The Princeton Companion to Mathematics

Zusatztext "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 Informationen zum Autor Edited by Timothy GowersJune Barrow-Green and Imre Leader, associate editors Klappentext "This is a wonderful book. The content is overwhelming. Every practicing mathematician, everyone who uses mathematics, and everyone who is interested in mathematics must have a copy of their own." --Simon A. Levin, Princeton University " The Princeton Companion to Mathematics fills a vital need. It is the only book of its kind." --Victor J. Katz, professor emeritus, University of the District of Columbia "I think that this is a wonderful book, completely different from anything that has been written before about mathematics and mathematicians." --Endre Süli, University of Oxford " The Princeton Companion to Mathematics is a much needed--and will become a much used--reference work. In fact, it will stand alone as the reference work in mathematics." --John J. Watkins, Colorado College Zusammenfassung Presents two hundred entries that introduce basic mathematical tools and vocabulary. This title traces the development of modern mathematics. It explains essential terms and concepts; examines core ideas in major areas of mathematics; and describes the achievements of scores of famous mathematicians.

Autorentext

Edited by Timothy Gowers June Barrow-Green and Imre Leader, associate editors

Klappentext

"This is a wonderful book. The content is overwhelming. Every practicing mathematician, everyone who uses mathematics, and everyone who is interested in mathematics must have a copy of their own."--Simon A. Levin, Princeton University

"The Princeton Companion to Mathematics fills a vital need. It is the only book of its kind."--Victor J. Katz, professor emeritus, University of the District of Columbia

"I think that this is a wonderful book, completely different from anything that has been written before about mathematics and mathematicians."--Endre Süli, University of Oxford

"The Princeton Companion to Mathematics is a much needed--and will become a much used--reference work. In fact, it will stand alone as the reference work in mathematics."--John J. Watkins, Colorado College

Zusammenfassung
Presents two hundred entries that introduce basic mathematical tools and vocabulary. This title traces the development of modern mathematics. It explains essential terms and concepts; examines core ideas in major areas of mathematics; and describes the achievements of scores of famous mathematicians.

Inhalt
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 Symmetr