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The Princeton Companion to Mathematics

Book Description

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, 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.

  • Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
  • Presents major ideas and branches of pure mathematics in a clear, accessible style
  • Defines and explains important mathematical concepts, methods, theorems, and open problems
  • Introduces the language of mathematics and the goals of mathematical research
  • Covers number theory, algebra, analysis, geometry, logic, probability, and more
  • Traces the history and development of modern mathematics
  • Profiles more than ninety-five mathematicians who influenced those working today
  • Explores the influence of mathematics on other disciplines
  • Includes bibliographies, cross-references, and a comprehensive index

Contributors 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

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Contributors
  7. Part I Introduction
    1. I.1 What Is Mathematics About?
    2. I.2 The Language and Grammar of Mathematics
    3. I.3 Some Fundamental Mathematical Definitions
    4. I.4 The General Goals of Mathematical Research
  8. Part II The Origins of Modern Mathematics
    1. II.1 From Numbers to Number Systems
    2. II.2 Geometry
    3. II.3 The Development of Abstract Algebra
    4. II.4 Algorithms
    5. II.5 The Development of Rigor in Mathematical Analysis
    6. II.6 The Development of the Idea of Proof
    7. II.7 The Crisis in the Foundations of Mathematics
  9. Part III Mathematical Concepts
    1. III.1 The Axiom of Choice
    2. III.2 The Axiom of Determinacy
    3. III.3 Bayesian Analysis
    4. III.4 Braid Groups
    5. III.5 Buildings
    6. III.6 Calabi–Yau Manifolds
    7. III.7 Cardinals
    8. III.8 Categories
    9. III.9 Compactness and Compactification
    10. III.10 Computational Complexity Classes
    11. III.11 Countable and Uncountable Sets
    12. III.12 C*-Algebras
    13. III.13 Curvature
    14. III.14 Designs
    15. III.15 Determinants
    16. III.16 Differential Forms and Integration
    17. III.17 Dimension
    18. III.18 Distributions
    19. III.19 Duality
    20. III.20 Dynamical Systems and Chaos
    21. III.21 Elliptic Curves
    22. III.22 The Euclidean Algorithm and Continued Fractions
    23. III.23 The Euler and Navier-Stokes Equations
    24. III.24 Expanders
    25. III.25 The Exponential and Logarithmic Functions
    26. III.26 The Fast Fourier Transform
    27. III.27 The Fourier Transform
    28. III.28 Fuchsian Groups
    29. III.29 Function Spaces
    30. III.30 Galois Groups
    31. III.31 The Gamma Function
    32. III.32 Generating Functions
    33. III.33 Genus
    34. III.34 Graphs
    35. III.35 Hamiltonians
    36. III.36 The Heat Equation
    37. III.37 Hilbert Spaces
    38. III.38 Homology and Cohomology
    39. III.39 Homotopy Groups
    40. III.40 The Ideal Class Group
    41. III.41 Irrational and Transcendental Numbers
    42. III.42 The Ising Model
    43. III.43 Jordan Normal Form
    44. III.44 Knot Polynomials
    45. III.45 K-Theory
    46. III.46 The Leech Lattice
    47. III.47 L-Functions
    48. III.48 Lie Theory
