The Princeton Companion to Mathematics / / ed. by Timothy Gowers, Imre Leader, June Barrow-Green.

• Superior document: Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013
• MitwirkendeR: Allan, G. R.,; Alon, Noga,; Andrews, George,; Archibald, Tom,; Atiyah, Michael,; Aubin, David,; Bagaria, Joan,; Ball, Keith,; Beardon, Alan F.,; Ben-Zvi, David D.,; Bergelson, Vitaly,; Bingham, Nicholas,; Bollobás, Béla,; Bos, Henk J. M.,; Branner, Bodil,; Bridson, Martin R.,; Burdman Feferman, Anita,; Burgess, John P.,; Buzzard, Kevin,; Cameron, Peter J.,; Chabert, Jean-Luc,; Cheng, Eugenia,; Cocks, Clifford,; Corry, Leo,; Coy, Wolfgang,; Crilly, Tony,; Cuomo, Serafina,; Dafermos, Mihalis,; Dasgupta, Partha,; Daubechies, Ingrid,; Dauben, Joseph W.,; Dawson, John W.,; Diaconis, Persi,; Ellenberg, Jordan S.,; Evans, Lawrence C.,; Fasanelli, Florence,; Feferman, Solomon,; Fefferman, Charles,; Fenster, Della,; Ferreirós, José,; Fisher, David,; Gannon, Terry,; Gardiner, A.,; Gillispie, Charles C.,; Goldreich, Oded,; Goldstein, Catherine,; Gouvêa, Fernando Q.,; Granville, Andrew,; Grattan-Guinness, Ivor,; Gray, Jeremy,; Green, Ben,; Grojnowski, Ian,; Guicciardini, Niccolò,; Harris, Michael,; Hashagen, Ulf,; Higson, Nigel,; Hodges, Andrew,; Hunger Parshall, Karen,; Johnson, F. E. A.,; Joshi, Mark,; Kedlaya, Kiran S.,; Kelly, Frank,; Klainerman, Sergiu,; Kleinberg, Jon,; Kleiner, Israel,; Klinowski, Jacek,; Knobloch, Eberhard,; Kollár, János,; Krivelevich, Michael,; Körner, T. W.,; Lax, Peter D.,; Le Gall, Jean-François,; Lickorish, W. B. R.,; Liebeck, Martin W.,; Lützen, Jesper,; MacHale,,; Mackay, Alan L.,; Majid, Shahn,; Maligranda, Lech,; Marker, David,; Mawhin, Jean,; Mazur, Barry,; McLarty, Colin,; Mohar, Bojan,; Neumann, Peter M.,; Nolan, Catherine,; Norris, James,; Osserman, Brian,; Palais, Richard S.,; Panza, Marco,; Paternain, Gabriel P.,; Peiffer, Jeanne,; Petri, Birgit,; Pomerance, Carl,; Pulte, Helmut,; Reed, Bruce,; Reed, Michael C.,; Rice, Adrian,; Robson, Eleanor,; Rodnianski, Igor,; Roe, John,; Ronan, Mark,; Sandifer, Edward,; Sauer, Tilman,; Schappacher, Norbert,; Schinzel, Andrzej,; Scholz, Erhard,; Siegmund-Schultze, Reinhard,; Slade, Gordon,; Spiegelhalter, David J.,; Stedall, Jacqueline,; Stubhaug, Arild,; Sudan, Madhu,; Tao, Terence,; Tappenden, Jamie,; Taubes, C. H.,; Thiele, Rüdiger,; Totaro, Burt,; Trefethen, Lloyd N.,; Weber, Richard,; Welsh, Dominic,; Wigderson, Avi,; Wilf, Herbert S.,; Wilkins, David,; Yandell, Benjamin H.,; Zaslow, Eric,; Zeilberger, Doron,; de Gandt, Francois,; van Dalen, Dirk,
• HerausgeberIn: Barrow-Green, June,; Gowers, Timothy,; Leader, Imre,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2010]; ©2009
• Year of Publication: 2010
• Edition: Core Textbook
• Language: English
• Physical Description: 1 online resource (1056 p.) :; Black-and-white illustrations throughout | Cross-references, bibliographies, index

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.Features nearly 200 entries, organized thematically and written by an international team of distinguished contributorsPresents major ideas and branches of pure mathematics in a clear, accessible styleDefines and explains important mathematical concepts, methods, theorems, and open problemsIntroduces the language of mathematics and the goals of mathematical researchCovers number theory, algebra, analysis, geometry, logic, probability, and moreTraces the history and development of modern mathematicsProfiles more than ninety-five mathematicians who influenced those working todayExplores the influence of mathematics on other disciplinesIncludes bibliographies, cross-references, and a comprehensive indexContributors incude:Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger

