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More About This Textbook
Overview
This is a oneofakind 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 musicand 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.
Contributors incude:
Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June BarrowGreen, Alan Beardon, David D. BenZvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, JeanLuc 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 GrattanGuinness, 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, JeanFrançois 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 SiegmundSchultze, 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
Editorial Reviews
bitplayer.org
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
International Sugar Journal
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.European Mathematical Society Newsletter
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.The Times  Ian Stewart
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.Science News  Tom Siegfried
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.London Mathematical Society  Robin Wilson
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 1000page 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.Notices of the American Mathematical Society  Simon Donaldson
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.Notices of the American Mathematical Society  Bryan Birch
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.Notices of the American Mathematical Society  Gil Kalai
The book is so rich and yet it is well done. A rare achievement indeed!Notices of the American Mathematical Society  Richard Kenyon
My advice to you, reader is to buy the book, open it to a random page, read, enjoy, and be enlightened.Bloomsbury Review  Gregory McNamee
Massive . . . endlessly fascinating.Choice  S.J. Colley
This volume is an enormous, farreaching effort to survey the current landscape of (pure) mathematics. Chief editor Gowers and associate editors BarrowGreen 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.Convergence  Frank Swetz
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.SIAM Review  Jonathan M. Borwein
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 wellintegrated and highly readable volume.MathBlog.com  Antonio Cangiano
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.bitplayer.org  Brian Hayes
I'm completely charmed. This is one of those books that makes you wish you had a desert island to be marooned on.Scientific Computing  John A. Wass
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.From the Publisher
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 numbercrunchers."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 shortlived, this wideranging 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 BarrowGreen 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 1000page 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 oneofakind 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, farreaching effort to survey the current landscape of (pure) mathematics. Chief editor Gowers and associate editors BarrowGreen 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 mathematicianprofessional 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 morewhich have led to a wellintegrated 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, MathBlog.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, bitplayer.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
Science
Although the editors' original goal of text that could be understood by anyone with a good background in high school mathematics provided shortlived, this wideranging 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.American Scientist
This impressive book represents an extremely ambitious and, I might add, highly successful attempt by Timothy Gowers and his coeditors, June BarrowGreen 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!Science News
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
London Mathematical Society
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 1000page 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
Mathematics Teacher
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 oneofakind reference for all things mathematics.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
Bloomsbury Review
Massive . . . endlessly fascinating.— Gregory McNamee
Choice
This volume is an enormous, farreaching effort to survey the current landscape of (pure) mathematics. Chief editor Gowers and associate editors BarrowGreen 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
Convergence
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
SIAM Review
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 wellintegrated and highly readable volume.— Jonathan M. Borwein
MathBlog.com
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
Scientific Computing
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
The Times
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 Economist
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 numbercrunchers.Library Journal
Edited by Gowers, a recipient of the Fields Medal (the most esteemed prize in mathematics), 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. Moreover, the volume explains core ideas in major areas of mathematics, describes the achievements of scores of famous mathematicians, and discusses the impact of mathematics on other disciplines such as biology, finance, and music.
—Al Vara
Product Details
Related Subjects
Meet the Author
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 BarrowGreen is lecturer in the history of mathematics at the Open University. Imre Leader is professor of pure mathematics at the University of Cambridge.
