The Princeton Companion to Mathematics
Gowers (Timothy), Barrow-Green (June) & Leader (Imre), Eds.
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Introductory Notes

Inside Cover Blurb
  1. 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.
  2. 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.
  3. Contents
    • 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.
  4. Contributors include1: 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, Bela Bollobas, 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, Jose Ferreiros, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvea, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolo 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, Janos Kollar, T. W. Korner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-Francois Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lutzen, 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, Rudiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, and Doron Zeilberger.

Amazon Reviews2
  1. The Princeton Companion to Mathematics makes a heroic attempt to keep [abstract concepts] to a minimum … and conveys the breadth, depth and diversity of mathematics. It is impressive and well written and it's good value for [the] money.
    → Ian Stewart, The Times
  2. This is a panoramic view of modern mathematics. It is tough going in some places, but much of it is surprisingly accessible. A must for budding number-crunchers.
    → The Economist
  3. Although the editors' original goal of text that could be understood by anyone with a good background in high school mathematics proved short-lived, this wide-ranging account should reward undergraduate and graduate students and anyone curious about math as well as help research mathematicians understand the work of their colleagues in other specialties. The editors note some advantages a carefully organized printed reference may enjoy over a collection of Web pages, and this impressive volume supports their claim.
    → Science
  4. This impressive book represents an extremely ambitious and, I might add, highly successful attempt by Timothy Gowers and his coeditors, June Barrow-Green and Imre Leader, to give a current account of the subject of mathematics. It has something for nearly everyone, from beginning students of mathematics who would like to get some sense of what the subject is all about, all the way to professional mathematicians who would like to get a better idea of what their colleagues are doing… If I had to choose just one book in the world to give an interested reader some idea of the scope, goals and achievements of modern mathematics, without a doubt this would be the one. So try it. I guarantee you'll like it!
    → American Scientist
  5. Accessible, technically precise and thorough account of all math's major aspects. Students of math will find this book a helpful reference for understanding their classes; students of everything else will find helpful guides to understanding how math describes it all.
    → Tom Siegfried, Science News
  6. Once in a while a book comes along that should be on every mathematician's bookshelf. This is such a book. Described as a 'companion', this 1000-page tome is an authoritative and informative reference work that is also highly pleasurable to dip into. Much of it can be read with benefit by undergraduate mathematicians, while there is a great deal to engage professional mathematicians of all persuasions.
    → Robin Wilson London Mathematical Society
  7. Imagine taking an overview of elementary and advanced mathematics, a history of mathematics and mathematicians, and a mathematical encyclopedia and combining them all into one comprehensive reference book. That is what Timothy Gowers, the 1998 Fields Medal laureate, has successfully accomplished in compiling and editing The Princeton Companion to Mathematics. At more than 1,000 pages and with nearly 200 entries written by some of the leading mathematicians of our time and specialists in their fields, this book is a one-of-a-kind reference for all things mathematics.
    → Mathematics Teacher
  8. Overall [The Princeton Companion to Mathematics] is an enormous achievement for which the authors deserve to be thanked. It contains a wealth of material, much of a kind one would not find elsewhere, and can be enjoyed by readers with many different backgrounds.
    → Simon Donaldson, Notices of the American Mathematical Society
  9. This is an enormously ambitious book, full of beautiful things; I would wish to keep it on my bedside table, but … of course it is far too large… To sum up, [The Princeton Companion to Mathematics] is really excellent. I know of no book that will give a young student a better idea of what mathematics is about. I am certain that this is the only single book that is likely to tell me what my colleagues are doing.
    → Bryan Birch, Notices of the American Mathematical Society
  10. The book is so rich and yet it is well done. A rare achievement indeed!