Table of Contents

Frontmatter -- Contents -- Preface -- Contributors -- 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 -- 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.15 Determinants -- III.16 Differential Forms and Integration -- III.17 Dimension -- III.18 Distributions -- III.19 Duality -- 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 -- 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 -- 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 -- 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 π -- 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 -- Part IV. Branches of Mathematics -- IV.1 Algebraic Numbers -- IV.2 Analytic Number Theory -- IV.3 Computational Number Theory -- IV.4 Algebraic Geometry -- 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 -- 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 NP Problem -- V.25 The Poincaré Conjecture -- 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 -- VI.2 Euclid -- VI.3 Archimedes -- VI.4 Apollonius -- VI.5 Abu Ja'far Muhammad ibn Mūsā al-Khwārizmī -- VI.6 Leonardo of Pisa (known as Fibonacci) -- VI.7 Girolamo Cardano -- VI.8 Rafael Bombelli -- VI.9 François Viète -- VI.10 Simon Stevin -- VI.11 René Descartes -- VI.12 Pierre Fermat -- VI.13 Blaise Pascal -- VI.14 Isaac Newton -- VI.15 Gottfried Wilhelm Leibniz -- VI.16 Brook Taylor -- VI.17 Christian Goldbach -- VI.18 The Bernoullis -- VI.19 Leonhard Euler -- VI.20 Jean Le Rond d'Alembert -- VI.21 Edward Waring -- VI.22 Joseph Louis Lagrange -- VI.23 Pierre-Simon Laplace -- VI.24 Adrien-Marie Legendre -- VI.25 Jean-Baptiste Joseph Fourier -- VI.26 Carl Friedrich Gauss -- VI.27 Siméon-Denis Poisson -- VI.28 Bernard Bolzano -- VI.29 Augustin-Louis Cauchy -- VI.30 August Ferdinand Möbius -- VI.31 Nicolai Ivanovich Lobachevskii -- VI.32 George Green -- VI.33 Niels Henrik Abel -- VI.34 János Bolyai -- VI.35 Carl Gustav Jacob Jacobi -- VI.36 Peter Gustav Lejeune Dirichlet -- VI.37 William Rowan Hamilton -- VI.38 Augustus De Morgan -- VI.39 Joseph Liouville -- VI.40 Ernst Eduard Kummer -- VI.41 Évariste Galois -- VI.42 James Joseph Sylvester -- VI.43 George Boole -- VI.44 Karl Weierstrass -- VI.45 Pafnuty Chebyshev -- VI.46 Arthur Cayley -- VI.47 Charles Hermite -- VI.48 Leopold Kronecker -- VI.49 Georg Friedrich Bernhard Riemann -- VI.50 Julius Wilhelm Richard Dedekind -- VI.51 Émile Léonard Mathieu -- VI.52 Camille Jordan -- VI.53 Sophus Lie -- VI.54 Georg Cantor -- VI.55 William Kingdon Clifford -- VI.56 Gottlob Frege -- VI.57 Christian Felix Klein -- VI.58 Ferdinand Georg Frobenius -- VI.59 Sofya (Sonya) Kovalevskaya -- VI.60 William Burnside -- VI.61 Jules Henri Poincaré -- VI.62 Giuseppe Peano -- VI.63 David Hilbert -- VI.64 Hermann Minkowski -- VI.65 Jacques Hadamard -- VI.66 Ivar Fredholm -- VI.67 Charles-Jean de la Vallée Poussin -- VI.68 Felix Hausdorff -- VI.69 Élie Joseph Cartan -- VI.70 Emile Borel -- VI.71 Bertrand Arthur William Russell -- VI.72 Henri Lebesgue -- VI.73 Godfrey Harold Hardy -- VI.74 Frigyes (Frédéric) Riesz -- VI.75 Luitzen Egbertus Jan Brouwer -- VI.76 Emmy Noether -- VI.77 Wacław Sierpiński -- VI.78 George Birkhoff -- VI.79 John Edensor Littlewood -- VI.80 Hermann Weyl -- VI.81 Thoralf Skolem -- VI.82 Srinivasa Ramanujan -- VI.83 Richard Courant -- VI.84 Stefan Banach -- VI.85 Norbert Wiener -- VI.86 Emil Artin -- VI.87 Alfred Tarski -- VI.88 Andrei Nikolaevich Kolmogorov -- VI.89 Alonzo Church -- VI.90 William Vallance Douglas Hodge -- VI.91 John von Neumann -- VI.92 Kurt Gödel -- VI.93 André Weil -- VI.94 Alan Turing -- VI.95 Abraham Robinson -- VI.96 Nicolas Bourbaki -- 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 -- 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 -- Part VIII. Final Perspectives -- VIII.1 The Art of Problem Solving -- VIII.2 "Why Mathematics?" You Might Ask -- VIII.3 The Ubiquity of Mathematics -- VIII.4 Numeracy -- VIII.5 Mathematics: An Experimental Science -- VIII.6 Advice to a Young Mathematician -- VIII.7 A Chronology of Mathematical Events -- Index