronology of Mathematical Events Index 1015

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