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

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Princeton, NJ : Princeton University Press, [2010]
1 online resource (1057 p.)
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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
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 / Gouvêa, Fernando Q.
II.2 Geometry / Gray, Jeremy
II.3 The Development of Abstract Algebra / Hunger Parshall, Karen
II.4 Algorithms / Chabert, Jean-Luc
II.5 The Development of Rigor in Mathematical Analysis / Archibald, Tom
II.6 The Development of the Idea of Proof / Corry, Leo
II.7 The Crisis in the Foundations of Mathematics / Ferreirós, José
Part III. Mathematical Concepts
III.1 The Axiom of Choice
III.2 The Axiom of Determinacy
III.3 Bayesian Analysis
III.4 Braid Groups / Johnson, F. E. A.
III.5 Buildings / Ronan, Mark
III.6 Calabi-Yau Manifolds / Zaslow, Eric
III.7 Cardinals
III.8 Categories / Cheng, Eugenia
III.9 Compactness and Compactification / Tao, Terence
III.10 Computational Complexity Classes
III.11 Countable and Uncountable Sets
III.12 C*-Algebras
III.13 Curvature
III.14 Designs / Cameron, Peter J.
III.15 Determinants
III.16 Differential Forms and Integration / Tao, Terence
III.17 Dimension
III.18 Distributions / Tao, Terence
III.19 Duality
III.20 Dynamical Systems and Chaos
III.21 Elliptic Curves / Ellenberg, Jordan S.
III.22 The Euclidean Algorithm and Continued Fractions / Ball, Keith
III.23 The Euler and Navier-Stokes Equations / Fefferman, Charles
III.24 Expanders / Wigderson, Avi
III.25 The Exponential and Logarithmic Functions
III.26 The Fast Fourier Transform
III.27 The Fourier Transform / Tao, Terence
III.28 Fuchsian Groups / Gray, Jeremy
III.29 Function Spaces / Tao, Terence
III.30 Galois Groups
III.31 The Gamma Function / Green, Ben
III.32 Generating Functions
III.33 Genus
III.34 Graphs
III.35 Hamiltonians / Tao, Terence
III.36 The Heat Equation / Rodnianski, Igor
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 / Green, Ben
III.42 The Ising Model
III.43 Jordan Normal Form
III.44 Knot Polynomials / Lickorish, W. B. R.
III.45 K-Theory
III.46 The Leech Lattice
III.47 L-Functions / Buzzard, Kevin
III.48 Lie Theory / Ronan, Mark
III.49 Linear and Nonlinear Waves and Solitons / Palais, Richard S.
III.50 Linear Operators and Their Properties
III.51 Local and Global in Number Theory / Gouvêa, Fernando Q.
III.52 The Mandelbrot Set
III.53 Manifolds
III.54 Matroids / Welsh, Dominic
III.55 Measures
III.56 Metric Spaces
III.57 Models of Set Theory
III.58 Modular Arithmetic / Green, Ben
III.59 Modular Forms / Buzzard, Kevin
III.60 Moduli Spaces
III.61 The Monster Group
III.62 Normed Spaces and Banach Spaces
III.63 Number Fields / Green, Ben
III.64 Optimization and Lagrange Multipliers / Ball, Keith
III.65 Orbifolds
III.66 Ordinals
III.67 The Peano Axioms
III.68 Permutation Groups / Liebeck, Martin W.
III.69 Phase Transitions
III.70 π
III.71 Probability Distributions / Norris, James
III.72 Projective Space
III.73 Quadratic Forms / Green, Ben
III.74 Quantum Computation
III.75 Quantum Groups / Majid, Shahn
III.76 Quaternions, Octonions, and Normed Division Algebras
III.77 Representations
III.78 Ricci Flow / Tao, Terence
III.79 Riemann Surfaces / Beardon, Alan F.
III.80 The Riemann Zeta Function
III.81 Rings, Ideals, and Modules
III.82 Schemes / Ellenberg, Jordan S.
III.83 The Schrödinger Equation / Tao, Terence
III.84 The Simplex Algorithm / Weber, Richard
III.85 Special Functions / Körner, T. W.
III.86 The Spectrum / Allan, G. R.
III.87 Spherical Harmonics
III.88 Symplectic Manifolds / Paternain, Gabriel P.
III.89 Tensor Products
III.90 Topological Spaces / Green, Ben
III.91 Transforms / Körner, T. W.
III.92 Trigonometric Functions / Green, Ben
III.93 Universal Covers
III.94 Variational Methods / Evans, Lawrence C.
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 / Mazur, Barry
IV.2 Analytic Number Theory / Granville, Andrew
IV.3 Computational Number Theory / Pomerance, Carl
IV.4 Algebraic Geometry / Kollár, János
IV.5 Arithmetic Geometry / Ellenberg, Jordan S.
IV.6 Algebraic Topology / Totaro, Burt
IV.7 Differential Topology / Taubes, C. H.
IV.8 Moduli Spaces / Ben-Zvi, David D.
