The Princeton Companion to Mathematics | ||||

Gowers (Timothy), Barrow-Green (June) & Leader (Imre), Eds. | ||||

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__Introductory Notes__

- This book focuses on
**Pure**mathematics; - For
**Applied**mathematics, the preface recommends "Penrose (Roger) - The Road to Reality: A Complete Guide to the Physical Universe", which it says is at the same level. - However, there seems to be a companion volume for
**Applied**mathematics: see Nick Higham, Ed. - The Princeton Companion to Applied Mathematics. - For the principal editor, see:-

→ Timothy Gowers - Cambridge DPMSS Home Page, and

→ Wikipedia: Timothy Gowers - I found a nice digitised pdf of the entire book on GITHub (of all pages). No doubt the publishers would object to this, if they knew, so as I possess the hardback I've downloaded the pdf before it disappears. I may be able to read some of it on my Kindle while out walking the dog.

- 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.
- 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.

__Contributors include___{1}: 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.

- 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - The book is so rich and yet it is well done. A rare achievement indeed!

→ Gil Kalai, Notices of the American Mathematical Society - 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 - Massive … endlessly fascinating.

→ Gregory McNamee, Bloomsbury Review - 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 - 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 - 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 - 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 - 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, Math-Blog.com - 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, bit-player.org - 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 - 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 - 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

- These are not very helpful.
- For more extensive reviews, see American Mathematical Society - reviews of The Princeton Companion to Mathematics (and various others on JSTOR).

- Princeton University Press (21 Oct 2008). Wonderful hardback.
- Received as a leaving present from HSBC when I retired at the end of 2010.

"

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

"

- What is mathematics about?
- The language and grammar of mathematics
- Some fundamental mathematical definitions
- The general goals of mathematical research

"

- From numbers to number systems
- Geometry
- The development of abstract algebra
- Algorithms
- The development of rigor in mathematical analysis
- The development of the idea of proof
- The crisis in the foundations of mathematics

"

- The axiom of choice
- The axiom of determinacy
- Bayesian analysis
- Braid groups
- Buildings
- Calabi-Yau manifolds
- Cardinals
- Categories
- Compactness and compactification
- Computational complexity classes
- Countable and uncountable sets
- C* - algebras
- Curvature
- Designs
- Determinants
- Differential forms and integration
- Dimension
- Distributions
- Duality
- Dynamical systems and chaos
- Elliptic curves
- The Euclidean algorithm and continued fractions
- The Euler and Navier-Stokes equations
- Expanders
- The exponential and logarithmic functions
- The fast Fourier transform
- The Fourier transform
- Fuchsian groups
- Function spaces
- Galois groups
- The gamma function
- Generating functions
- Genus
- Graphs
- Hamiltonians
- The heat equation
- Hilbert spaces
- Homology and cohomology
- Homotopy Groups
- The ideal class group
- Irrational and transcendental numbers
- The Ising model
- Jordan normal form
- Knot polynomials
- K-theory
- The leech lattice
- L-function
- Lie theory
- Linear and nonlinear waves and solitons
- Linear operators and their properties
- Local and global in number theory
- The Mandelbrot set
- Manifolds
- Matroids
- Measures
- Metric spaces
- Models of set theory
- Modular arithmetic
- Modular forms
- Moduli spaces
- The monster group
- Normed spaces and banach spaces
- Number fields
- Optimization and Lagrange multipliers
- Orbifolds
- Ordinals
- The Peano axioms
- Permutation groups
- Phase transitions
- [pi]
- Probability distributions
- Projective space
- Quadratic forms
- Quantum computation
- Quantum groups
- Quaternions, octonions, and normed division algebras
- Representations
- Ricci flow
- Riemann surfaces
- The Riemann zeta function
- Rings, ideals, and modules
- Schemes
- The Schrödinger equation
- The simplex algorithm
- Special functions
- The spectrum
- Spherical harmonics
- Symplectic manifolds
- Tensor products
- Topological spaces
- Transforms
- Trigonometric functions
- Universal covers
- Variational methods
- Varieties
- Vector bundles
- Von Neumann algebras
- Wavelets
- The Zermelo-Fraenkel axioms

"

- Algebraic numbers
- Analytic number theory
- Computational number theory
- Algebraic geometry
- Arithmetic geometry
- Algebraic topology
- Differential topology
- Moduli spaces
- Representation theory
- Geometric and combinatorial group theory
- Harmonic analysis
- Partial differential equations
- General relativity and the Einstein equations
- Dynamics
- Operator algebras
- Mirror symmetry
- Vertex operator algebras
- Enumerative and algebraic combinatorics
- Extremal and probabilistic combinatorics
- Computational complexity
- Numerical analysis
- Set theory
- Logic and model theory
- Stochastic processes
- Probabilistic models of critical phenomena
- High-dimensional geometry and its probabilistic analogues

