Princeton Companion to Mathematics

Timothy Gowers is the Rouse Ball Professor of Mathematics at the University of Cambridge. He received the Fields Medal in 1998, and is the author of "Mathematics: A Very Short Introduction". June Barrow-Green is lecturer in the history of mathematics at the Open University. Imre Leader is professor of pure mathematics at the University of Cambridge.

• Preface ix Contributors xvii Part I Introduction I.1 What Is Mathematics About? 1 I.2 The Language and Grammar of Mathematics 8 I.3 Some Fundamental Mathematical Definitions 16 I.4 The General Goals of Mathematical Research 48 Part II The Origins of Modern Mathematics II.1 From Numbers to Number Systems 77 II.2 Geometry 83 II.3 The Development of Abstract Algebra 95 II.4 Algorithms 106 II.5 The Development of Rigor in Mathematical Analysis 117 II.6 The Development of the Idea of Proof 129 II.7 The Crisis in the Foundations of Mathematics 142 Part III Mathematical Concepts III.1 The Axiom of Choice 157 III.2 The Axiom of Determinacy 159 III.3 Bayesian Analysis 159 III.4 Braid Groups 160 III.5 Buildings 161 III.6 Calabi-Yau Manifolds 163 III.7 Cardinals 165 III.8 Categories 165 III.9 Compactness and Compactification 167 III.10 Computational Complexity Classes 169 III.11 Countable and Uncountable Sets 170 III.12 C*-Algebras 172 III.13 Curvature 172 III.14 Designs 172 III.15 Determinants 174 III.16 Differential Forms and Integration 175 III.17 Dimension 180 III.18 Distributions 184 III.19 Duality 187 III.20 Dynamical Systems and Chaos 190 III.21 Elliptic Curves 190 III.22 The Euclidean Algorithm and Continued Fractions 191 III.23 The Euler and Navier-Stokes Equations 193 III.24 Expanders 196 III.25 The Exponential and Logarithmic Functions 199 III.26 The Fast Fourier Transform 202 III.27 The Fourier Transform 204 III.28 Fuchsian Groups 208 III.29 Function Spaces 210 III.30 Galois Groups 213 III.31 The Gamma Function 213 III.32 Generating Functions 214 III.33 Genus 215 III.34 Graphs 215 III.35 Hamiltonians 215 III.36 The Heat Equation 216 III.37 Hilbert Spaces 219 III.38 Homology and Cohomology 221 III.39 Homotopy Groups 221 III.40 The Ideal Class Group 221 III.41 Irrational and Transcendental Numbers 222 III.42 The Ising Model 223 III.43 Jordan Normal Form 223 III.44 Knot Polynomials 225 III.45 K-Theory 227 III.46 The Leech Lattice 227 III.47 L-Functions 228 III.48 Lie Theory 229 III.49 Linear and Nonlinear Waves and Solitons 234 III.50 Linear Operators and Their Properties 239 III.51 Local and Global in Number Theory 241 III.52 The Mandelbrot Set 244 III.53 Manifolds 244 III.54 Matroids 244 III.55 Measures 246 III.56 Metric Spaces 247 III.57 Models of Set Theory 248 III.58 Modular Arithmetic 249 III.59 Modular Forms 250 III.60 Moduli Spaces 252 III.61 The Monster Group 252 III.62 Normed Spaces and Banach Spaces 252 III.63 Number Fields 254 III.64 Optimization and Lagrange Multipliers 255 III.65 Orbifolds 257 III.66 Ordinals 258 III.67 The Peano Axioms 258 III.68 