IV.9 Representation Theory / Grojnowski, Ian
IV.10 Geometric and Combinatorial Group Theory / Bridson, Martin R.
IV.11 Harmonic Analysis / Tao, Terence
IV.12 Partial Differential Equations / Klainerman, Sergiu
IV.13 General Relativity and the Einstein Equations / Dafermos, Mihalis
IV.14 Dynamics / Branner, Bodil
IV.15 Operator Algebras / Higson, Nigel / Roe, John
IV.16 Mirror Symmetry / Zaslow, Eric
IV.17 Vertex Operator Algebras / Gannon, Terry
IV.18 Enumerative and Algebraic Combinatorics / Zeilberger, Doron
IV.19 Extremal and Probabilistic Combinatorics / Alon, Noga / Krivelevich, Michael
IV.20 Computational Complexity / Goldreich, Oded / Wigderson, Avi
IV.21 Numerical Analysis / Trefethen, Lloyd N.
IV.22 Set Theory / Bagaria, Joan
IV.23 Logic and Model Theory / Marker, David
IV.24 Stochastic Processes / Le Gall, Jean-François
IV.25 Probabilistic Models of Critical Phenomena / Slade, Gordon
IV.26 High-Dimensional Geometry and Its Probabilistic Analogues / Ball, Keith
Part V. Theorems and Problems
V.1 The ABC Conjecture
V.2 The Atiyah-Singer Index Theorem / Higson, Nigel / Roe, John
V.3 The Banach-Tarski Paradox / Körner, T. W.
V.4 The Birch-Swinnerton-Dyer Conjecture
V.5 Carleson's Theorem / Fefferman, Charles
V.6 The Central Limit Theorem
V.7 The Classification of Finite Simple Groups / Liebeck, Martin W.
V.8 Dirichlet's Theorem
V.9 Ergodic Theorems / Bergelson, Vitaly
V.10 Fermat's Last Theorem
V.11 Fixed Point Theorems
V.12 The Four-Color Theorem / Mohar, Bojan
V.13 The Fundamental Theorem of Algebra
V.14 The Fundamental Theorem of Arithmetic
V.15 Gödel's Theorem / Cameron, Peter J.
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 / Liebeck, Martin W.
V.22 Liouville's Theorem and Roth's Theorem
V.23 Mostow's Strong Rigidity Theorem / Fisher, David
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 / Kedlaya, Kiran S.
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 / Reed, Bruce
V.33 The Three-Body Problem
V.34 The Uniformization Theorem
V.35 The Weil Conjectures / Osserman, Brian
Part VI.
VI.1 Pythagoras / Cuomo, Serafina
VI.2 Euclid / Cuomo, Serafina
VI.3 Archimedes / Cuomo, Serafina
VI.4 Apollonius / Cuomo, Serafina
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 / Stedall, Jacqueline
VI.10 Simon Stevin
VI.11 René Descartes / Bos, Henk J. M.
VI.12 Pierre Fermat / Goldstein, Catherine
VI.13 Blaise Pascal
VI.14 Isaac Newton / Guicciardini, Niccolò
VI.15 Gottfried Wilhelm Leibniz / Knobloch, Eberhard
VI.16 Brook Taylor
VI.17 Christian Goldbach
VI.18 The Bernoullis / Peiffer, Jeanne
VI.19 Leonhard Euler / Sandifer, Edward
VI.20 Jean Le Rond d'Alembert / de Gandt, Francois
VI.21 Edward Waring
VI.22 Joseph Louis Lagrange / Panza, Marco
VI.23 Pierre-Simon Laplace / Gillispie, Charles C.
VI.24 Adrien-Marie Legendre / Grattan-Guinness, Ivor
VI.25 Jean-Baptiste Joseph Fourier / Grattan-Guinness, Ivor
VI.26 Carl Friedrich Gauss / Gray, Jeremy
VI.27 Siméon-Denis Poisson / Grattan-Guinness, Ivor
VI.28 Bernard Bolzano
VI.29 Augustin-Louis Cauchy / Grattan-Guinness, Ivor
VI.30 August Ferdinand Möbius / Gray, Jeremy
VI.31 Nicolai Ivanovich Lobachevskii / Gray, Jeremy
VI.32 George Green
VI.33 Niels Henrik Abel / Neumann, Peter M.
VI.34 János Bolyai / Gray, Jeremy
VI.35 Carl Gustav Jacob Jacobi / Pulte, Helmut
VI.36 Peter Gustav Lejeune Dirichlet / Hashagen, Ulf
VI.37 William Rowan Hamilton / Wilkins, David
VI.38 Augustus De Morgan
VI.39 Joseph Liouville / Lützen, Jesper
VI.40 Ernst Eduard Kummer
VI.41 Évariste Galois / Neumann, Peter M.