    → Gil Kalai, Notices of the American Mathematical Society
  11. My advice to you, reader is to buy the book, open it to a random page, read, enjoy, and be enlightened.
    → Richard Kenyon, Notices of the American Mathematical Society
  12. Massive … endlessly fascinating.
    → Gregory McNamee, Bloomsbury Review
  13. This volume is an enormous, far-reaching effort to survey the current landscape of (pure) mathematics. Chief editor Gowers and associate editors Barrow-Green and Leader have enlisted scores of leading mathematicians worldwide to produce a gorgeous volume of longer essays and short, specific articles that convey some of the dense fabric of ideas and techniques of modern mathematics… This volume should be on the shelf of every university and public library, and of every mathematician--professional and amateur alike.
    → S.J. Colley, Choice
  14. The Princeton Companion to Mathematics is a friendly, informative reference book that attempts to explain what mathematics is about and what mathematicians do. Over 200 entries by a panel of experts span such topics as: the origins of modern mathematics; mathematical concepts; branches of mathematics; mathematicians that contributed to the present state of the discipline; theorems and problems; the influences of mathematics and some perspectives. Its presentations are selective, satisfying, and complete within themselves but not overbearingly comprehensive. Any reader from a curious high school student to an experienced mathematician seeking information on a particular mathematical subject outside his or her field will find this book useful. The writing is clear and the examples and illustrations beneficial.
    → Frank Swetz, Convergence
  15. Every research mathematician, every university student of mathematics, and every serious amateur of mathematical science should own a least one copy of The Companion. Indeed, the sheer weight of the volume suggests that it is advisable to own two: one for work and one at home… Even an academic sourpuss should be pleased with the attention to detail of The Companion's publishers, editors, and authors and with many judicious decisions about the level of exposition, level of detail, what to include and what to omit, and much more--which have led to a well-integrated and highly readable volume.
    → Jonathan M. Borwein, SIAM Review
  16. Edited by Gowers, a recipient of the Fields Medal, this volume contains almost 200 entries, commissioned especially for this book from the world's leading mathematicians. It introduces basic mathematical tools and vocabulary, traces the development of modern mathematics, defines essential terms and concepts, and puts them in context… Packed with information 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.
    → Library Journal
  17. The book I'm talking about is The Princeton Companion to Mathematics. If you are in an absolute rush, the short version of my post today is, buy this book. You don't have to click on the link with my referral if you don't want to, seriously just pick up a copy of this book, I can guarantee you that it will be love at first sight… The Princeton Companion to Mathematics is not only a beautiful book from an aesthetic standpoint, with its heavy, high quality pages and sturdy binding, but above all it's a monumental piece of work. I have never seen a book like this before… [T]he bible of mathematics… I believe this is the kind of book that will still be in use a hundred years from now.
    → Antonio Cangiano,
  18. I'm completely charmed. This is one of those books that makes you wish you had a desert island to be marooned on.
    → Brian Hayes,
  19. This has been a long time coming, but the wait was worth it! After many years of slogging through textbooks that presented too many proofs and demonstrations that were left to the student or lacking numerous intermediate steps, after encountering numerous 'introductions' that were obtuse and highly theoretical and after digesting far too many explanations with maximal equations and minimal verbiage, we arrive at the happy medium. This book is a companion in every sense of the word and a very friendly one at that… For a comprehensive overview of many areas of mathematics in a readable format, there has never been anything quite like this. I would urge a trip to the local library to have a look.
    → John A. Wass, Scientific Computing
  20. This book is supremely accessible. Many in the sugar industry with a fairly good grasp of mathematics will probably not struggle with it, and will invariably marvel at its richness and diversity. [A] great companion.
    → International Sugar Journal
  21. The book contains some valuable surveys of the main branches of mathematics that are written in an accessible style. Hence, it is recommended both to students of mathematics and researchers seeking to understand areas outside their specialties.
    → European Mathematical Society Newsletter