Table of Contents
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 CalabiYau 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 NavierStokes 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 KTheory 227
III.46 The Leech Lattice 227
III.47 LFunctions 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 ZermeloFraenkel 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 HighDimensional Geometry and Its Probabilistic Analogues 670
Part V Theorems and Problems
V.1 The ABC Conjecture 681
V.2 The AtiyahSinger Index Theorem 681
V.3 The BanachTarski Paradox 684
V.4 The BirchSwinnertonDyer 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 FourColor 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 PolynomialGrowth 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 RiemannRoch Theorem 723
V.32 The RobertsonSeymour Theorem 725
V.33 The ThreeBody 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 alKhwarizmi (800847) 736
VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170ca. 1250) 737
VI.7 Girolamo Cardano (15011576) 737
VI.8 Rafael Bombelli (1526after 1572) 737
VI.9 François Viète (15401603) 737
VI.10 Simon Stevin (15481620) 738
VI.11 René Descartes (15961650) 739
VI.12 Pierre Fermat (160?1665) 740
VI.13 Blaise Pascal (16231662) 741
VI.14 Isaac Newton (16421727) 742
VI.15 Gottfried Wilhelm Leibniz (16461716) 743
VI.16 Brook Taylor (16851731) 745
VI.17 Christian Goldbach (16901764) 745
VI.18 The Bernoullis (fl. 18th century) 745
VI.19 Leonhard Euler (17071783) 747
VI.20 Jean Le Rond d'Alembert (17171783) 749
VI.21 Edward Waring (ca. 17351798) 750
VI.22 Joseph Louis Lagrange (17361813) 751
VI.23 PierreSimon Laplace (17491827) 752
VI.24 AdrienMarie Legendre (17521833) 754
VI.25 JeanBaptiste Joseph Fourier (17681830) 755
VI.26 Carl Friedrich Gauss (17771855) 755
VI.27 SiméonDenis Poisson (17811840) 757
VI.28 Bernard Bolzano (17811848) 757
VI.29 AugustinLouis Cauchy (17891857) 758
VI.30 August Ferdinand Möbius (17901868) 759
VI.31 Nicolai Ivanovich Lobachevskii (17921856) 759
VI.32 George Green (17931841) 760
VI.33 Niels Henrik Abel (18021829) 760
VI.34 János Bolyai (18021860) 762
VI.35 Carl Gustav Jacob Jacobi (18041851) 762
VI.36 Peter Gustav Lejeune Dirichlet (18051859) 764
VI.37 William Rowan Hamilton (18051865) 765
VI.38 Augustus De Morgan (18061871) 765
VI.39 Joseph Liouville (18091882) 766
VI.40 Eduard Kummer (18101893) 767
VI.41 Évariste Galois (18111832) 767
VI.42 James Joseph Sylvester (18141897) 768
VI.43 George Boole (18151864) 769
VI.44 Karl Weierstrass (18151897) 770
VI.45 Pafnuty Chebyshev (18211894) 771
VI.46 Arthur Cayley (18211895) 772
VI.47 Charles Hermite (18221901) 773
VI.48 Leopold Kronecker (18231891) 773
VI.49 Georg Friedrich Bernhard Riemann (18261866) 774
VI.50 Julius Wilhelm Richard Dedekind (18311916) 776
VI.51 Émile Léonard Mathieu (18351890) 776
VI.52 Camille Jordan (18381922) 777
VI.53 Sophus Lie (18421899) 777
VI.54 Georg Cantor (18451918) 778
VI.55 William Kingdon Clifford (18451879) 780
VI.56 Gottlob Frege (18481925) 780
VI.57 Christian Felix Klein (18491925) 782
VI.58 Ferdinand Georg Frobenius (18491917) 783
VI.59 Sofya (Sonya) Kovalevskaya (18501891) 784
VI.60 William Burnside (18521927) 785
VI.61 Jules Henri Poincaré (18541912) 785 [Illustration credit: Portrait courtesy of Henri Poincaré Archives (CNRS,UMR 7117, Nancy)]
VI.62 Giuseppe Peano (18581932) 787
VI.63 David Hilbert (18621943) 788
VI.64 Hermann Minkowski (18641909) 789
VI.65 Jacques Hadamard (18651963) 790
VI.66 Ivar Fredholm (18661927) 791
VI.67 CharlesJean de la Vallée Poussin (18661962) 792
VI.68 Felix Hausdorff (18681942) 792
VI.69 Élie Joseph Cartan (18691951) 794
VI.70 Emile Borel (18711956) 795
VI.71 Bertrand Arthur William Russell (18721970) 795
VI.72 Henri Lebesgue (18751941) 796
VI.73 Godfrey Harold Hardy (18771947) 797
VI.74 Frigyes (Frédéric) Riesz (18801956) 798
VI.75 Luitzen Egbertus Jan Brouwer (18811966) 799
VI.76 Emmy Noether (18821935) 800
VI.77 Wac?aw Sierpinski (18821969) 801
VI.78 George Birkhoff (18841944) 802
VI.79 John Edensor Littlewood (18851977) 803
VI.80 Hermann Weyl (18851955) 805
VI.81 Thoralf Skolem (18871963) 806
VI.82 Srinivasa Ramanujan (18871920) 807
VI.83 Richard Courant (18881972) 808
VI.84 Stefan Banach (18921945) 809
VI.85 Norbert Wiener (18941964) 811
VI.86 Emil Artin (18981962) 812
VI.87 Alfred Tarski (19011983) 813
VI.88 Andrei Nikolaevich Kolmogorov (19031987) 814
VI.89 Alonzo Church (19031995) 816
VI.90 William Vallance Douglas Hodge (19031975) 816
VI.91 John von Neumann (19031957) 817
VI.92 Kurt Gödel (19061978) 819
VI.93 André Weil (19061998) 819
VI.94 Alan Turing (19121954) 821
VI.95 Abraham Robinson (19181974) 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