VI.42 James Joseph Sylvester / Hunger Parshall, Karen
VI.43 George Boole / MacHale
VI.44 Karl Weierstrass / Kleiner, Israel
VI.45 Pafnuty Chebyshev
VI.46 Arthur Cayley / Crilly, Tony
VI.47 Charles Hermite / Archibald, Tom
VI.48 Leopold Kronecker / Schappacher, Norbert / Petri, Birgit
VI.49 Georg Friedrich Bernhard Riemann / Gray, Jeremy
VI.50 Julius Wilhelm Richard Dedekind / Ferreirós, José
VI.51 Émile Léonard Mathieu
VI.52 Camille Jordan
VI.53 Sophus Lie / Stubhaug, Arild
VI.54 Georg Cantor / Dauben, Joseph W.
VI.55 William Kingdon Clifford / Gray, Jeremy
VI.56 Gottlob Frege / Tappenden, Jamie
VI.57 Christian Felix Klein / Thiele, Rüdiger
VI.58 Ferdinand Georg Frobenius / Neumann, Peter M.
VI.59 Sofya (Sonya) Kovalevskaya
VI.60 William Burnside / Neumann, Peter M.
VI.61 Jules Henri Poincaré
VI.62 Giuseppe Peano / Ferreirós, José
VI.63 David Hilbert / Yandell, Benjamin H.
VI.64 Hermann Minkowski / Sauer, Tilman
VI.65 Jacques Hadamard
VI.66 Ivar Fredholm
VI.67 Charles-Jean de la Vallée Poussin / Mawhin, Jean
VI.68 Felix Hausdorff / Scholz, Erhard
VI.69 Élie Joseph Cartan / Gray, Jeremy
VI.70 Emile Borel
VI.71 Bertrand Arthur William Russell / Grattan-Guinness, Ivor
VI.72 Henri Lebesgue / Siegmund-Schultze, Reinhard
VI.73 Godfrey Harold Hardy / Bollobás, Béla
VI.74 Frigyes (Frédéric) Riesz / Grattan-Guinness, Ivor
VI.75 Luitzen Egbertus Jan Brouwer / van Dalen, Dirk
VI.76 Emmy Noether / McLarty, Colin
VI.77 Wacław Sierpiński / Schinzel, Andrzej
VI.78 George Birkhoff
VI.79 John Edensor Littlewood / Bollobás, Béla
VI.80 Hermann Weyl / Scholz, Erhard
VI.81 Thoralf Skolem / Gray, Jeremy
VI.82 Srinivasa Ramanujan / Andrews, George
VI.83 Richard Courant / Lax, Peter D.
VI.84 Stefan Banach / Maligranda, Lech
VI.85 Norbert Wiener / Siegmund-Schultze, Reinhard
VI.86 Emil Artin / Fenster, Della
VI.87 Alfred Tarski / Burdman Feferman, Anita / Feferman, Solomon
VI.88 Andrei Nikolaevich Kolmogorov / Bingham, Nicholas
VI.89 Alonzo Church
VI.90 William Vallance Douglas Hodge / Atiyah, Michael
VI.91 John von Neumann / Coy, Wolfgang
VI.92 Kurt Gödel / Dawson, John W.
VI.93 André Weil / Schappacher, Norbert / Petri, Birgit
VI.94 Alan Turing / Hodges, Andrew
VI.95 Abraham Robinson / Dauben, Joseph W.
VI.96 Nicolas Bourbaki / Aubin, David
Part VII. The Influence of Mathematics
VII.1 Mathematics and Chemistry / Klinowski, Jacek / Mackay, Alan L.
VII.2 Mathematical Biology / Reed, Michael C.
VII.3 Wavelets and Applications / Daubechies, Ingrid
VII.4 The Mathematics of Traffic in Networks / Kelly, Frank
VII.5 The Mathematics of Algorithm Design / Kleinberg, Jon
VII.6 Reliable Transmission of Information / Sudan, Madhu
VII.7 Mathematics and Cryptography / Cocks, Clifford
VII.8 Mathematics and Economic Reasoning / Dasgupta, Partha
VII.9 The Mathematics of Money / Joshi, Mark
VII.10 Mathematical Statistics / Diaconis, Persi
VII.11 Mathematics and Medical Statistics / Spiegelhalter, David J.
VII.12 Analysis, Mathematical and Philosophical / Burgess, John P.
VII.13 Mathematics and Music / Nolan, Catherine
VII.14 Mathematics and Art / Fasanelli, Florence
Part VIII. Final Perspectives
VIII.1 The Art of Problem Solving / Gardiner, A.
VIII.2 "Why Mathematics?" You Might Ask / Harris, Michael
VIII.3 The Ubiquity of Mathematics / Körner, T. W.
VIII.4 Numeracy / Robson, Eleanor
VIII.5 Mathematics: An Experimental Science / Wilf, Herbert S.
VIII.6 Advice to a Young Mathematician
VIII.7 A Chronology of Mathematical Events / Rice, Adrian
Description based upon print version of record.
Includes bibliographical references.
Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)
Princeton University.
Credo Reference (Firm)
Barrow-Green, June, editor.
Gowers, Timothy, editor.
Leader, Imre, editor.
Publisher Number:
10.1515/9781400830398 doi
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