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, 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, Doron Zeilberger

I think that this is a wonderful book, completely different from anything that has been written before about mathematics and mathematicians.

fills a vital need. It is the only book of its kind.

is a much needed--and will become a much used--reference work. In fact, it will stand alone as the reference work in mathematics.

What is a modular form? What is the point of symplectic geometry? This one-of-a-kind reference illuminates modern pure mathematics in all its diversity. More than 200 articles, organized thematically and written by many of the world's leading mathematicians, explain the major ideas and branches of mathematics in a clear, accessible style. Presenting not only definitions but also motivation and context for concepts, methods, theorems, and other topics of mathematical study, this is an indispensable resource for everyone with a serious interest in the field.

1. What is Mathematics?Introduces the language and grammar of mathematicsfundamental definitionsgeneral goals of mathematical researchthe subject matter of mathematics

2. The Fundamental Ideas of MathematicsExplores algebraalgorithmsgeometryhow analysis became rigorousnumbersthe crisis in the foundations of mathematics the development of the idea of proof

3. Mathematical ObjectsDefines and explains more than 75 mathematical objects, concepts, and buzzwords, from axiom of choice to zeta function

4. Branches of MathematicsIncludes detailed coverage of algebraalgebraic geometryanalysiscombinatoricscomputationgeometrylogic and set theorynumber theoryprobability

5. MathematiciansProfiles 70 mathematicians, from Apollonius to Weyl, who influenced the field

6. Theorems and ProblemsDiscusses notable theorems and open problems, from the four-color theorem to the Reimann hypothesis

7. The Influence of MathematicsCovers the intellectual and practical influence of mathematics on other disciplines such as analytic philosophy, art, biology, chemistry, economics, finance, and music

8. MiscellaneousAdvice to a YoungMathematicianChronology of MathematicsComputer Experiments in MathematicsThe Art of Problem Solvingand more

CONTRIBUTORS (partial list):

Graham Allan ?? Noga Alon ?? George Andrews ?? Tom Archibald ?? Michael Atiyah ?? David Aubin ?? Joan Bagaria ?? Keith Ball ?? Alan Beardon ?? David Ben-Zvi ?? Vitaly Bergelson ?? Nick Bingham ?? Bela Bollabas ?? Henk Bos ?? Bodil Branner ?? Haim Brezis ?? Martin Bridson ?? John Burgess ?? Kevin Buzzard ?? Peter Cameron ?? Jean-Luc Chabert ?? Eugenia Cheng ?? Clifford Cocks ?? Alain Connes ?? Leo Corry ?? Wolfgang Coy ?? Tony Crilly ?? Serafina Cuomo ?? Mihalis Dafermos ?? Partha Dasgupta ?? Ingrid Daubechies ?? Joseph Dauben ?? John Dawson ?? Francois de Gandt ?? Persi Diaconis ?? Robbert Dijkgraaf ?? Jordan Ellenberg ?? Florence Fasanelli ?? Solomon Feferman ?? Anita Feferman ?? Charles Fefferman ?? Jose Ferreiros ?? Terry Gannon ?? Tony Gardiner ?? Charles Gillespie ?? Oded Goldreich ?? Catherine Goldstein ?? Francois Golse ?? Fernando Gouvea ?? Andrew Granville ?? Ivor Grattan-Guinness ?? Jeremy Gray ?? Ben Green ?? Niccolo Guicciardini ?? Michael Harris ?? Ulf Hashagen ?? Nigel Higson ?? Andrew Hodges ?? Kollar Janos ?? Frank Johnson ?? Mark Joshi ?? Dominic Joyce ?? Jean-Pierre Kahane ?? Frank Kelly ?? Sergiu Klainerman ?? Jon Kleinberg ?? Israel Kleiner ?? Jacek Klinowski ?? Eberhard Knobloch ?? Joe Kohn ?? Tom Korner ?? Peter Lax ?? Jean-Francois Le Gall ?? Raymond Lickorish ?? Martin Liebeck ?? Jesper Lutzen ?? Des MacHale ?? Lech Maligranda ?? David Marker ?? Adrian Mathias ?? Jean Mawhin ?? Barry Mazur ?? Colin McLarty ?? Peter Neumann ?? Catherine Nolan ?? James Norris ?? Marco Panza ?? Karen Parshall ?? Jeanne Peiffer ?? CarlPomerance ?? Helmut Pulte ?? Bruce Reed ?? Mike Reed ?? Adrian Rice ?? Jeremy Rickard ?? Eleanor Robson ?? Igor Rodnianski ?? John Roe ?? Mark Ronan ?? Edward Sandifer ?? Norbert Schappacher ?? Andrzej Schinzel ?? Erhard Scholz ?? Joachim Schwermer ?? Gordon Slade ?? David Spiegelhalter ?? Neil Strickland ?? Arild Stubhaug ?? Madhu Sudan ?? Terence Tao ?? Jamie Tappenden ?? Cliff Taubes ?? Rudiger Thiele ?? Nick Trefethen ?? Dirk van Dalen ?? Richard Weber ?? Dominic Welsh ?? Avi Wigderson ?? Herbert Wilf ?? David Wilkins ?? Ben Yandell ?? Doron Zeilberger

If you would like to receive information later this year about a pre-publication offer for this book, please send an email to: webmaster@pupress.princeton.edu

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 Schrödinger 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 Gödel'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 Poincaré 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 François Viète (1540-1603) 737

VI.10 Simon Stevin (1548-1620) 738

VI.11 René 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 Siméon-Denis Poisson (1781-1840) 757

VI.28 Bernard Bolzano (1781-1848) 757

VI.29 Augustin-Louis Cauchy (1789-1857) 758

VI.30 August Ferdinand Möbius (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 János 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 Évariste 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 Émile Léonard 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 Poincaré (1854-1912) 785

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 Vallée Poussin (1866-1962) 792

VI.68 Felix Hausdorff (1868-1942) 792

VI.69 Élie 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 (Frédéric) 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 Gödel (1906-1978) 819

VI.93 André 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