Permutation Groups 259 III.69 Phase Transitions 261 III.70 p 261 III.71 Probability Distributions 263 III.72 Projective Space 267 III.73 Quadratic Forms 267 III.74 Quantum Computation 269 III.75 Quantum Groups 272 III.76 Quaternions, Octonions, and Normed Division Algebras 275 III.77 Representations 279 III.78 Ricci Flow 279 III.79 Riemann Surfaces 282 III.80 The Riemann Zeta Function 283 III.81 Rings, Ideals, and Modules 284 III.82 Schemes 285 III.83 The Schrodinger Equation 285 III.84 The Simplex Algorithm 288 III.85 Special Functions 290 III.86 The Spectrum 294 III.87 Spherical Harmonics 295 III.88 Symplectic Manifolds 297 III.89 Tensor Products 301 III.90 Topological Spaces 301 III.91 Transforms 303 III.92 Trigonometric Functions 307 III.93 Universal Covers 309 III.94 Variational Methods 310 III.95 Varieties 313 III.96 Vector Bundles 313 III.97 Von Neumann Algebras 313 III.98 Wavelets 313 III.99 The Zermelo-Fraenkel Axioms 314 Part IV Branches of Mathematics IV.1 Algebraic Numbers 315 IV.2 Analytic Number Theory 332 IV.3 Computational Number Theory 348 IV.4 Algebraic Geometry 363 IV.5 Arithmetic Geometry 372 IV.6 Algebraic Topology 383 IV.7 Differential Topology 396 IV.8 Moduli Spaces 408 IV.9 Representation Theory 419 IV.10 Geometric and Combinatorial Group Theory 431 IV.11 Harmonic Analysis 448 IV.12 Partial Differential Equations 455 IV.13 General Relativity and the Einstein Equations 483 IV.14 Dynamics 493 IV.15 Operator Algebras 510 IV.16 Mirror Symmetry 523 IV.17 Vertex Operator Algebras 539 IV.18 Enumerative and Algebraic Combinatorics 550 IV.19 Extremal and Probabilistic Combinatorics 562 IV.20 Computational Complexity 575 IV.21 Numerical Analysis 604 IV.22 Set Theory 615 IV.23 Logic and Model Theory 635 IV.24 Stochastic Processes 647 IV.25 Probabilistic Models of Critical Phenomena 657 IV.26 High-Dimensional Geometry and Its Probabilistic Analogues 670 Part V Theorems and Problems V.1 The ABC Conjecture 681 V.2 The Atiyah-Singer Index Theorem 681 V.3 The Banach-Tarski Paradox 684 V.4 The Birch-Swinnerton-Dyer Conjecture 685 V.5 Carleson's Theorem 686 V.6 The Central Limit Theorem 687 V.7 The Classification of Finite Simple Groups 687 V.8 Dirichlet's Theorem 689 V.9 Ergodic Theorems 689 V.10 Fermat's Last Theorem 691 V.11 Fixed Point Theorems 693 V.12 The Four-Color Theorem 696 V.13 The Fundamental Theorem of Algebra 698 V.14 The Fundamental Theorem of Arithmetic 699 V.15 Godel's Theorem 700 V.16 Gromov's Polynomial-Growth Theorem 702 V.17 Hilbert's Nullstellensatz 703 V.18 The Independence of the Continuum Hypothesis 703 V.19 Inequalities 703 V.20 The Insolubility of the Halting Problem 706 V.21 The Insolubility of the Quintic 708 V.22 Liouville's Theorem and Roth's Theorem 710 V.23 Mostow's Strong Rigidity Theorem 711 V.24 The P versus NP Problem 713 V.25 The Poincare Conjecture 714 V.26 The Prime Number Theorem and the Riemann Hypothesis 714 V.27 Problems and Results in Additive Number Theory 715 V.28 From Quadratic Reciprocity to Class Field Theory 718 V.29 Rational Points on Curves and the Mordell Conjecture 720 V.30 The Resolution of Singularities 722 V.31 The Riemann-Roch Theorem 723 V.32 The Robertson-Seymour Theorem 725 V.33 The Three-Body Problem 726 V.34 The Uniformization Theorem 728 V.35 The Weil Conjectures 729 Part VI Mathematicians VI.1 Pythagoras (ca. 569 B.C.E.