"

- The ABC conjecture
- The Atiyah-Singer index theorem
- The Banach-Tarski paradox
- The Birch-Swinnerton-Dyer conjecture
- Carleson's theorem
- The central limit theorem
- The classification of finite simple groups
- Dirichlet's theorem
- Ergodic theorems
- Fermat's last theorem
- Fixed point theorems
- The four-color theorem
- The fundamental theorem of algebra
- The fundamental theorem of arithmetic
- Gödel's theorem
- Gromov's polynomial-growth theorem
- Hilbert's nullstellensatz
- The independence of the continuum hypothesis
- Inequalities
- The insolubility of the halting problem
- The insolubility of the quintic
- Liouville's theorem and Roth's theorem
- Mostow's strong rigidity theorem
- The p versus NP problem
- The Poincaré conjecture
- The prime number theorem and the Riemann hypothesis
- Problems and results in additive number theory
- From quadratic reciprocity to class field theory
- Rational points on curves and the Mordell conjecture
- The resolution of singularities
- The Riemann-Roch theorem
- The Robertson-Seymour theorem
- The three-body problem
- The uniformization theorem
- The Weil conjecture

"

- Pythagoras
- Euclid
- Archimedes
- Apollonius
- Abu Jaʾfar Muhammad ibn Mūsā al-Khwārizmī
- Leonardo of Pisa (known as Fibonacci)
- Girolamo Cardano
- Rafael Bombelli
- François Viète
- Simon Stevin
- René Descartes
- Pierre Fermat
- Blaise Pascal
- Isaac Newton
- Gottfried Wilhelm Leibniz
- Brook Taylor
- Christian Goldbach
- The Bernoullis
- Leonhard Euler
- Jean Le Rond d'Alembert
- Edward Waring
- Joseph Louis Lagrange
- Pierre-Simon Laplace
- Adrien-Marie Legendre
- Jean-Baptiste Joseph Fourier
- Carl Friedrich Gauss
- Siméon-Denis Poisson
- Bernard Bolzano
- Augustin-Louis Cauchy
- August Ferdinand Möbius
- Nicolai Ivanovich Lobachevskii
- George Green
- Niels Henrik Abel
- János Bolyai
- Carl Gustav Jacob Jacobi
- Peter Gustav Lejeune Dirichlet
- William Rowan Hamilton
- Augustus De Morgan
- Joseph Liouville
- Eduard Kummer
- 1Évariste Galois
- James Joseph Sylvester
- George Boole
- Karl Weierstrass
- Pafnuty Chebyshev
- Arthur Cayley
- Charles Hermite
- Leopold Kronecker
- Georg Friedrich Bernhard Riemann
- Julius Wilhelm Richard Dedekind
- Émile Léonard Mathieu
- Camille Jordan
- Sophus Lie
- Georg Cantor
- William Kingdon Clifford
- Gottlob Frege
- Christian Felix Klein
- Ferdinand Georg Frobenius
- Sofya (Sonya) Kovalevskaya
- William Burnside
- Jules Henri Poincaré
- Giuseppe Peano
- David Hilbert
- Hermann Minkowski
- Jacques Hadamard
- Ivar Fredholm
- Charles-Jean de la Vallée Poussin
- Felix Hausdorff
- Élie Joseph Cartan
- Emile Borel
- Bertrand Arthur William Russell
- Henri Lebesgue
- Godfrey Harold Hardy
- Frigyes (Frédéric) Riesz
- Luitzen Egbertus Jan Brouwer
- Emmy Noether
- Wacław Sierpiński
- George Birkhoff
- John Edensor Littlewood
- Hermann Weyl
- Thoralf Skolem
- Srinivasa Ramanujan
- Richard Courant
- Stefan Banach
- Norbert Wiener
- Emil Artin
- Alfred Tarski
- Andrei Nikolaevich Kolmogorov
- Alonzo Church
- William Vallance Douglas Hodge
- John von Neumann
- Kurt Gödel
- André Weil
- Alan Turing
- Abraham Robinson
- Nicolas Bourbaki

"

- Mathematics and chemistry
- Mathematical biology
- Wavelets and applications
- The mathematics of traffic in networks
- The mathematics of algorithm design
- Reliable transmission of information
- Mathematics and cryptography
- Mathematics and economic reasoning
- The mathematics of money
- Mathematical statistics
- Mathematics and medical statistics
- Analysis, mathematical and philosophical
- Mathematics and music
- Mathematics and art

"

- The art of problem solving
- "Why mathematics?" you might ask
- The ubiquity of mathematics
- Numeracy
- Mathematics: an experimental science
- Advice to a young mathematician
- A chronology of mathematical events

- Blue: Text by me; © Theo Todman, 2024
- Mauve: Text by correspondent(s) or other author(s); © the author(s)

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