    49. III.49 Linear and Nonlinear Waves and Solitons
    50. III.50 Linear Operators and Their Properties
    51. III.51 Local and Global in Number Theory
    52. III.52 The Mandelbrot Set
    53. III.53 Manifolds
    54. III.54 Matroids
    55. III.55 Measures
    56. III.56 Metric Spaces
    57. III.57 Models of Set Theory
    58. III.58 Modular Arithmetic
    59. III.59 Modular Forms
    60. III.60 Moduli Spaces
    61. III.61 The Monster Group
    62. III.62 Normed Spaces and Banach Spaces
    63. III.63 Number Fields
    64. III.64 Optimization and Lagrange Multipliers
    65. III.65 Orbifolds
    66. III.66 Ordinals
    67. III.67 The Peano Axioms
    68. III.68 Permutation Groups
    69. III.69 Phase Transitions
    70. III.70 π
    71. III.71 Probability Distributions
    72. III.72 Projective Space
    73. III.73 Quadratic Forms
    74. III.74 Quantum Computation
    75. III.75 Quantum Groups
    76. III.76 Quaternions, Octonions, and Normed Division Algebras
    77. III.77 Representations
    78. III.78 Ricci Flow
    79. III.79 Riemann Surfaces
    80. III.80 The Riemann Zeta Function
    81. III.81 Rings, Ideals, and Modules
    82. III.82 Schemes
    83. III.83 The Schrödinger Equation
    84. III.84 The Simplex Algorithm
    85. III.85 Special Functions
    86. III.86 The Spectrum
    87. III.87 Spherical Harmonics
    88. III.88 Symplectic Manifolds
    89. III.89 Tensor Products
    90. III.90 Topological Spaces
    91. III.91 Transforms
    92. III.92 Trigonometric Functions
    93. III.93 Universal Covers
    94. III.94 Variational Methods
    95. III.95 Varieties
    96. III.96 Vector Bundles
    97. III.97 Von Neumann Algebras
    98. III.98 Wavelets
    99. III.99 The Zermelo–Fraenkel Axioms
  10. Part IV Branches of Mathematics
    1. IV.1 Algebraic Numbers
    2. IV.2 Analytic Number Theory
    3. IV.3 Computational Number Theory
    4. IV.4 Algebraic Geometry
    5. IV.5 Arithmetic Geometry
    6. IV.6 Algebraic Topology
    7. IV.7 Differential Topology
    8. IV.8 Moduli Spaces
    9. IV.9 Representation Theory
    10. IV.10 Geometric and Combinatorial Group Theory
    11. IV.11 Harmonic Analysis
    12. IV.12 Partial Differential Equations
    13. IV.13 General Relativity and the Einstein Equations
    14. IV.14 Dynamics
    15. IV.15 Operator Algebras
    16. IV.16 Mirror Symmetry
    17. IV.17 Vertex Operator Algebras
    18. IV.18 Enumerative and Algebraic Combinatorics
    19. IV.19 Extremal and Probabilistic Combinatorics
    20. IV.20 Computational Complexity
    21. IV.21 Numerical Analysis
    22. IV.22 Set Theory
    23. IV.23 Logic and Model Theory
    24. IV.24 Stochastic Processes
    25. IV.25 Probabilistic Models of Critical Phenomena
    26. IV.26 High-Dimensional Geometry and Its Probabilistic Analogues
  11. Part V Theorems and Problems
    1. V.1 The ABC Conjecture
    2. V.2 The Atiyah–Singer Index Theorem
    3. V.3 The Banach–Tarski Paradox
    4. V.4 The Birch–Swinnerton-Dyer Conjecture
    5. V.5 Carleson’s Theorem
    6. V.6 The Central Limit Theorem
    7. V.7 The Classification of Finite Simple Groups
    8. V.8 Dirichiet’s Theorem
    9. V.9 Ergodic Theorems
    10. V.10 Fermat’s Last Theorem
    11. V.11 Fixed Point Theorems