-ca. 494 B.C.E.) 733 VI.2 Euclid (ca. 325 B.C.E.-ca. 265 B.C.E.) 734 VI.3 Archimedes (ca. 287 B.C.E.-212 B.C.E.) 734 VI.4 Apollonius (ca. 262 B.C.E.-ca. 190 B.C.E.) 735 VI.5 Abu Ja'far Muhammad ibn Musa al-Khwarizmi (800-847) 736 VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170-ca. 1250) 737 VI.7 Girolamo Cardano (1501-1576) 737 VI.8 Rafael Bombelli (1526-after 1572) 737 VI.9 Francois Viete (1540-1603) 737 VI.10 Simon Stevin (1548-1620) 738 VI.11 Rene Descartes (1596-1650) 739 VI.12 Pierre Fermat (160?-1665) 740 VI.13 Blaise Pascal (1623-1662) 741 VI.14 Isaac Newton (1642-1727) 742 VI.15 Gottfried Wilhelm Leibniz (1646-1716) 743 VI.16 Brook Taylor (1685-1731) 745 VI.17 Christian Goldbach (1690-1764) 745 VI.18 The Bernoullis (fl. 18th century) 745 VI.19 Leonhard Euler (1707-1783) 747 VI.20 Jean Le Rond d'Alembert (1717-1783) 749 VI.21 Edward Waring (ca. 1735-1798) 750 VI.22 Joseph Louis Lagrange (1736-1813) 751 VI.23 Pierre-Simon Laplace (1749-1827) 752 VI.24 Adrien-Marie Legendre (1752-1833) 754 VI.25 Jean-Baptiste Joseph Fourier (1768-1830) 755 VI.26 Carl Friedrich Gauss (1777-1855) 755 VI.27 Simeon-Denis Poisson (1781-1840) 757 VI.28 Bernard Bolzano (1781-1848) 757 VI.29 Augustin-Louis Cauchy (1789-1857) 758 VI.30 August Ferdinand Mobius (1790-1868) 759 VI.31 Nicolai Ivanovich Lobachevskii (1792-1856) 759 VI.32 George Green (1793-1841) 760 VI.33 Niels Henrik Abel (1802-1829) 760 VI.34 Janos Bolyai (1802-1860) 762 VI.35 Carl Gustav Jacob Jacobi (1804-1851) 762 VI.36 Peter Gustav Lejeune Dirichlet (1805-1859) 764 VI.37 William Rowan Hamilton (1805-1865) 765 VI.38 Augustus De Morgan (1806-1871) 765 VI.39 Joseph Liouville (1809-1882) 766 VI.40 Eduard Kummer (1810-1893) 767 VI.41 Evariste Galois (1811-1832) 767 VI.42 James Joseph Sylvester (1814-1897) 768 VI.43 George Boole (1815-1864) 769 VI.44 Karl Weierstrass (1815-1897) 770 VI.45 Pafnuty Chebyshev (1821-1894) 771 VI.46 Arthur Cayley (1821-1895) 772 VI.47 Charles Hermite (1822-1901) 773 VI.48 Leopold Kronecker (1823-1891) 773 VI.49 Georg Friedrich Bernhard Riemann (1826-1866) 774 VI.50 Julius Wilhelm Richard Dedekind (1831-1916) 776 VI.51 Emile Leonard Mathieu (1835-1890) 776 VI.52 Camille Jordan (1838-1922) 777 VI.53 Sophus Lie (1842-1899) 777 VI.54 Georg Cantor (1845-1918) 778 VI.55 William Kingdon Clifford (1845-1879) 780 VI.56 Gottlob Frege (1848-1925) 780 VI.57 Christian Felix Klein (1849-1925) 782 VI.58 Ferdinand Georg Frobenius (1849-1917) 783 VI.59 Sofya (Sonya) Kovalevskaya (1850-1891) 784 VI.60 William Burnside (1852-1927) 785 VI.61 Jules Henri Poincare (1854-1912) 785 [Illustration credit: Portrait courtesy of Henri Poincare Archives (CNRS,UMR 7117, Nancy)] VI.62 Giuseppe Peano (1858-1932) 787 VI.63 David Hilbert (1862-1943) 788 VI.64 Hermann Minkowski (1864-1909) 789 VI.65 Jacques Hadamard (1865-1963) 790 VI.66 Ivar Fredholm (1866-1927) 791 VI.67 Charles-Jean de la Vallee Poussin (1866-1962) 792 VI.68 Felix Hausdorff (1868-1942) 792 VI.69 Elie Joseph Cartan (1869-1951) 794 VI.70 Emile Borel (1871-1956) 795 VI.71 Bertrand Arthur William Russell (1872-1970) 795 VI.72 Henri Lebesgue (1875-1941) 796 VI.73 Godfrey Harold Hardy (1877-1947) 797 VI.74 Frigyes (Frederic) Riesz (1880-1956) 798 VI.75 Luitzen Egbertus Jan Brouwer (1881-1966) 799 VI.76 Emmy Noether (1882-1935) 800 VI.77 Wac?aw Sierpinski (1882-1969) 801 VI.78 George Birkhoff (1884-1944) 802 VI.79 John Edensor Littlewood (1885-1977) 803 VI.80 Hermann Weyl (1885-1955) 805 VI.81 Thoralf Skolem (1887-1963) 806 VI.82 Srinivasa Ramanujan (1887-1920) 807 VI.83 Richard Courant (1888-1972) 808 VI.84 Stefan Banach (1892-1945) 809 VI.85 Norbert Wiener (1894-1964) 811 VI.86 Emil Artin (1898-1962) 812 VI.87 Alfred Tarski (1901-1983) 813 VI.88 Andrei Nikolaevich Kolmogorov (1903-1987) 814 VI.89 Alonzo Church (1903-1995) 816 VI.90 William Vallance Douglas Hodge (1903-1975) 816 VI.91 John von Neumann (1903-1957) 817 VI.92 Kurt Godel (1906-1978) 819 VI.93 Andre Weil (1906-1998) 819 VI.94 Alan Turing (1912-1954) 821 VI.95 Abraham Robinson (1918-1974) 822 VI.96 Nicolas Bourbaki (1935-) 823 Part VII The Influence of Mathematics VII.1 Mathematics and Chemistry 827 VII.2 Mathematical Biology 837 VII.3 Wavelets and Applications 848 VII.4 The Mathematics of Traffic in Networks 862 VII.5 The Mathematics of Algorithm Design 871 VII.6 Reliable Transmission of Information 878 VII.7 Mathematics and Cryptography 887 VII.8 Mathematics and Economic Reasoning 895 VII.9 The Mathematics of Money 910 VII.10 Mathematical Statistics 916 VII.11 Mathematics and Medical Statistics 921 VII.12 Analysis, Mathematical and Philosophical 928 VII.13 Mathematics and Music 935 VII.14 Mathematics and Art 944 Part VIII Final Perspectives VIII.1 The Art of Problem Solving 955 VIII.2 "Why Mathematics?" You Might Ask 966 VIII.3 The Ubiquity of Mathematics 977 VIII.4 Numeracy 983 VIII.5 Mathematics: An Experimental Science 991 VIII.6 Advice to a Young Mathematician 1000 VIII.7 A Chronology of Mathematical Events 1010 Index 1015

Winner of the 2011 Euler Book Prize, Mathematical Association of America Honorable Mention for the 2008 PROSE Award for Single Volume Reference/Science, Association of American Publishers One of Choice's Outstanding Academic Titles for 2009 "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 (Best Books of 2008) "Although the editors' original goal of text that could be understood by anyone with a good background in high school mathematics provided 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 man 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 that could only be possible relays, since 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