In-Page Footnotes ("Gowers (Timothy), Barrow-Green (June) & Leader (Imre), Eds. - The Princeton Companion to Mathematics")

Footnote 1: List taken from Amazon Product Description.

Footnote 2:
Book Comment
  • Princeton University Press (21 Oct 2008). Wonderful hardback.
  • Received as a leaving present from HSBC when I retired at the end of 2010.

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Preface"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

Paper Comment

For the full text on-line see Gowers, Etc - The Princeton Companion to Mathematics: Preface.

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part I - Introduction"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

  1. What is mathematics about?
  2. The language and grammar of mathematics
  3. Some fundamental mathematical definitions
  4. The general goals of mathematical research

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part II - The Origins of Modern Mathematics"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

  1. From numbers to number systems
  2. Geometry
  3. The development of abstract algebra
  4. Algorithms
  5. The development of rigor in mathematical analysis
  6. The development of the idea of proof
  7. The crisis in the foundations of mathematics

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part III - Mathematical Concepts"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

  1. The axiom of choice
  2. The axiom of determinacy
  3. Bayesian analysis
  4. Braid groups
  5. Buildings
  6. Calabi-Yau manifolds
  7. Cardinals
  8. Categories
  9. Compactness and compactification
  10. Computational complexity classes
  11. Countable and uncountable sets
  12. C* - algebras
  13. Curvature
  14. Designs
  15. Determinants
  16. Differential forms and integration
  17. Dimension
  18. Distributions
  19. Duality
  20. Dynamical systems and chaos
  21. Elliptic curves
  22. The Euclidean algorithm and continued fractions
  23. The Euler and Navier-Stokes equations
  24. Expanders
  25. The exponential and logarithmic functions
  26. The fast Fourier transform
  27. The Fourier transform
  28. Fuchsian groups
  29. Function spaces
  30. Galois groups
  31. The gamma function
  32. Generating functions
  33. Genus
  34. Graphs
  35. Hamiltonians
  36. The heat equation
  37. Hilbert spaces
  38. Homology and cohomology
  39. Homotopy Groups
  40. The ideal class group
  41. Irrational and transcendental numbers
  42. The Ising model
  43. Jordan normal form
  44. Knot polynomials
  45. K-theory
  46. The leech lattice
  47. L-function
  48. Lie theory
  49. Linear and nonlinear waves and solitons
  50. Linear operators and their properties
  51. Local and global in number theory
  52. The Mandelbrot set
  53. Manifolds
  54. Matroids
  55. Measures
  56. Metric spaces
  57. Models of set theory
  58. Modular arithmetic
  59. Modular forms
  60. Moduli spaces
  61. The monster group
  62. Normed spaces and banach spaces
  63. Number fields
  64. Optimization and Lagrange multipliers
  65. Orbifolds
  66. Ordinals
  67. The Peano axioms
  68. Permutation groups
  69. Phase transitions
  70. [pi]
  71. Probability distributions
  72. Projective space
  73. Quadratic forms
  74. Quantum computation
  75. Quantum groups
  76. Quaternions, octonions, and normed division algebras
  77. Representations
  78. Ricci flow
  79. Riemann surfaces
  80. The Riemann zeta function
  81. Rings, ideals, and modules
  82. Schemes
  83. The Schrödinger equation
  84. The simplex algorithm
  85. Special functions
  86. The spectrum
  87. Spherical harmonics
  88. Symplectic manifolds
  89. Tensor products
  90. Topological spaces
  91. Transforms
  92. Trigonometric functions
  93. Universal covers
  94. Variational methods
  95. Varieties
  96. Vector bundles
  97. Von Neumann algebras
  98. Wavelets
  99. The Zermelo-Fraenkel axioms

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part IV - Branches of Mathematics"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

  1. Algebraic numbers
  2. Analytic number theory
  3. Computational number theory
  4. Algebraic geometry
  5. Arithmetic geometry
  6. Algebraic topology
  7. Differential topology
  8. Moduli spaces
  9. Representation theory
  10. Geometric and combinatorial group theory
  11. Harmonic analysis
  12. Partial differential equations
  13. General relativity and the Einstein equations
  14. Dynamics
  15. Operator algebras
  16. Mirror symmetry
  17. Vertex operator algebras
  18. Enumerative and algebraic combinatorics
  19. Extremal and probabilistic combinatorics
  20. Computational complexity
  21. Numerical analysis
  22. Set theory
  23. Logic and model theory
  24. Stochastic processes
  25. Probabilistic models of critical phenomena
  26. High-dimensional geometry and its probabilistic analogues

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part V - Theorems and Problems"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

  1. The ABC conjecture
  2. The Atiyah-Singer index theorem
  3. The Banach-Tarski paradox
  4. The Birch-Swinnerton-Dyer conjecture
  5. Carleson's theorem
  6. The central limit theorem
  7. The classification of finite simple groups
  8. Dirichlet's theorem
  9. Ergodic theorems
  10. Fermat's last theorem
  11. Fixed point theorems
  12. The four-color theorem
  13. The fundamental theorem of algebra
  14. The fundamental theorem of arithmetic
  15. Gödel's theorem
  16. Gromov's polynomial-growth theorem
  17. Hilbert's nullstellensatz
  18. The independence of the continuum hypothesis
  19. Inequalities
  20. The insolubility of the halting problem
  21. The insolubility of the quintic
  22. Liouville's theorem and Roth's theorem
  23. Mostow's strong rigidity theorem
  24. The p versus NP problem
  25. The Poincaré conjecture
  26. The prime number theorem and the Riemann hypothesis
  27. Problems and results in additive number theory
  28. From quadratic reciprocity to class field theory
  29. Rational points on curves and the Mordell conjecture
  30. The resolution of singularities
  31. The Riemann-Roch theorem
  32. The Robertson-Seymour theorem
  33. The three-body problem
  34. The uniformization theorem
  35. The Weil conjecture

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part VI - Mathematicians"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

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

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part VII - The Influence of Mathematics"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

  1. Mathematics and chemistry
  2. Mathematical biology
  3. Wavelets and applications
  4. The mathematics of traffic in networks
  5. The mathematics of algorithm design
  6. Reliable transmission of information
  7. Mathematics and cryptography
  8. Mathematics and economic reasoning
  9. The mathematics of money
  10. Mathematical statistics
  11. Mathematics and medical statistics
  12. Analysis, mathematical and philosophical
  13. Mathematics and music
  14. Mathematics and art

"Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics: Part VIII - Final Perspectives"

Source: Gowers (Timothy), Barrow-Green (June) & Leader (Imre) - The Princeton Companion to Mathematics

  1. The art of problem solving
  2. "Why mathematics?" you might ask
  3. The ubiquity of mathematics
  4. Numeracy
  5. Mathematics: an experimental science
  6. Advice to a young mathematician
  7. A chronology of mathematical events

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