    12. V.12 The Four-Color Theorem
    13. V.13 The Fundamental Theorem of Algebra
    14. V.14 The Fundamental Theorem of Arithmetic
    15. V.15 Gödel’s Theorem
    16. V.16 Gromov’s Polynomial-Growth Theorem
    17. V.17 Hilbert’s Nullstellensatz
    18. V.18 The Independence of the Continuum Hypothesis
    19. V.19 Inequalities
    20. V.20 The Insolubility of the Halting Problem
    21. V.21 The Insolubility of the Quintic
    22. V.22 Liouvflle’s Theorem and Roth’s Theorem
    23. V.23 Mostow’s Strong Rigidity Theorem
    24. V.24 The P versus NP Problem
    25. V.25 The Poincaré Conjecture
    26. V.26 The Prime Number Theorem and the Riemann Hypothesis
    27. V.27 Problems and Results in Additive Number Theory
    28. V.28 From Quadratic Reciprocity to Class Field Theory
    29. V.29 Rational Points on Curves and the Mordell Conjecture
    30. V.30 The Resolution of Singularities
    31. V.31 The Riemann–Roch Theorem
    32. V.32 The Robertson–Seymour Theorem
    33. V.33 The Three-Body Problem
    34. V.34 The Uniformization Theorem
    35. V.35 The Weil Conjectures
  12. Part VI Mathematicians
    1. VI.1 Pythagoras (ca. 569 B.C.E.–ca.494 B.C.E.)
    2. VI.2 Euclid (ca. 325 B.C.E.–ca. 265 B.C.E.)
    3. VI.3 Archimedes (ca. 287 B.C.E.–212 B.C.E.)
    4. VI.4 Apollonius (ca. 262 B.C.E.–ca. 190 B.C.E.)
    5. VI.5 Abu Ja’far Muhammad ibn Mūsā al-Khwārizmī (800–847)
    6. VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170–ca. 1250)
    7. VI.7 Girolamo Cardano (1501–1576)
    8. VI.8 Rafael Bombelli (1526-after 1572)
    9. VI.9 François Viète (1540–1603)
    10. VI.10 Simon Stevin (1548–1620)
    11. VI.11 René Descartes (1596–1650)
    12. VI.12 Pierre Fermat (160?-1665)
    13. VI.13 Blaise Pascal (1623–1662)
    14. VI.14 Isaac Newton (1642–1727)
    15. VI.15 Gottfried Wilhelm Leibniz (1646–1716)
    16. VI.16 Brook Taylor (1685–1731)
    17. VI.17 Christian Goldbach (1690–1764)
    18. VI.18 The Bernoullis (f1.18th century)
    19. VI.19 Leonhard Euler (1707–1783)
    20. VI.20 Jean Le Rond d’Alembert (1717–1783)
    21. VI.21 Edward Waring (ca.1735–1798)
    22. VI.22 Joseph Louis Lagrange (1736–1813)
    23. VI.23 Pierre-Simon Laplace (1749–1827)
    24. VI.24 Adrien-Marie Legendre (1752–1833)
    25. VI.25 Jean-Baptiste Joseph Fourier (1768–1830)
    26. VI.26 Carl Friedrich Gauss (1777–1855)
    27. VI.27 Siméon-Denis Poisson (1781–1840)
    28. VI.28 Bernard Bolzano (1781–1848)
    29. VI.29 Augustin-Louis Cauchy (1789–1857)
    30. VI.30 August Ferdinand Möbius (1790–1868)
    31. VI.31 Nicolai Ivanovich Lobachevskii (1792–1856)
    32. VI.32 George Green (1793–1841)
    33. VI.33 Niels Henrik Abel (1802–1829)
    34. VI.34 János Bolyai (1802–1860)
    35. VI.35 Carl Gustav Jacob Jacobi (1804–1851)
    36. VI.36 Peter Gustav Lejeune Dirichlet (1805–1859)
    37. VI.37 William Rowan Hamilton (1805–1865)
    38. VI.38 Augustus De Morgan (1806–1871)
    39. VI.39 Joseph Liouville (1809–1882)
    40. VI.40 Ernst Eduard Kummer (1810–1893)
    41. VI.41 Évariste Galois (1811–1832)
    42. VI.42 James Joseph Sylvester (1814–1897)
    43. VI.43 George Boole (1815–1864)
    44. VI.44 Karl Weierstrass (1815–1897)
    45. VI.45 Pafnuty Chebyshev (1821–1894)
    46. VI.46 Arthur Cayley (1821–1895)
    47. VI.47 Charles Hermite (1822–1901)
    48. VI.48 Leopold Kronecker (1823–1891)
    49. VI.49 Georg Friedrich Bernhard Riemann (1826–1866)
    50. VI.50 Julius Wilhelm Richard Dedekind (1831–1916)
    51. VI.51 Émile Léonard Mathieu (1835–1890)
    52. VI.52 Camille Jordan (1838–1922)
    53. VI.53 Sophus Lie (1842–1899)
    54. VI.54 Georg Cantor (1845–1918)
    55. VI.55 William Kingdon Clifford (1845–1879)
    56. VI.56 Gottlob Frege (1848–1925)
    57. VI.57 Christian Felix Klein (1849–1925)
    58. VI.58 Ferdinand Georg Frobenius (1849–1917)
    59. VI.59 Sofya (Sonya) Kovalevskaya (1850–1891)
    60. VI.60 William Burnside (1852–1927)
    61. VI.61 Jules Henri Poincaré (1854–1912)
    62. VI.62 Giuseppe Peano (1858–1932)
    63. VI.63 David Hilbert (1862–1943)
    64. VI.64 Hermann Minkowski (1864–1909)
    65. VI.65 Jacques Hadamard (1865–1963)
    66. VI.66 Ivar Fredholm (1866–1927)
    67. VI.67 Charles-Jean de la Vallée Poussin (1866–1962)
    68. VI.68 Felix Hausdorff (1868–1942)
    69. VI.69 Élie Joseph Cartan (1869–1951)
    70. VI.70 Emile Borel (1871–1956)
    71. VI.71 Bertrand Arthur William Russell (1872–1970)
    72. VI.72 Henri Lebesgue (1875–1941)
    73. VI.73 Godfrey Harold Hardy (1877–1947)
    74. VI.74 Frigyes (Frédéric) Riesz (1880–1956)
    75. VI.75 Luitzen Egbertus Jan Brouwer (1881–1966)
    76. VI.76 Emmy Noether (1882–1935)
    77. VI.77 Waclaw Sierpiński (1882–1969)
    78. VI.78 George Birkhoff (1884–1944)
    79. VI.79 John Edensor Littlewood (1885–1977)
    80. VI.80 Hermann Weyl (1885–1955)
    81. VI.81 Thoralf Skolem (1887–1963)
    82. VI.82 Srinwasa Ramanujan (1887–1920)
    83. VI.83 Richard Courant (1888–1972)
    84. VI.84 Stefan Banach (1892–1945)
    85. VI.85 Norbert Wiener (1894–1964)
    86. VI.86 Emil Artin (1898–1962)
    87. VI.87 Alfred Tarski (1901–1983)
    88. VI.88 Andrei Nikolaevich Kolmogorov (1903–1987)
    89. VI.89 Alonzo Church (1903–1995)
    90. VI.90 William Valiance Douglas Hodge (1903–1975)
    91. VI.91 John von Neumann (1903–1957)
    92. VI.92 Kurt Gödel (1906–1978)
    93. VI.93 André Weil (1906–1998)
    94. VI.94 Alan Turing (1912–1954)
    95. VI.95 Abraham Robinson (1918–1974)
    96. VI.96 Nicolas Bourbaki (1935–)
  13. Part VII The Influence of Mathematics
    1. VII.1 Mathematics and Chemistry
    2. VII.2 Mathematical Biology
    3. VII.3 Wavelets and Applications
    4. VII.4 The Mathematics of Traffic in Networks
    5. VII.5 The Mathematics of Algorithm Design
    6. VII.6 Reliable Transmission of Information
    7. VII.7 Mathematics and Cryptography
    8. VII.8 Mathematics and Economic Reasoning
    9. VII.9 The Mathematics of Money
    10. VII.10 Mathematical Statistics
    11. VII.11 Mathematics and Medical Statistics
    12. VII.12 Analysis, Mathematical and Philosophical
    13. VII.13 Mathematics and Music
    14. VII.14 Mathematics and Art
  14. Part VIII Final Perspectives
    1. VIII.1 The Art of Problem Solving
    2. VIII.2 “Why Mathematics” You Might Ask
    3. VIII.3 The Ubiquity of Mathematics
    4. VIII.4 Numeracy
    5. VIII.5 Mathematics: An Experimental Science
    6. VIII.6 Advice to a Young Mathematician
    7. VIII.7 A Chronology of Mathematical Events
  15. Index