the-princeton-companion-to-mathematics-table-of-contents.txt

the-princeton-companion-to-mathematics-table-of-contents.txt

1 "Cover" 1
1 "Part I" 2
1 "I.1 What Is Mathematics About?" 2
1 "I.2 The Language and Grammar of Mathematics" 8
1 "I.3 Some Fundamental Mathematical Definitions" 17
1 "I.4 The General Goals of Mathematical Research" 46
1 "Part II The Origins of Modern" 74
1 "II.1 From Numbers to Number Systems" 74
1 "II.2 Geometry" 80
1 "II.3 The Development of Abstract Algebra" 93
1 "II.4 Algorithms" 104
1 "II.5 The Development of Rigor in Mathematical Analysis" 115
1 "II.6 The Development of the Idea of Proof" 127
1 "II.7 The Crisis in the Foundations of Mathematics" 140
1 "Part III Mathematical" 156
1 "III.1 The Axiom of Choice" 156
1 "III.2 The Axiom of Determinacy" 158
1 "III.3 Bayesian Analysis" 158
1 "III.4 Braid Groups" 159
1 "III.5 Buildings" 160
1 "III.6 Calabi-Yau Manifolds" 161
1 "III.7 Cardinals" 164
1 "III.8 Categories" 164
1 "III.9 Compactness and Compactification" 166
1 "III.10 Computational Complexity Classes" 168
1 "III.11 Countable and Uncountable Sets" 169
1 "III.12 C*-Algebras" 170
1 "III.13 Curvature" 171
1 "III.14 Designs" 171
1 "III.15 Determinants" 172
1 "III.16 Differential Forms and Integration" 173
1 "III.17 Dimension" 179
1 "III.18 Distributions" 183
1 "III.19 Duality" 186
1 "III.20 Dynamical Systems and Chaos" 188
1 "III.21 Elliptic Curves" 189
1 "III.22 The Euclidean Algorithm and Continued Fractions" 190
1 "III.23 The Euler and Navier-Stokes Equations" 192
1 "III.24 The Exponential and Logarithmic Functions" 195
1 "III.25 Expanders" 198
1 "III.26 The Fast Fourier Transform" 201
1 "III.27 The Fourier Transform" 203
1 "III.28 Fuchsian Groups" 206
1 "III.29 Function Spaces" 208
1 "III.30 Galois Groups" 211
1 "III.31 The Gamma Function" 212
1 "III.32 Generating Functions" 213
1 "III.33 Genus" 213
1 "III.34 Graphs" 214
1 "III.35 Hamiltonians" 214
1 "III.36 The Heat Equation" 215
1 "III.37 Hilbert Spaces" 218
1 "III.38 Holomorphic Functions" 219
1 "III.39 Homology and Cohomology" 219
1 "III.40 Homotopy Groups" 220
1 "III.41 The Hyperbolic Plane" 220
1 "III.42 The Ideal Class Group" 220
1 "III.43 Irrational and Transcendental Numbers" 221
1 "III.44 The Ising Model" 222
1 "III.45 Jordan Normal Form" 222
1 "III.46 Knot Polynomials" 224
1 "III.47 K-Theory" 226
1 "III.48 The Leech Lattice" 226
1 "III.49 L-Functions" 227
1 "III.50 Lie Theory" 228
1 "III.51 Linear and Nonlinear Waves and Solitons" 233
1 "III.52 Linear Operators and Their Properties" 238
1 "III.53 Local and Global in Number Theory" 240
1 "III.54 Optimization and Lagrange Multipliers" 243
1 "III.55 The Mandelbrot Set" 245
1 "III.56 Manifolds" 245
1 "III.57 Matroids" 246
1 "III.58 Measures" 247
1 "III.59 Metric Spaces" 249
1 "III.60 Models of Set Theory" 250
1 "III.61 Modular Arithmetic" 250
1 "III.62 Modular Forms" 252
1 "III.63 Moduli Spaces" 253
1 "III.64 The Monster Group" 253
1 "III.65 Normed Spaces and Banach Spaces" 254
1 "III.66 Number Fields" 255
1 "III.67 Orbifolds" 256
1 "III.68 Ordinals" 257
1 "III.69 The Peano Axioms" 257
1 "III.70 Permutation Groups" 258
1 "III.71 Phase Transitions" 260
1 "III.72 ?" 260
1 "III.73 Probability Distributions" 261
1 "III.74 Projective Space" 266
1 "III.75 Quadratic Forms" 266
1 "III.76 Quantum Computation" 267
1 "III.77 Quantum Groups" 270
1 "III.78 Quaternions, Octonions, and Normed Division Algebras" 274
1 "III.79 Representations" 277
1 "III.80 Ricci Flow" 278
1 "III.81 Riemannian Metrics" 280
1 "III.82 Riemann Surfaces" 281
1 "III.83 Rings, Ideals, and Modules" 283
1 "III.84 Schemes" 284
1 "III.85 The Schrodinger Equation" 285
1 "III.86 The Simplex Algorithm" 287
1 "III.87 Special Functions" 289
1 "III.88 The Spectrum" 293
1 "III.89 Spherical Harmonics" 294
1 "III.90 Symplectic Manifolds" 296
1 "III.91 Tensor Products" 300
1 "III.92 Topological Spaces" 300
1 "III.93 Transforms" 302
1 "III.94 Trigonometric Functions" 306
1 "III.95 Variational Methods" 308
1 "III.96 Varieties" 310
1 "III.97 Vector Bundles" 311
1 "III.98 Von Neumann Algebras" 311
1 "III.99 Wavelets" 311
1 "III.100 Zeta Functions" 312
1 "III.101 The Zermelo-Fraenkel Axioms" 312
1 "Part IV Branches of" 314
1 "IV.1 Set Theory" 314
1 "IV.2 Logic and Model Theory" 333
1 "IV.3 Algebraic Numbers" 345
1 "IV.4 Analytic Number Theory" 362
1 "IV.5 Computational Number Theory" 379
1 "IV.6 Arithmetic Geometry" 393
1 "IV.7 Algebraic Geometry" 404
1 "IV.8 Moduli Spaces" 413
1 "IV.9 Differential Topology" 425
1 "IV.10 Algebraic Topology" 437
1 "IV.11 Geometric and Combinatorial Group Theory" 450
1 "IV.12 Representation Theory" 467
1 "IV.13 Vertex Operator Algebras" 479
1 "IV.14 Mirror Symmetry" 490
1 "IV.15 Dynamics" 506
1 "IV.16 Partial Differential Equations" 523
1 "IV.17 General Relativity and the Einstein Equations" 550
1 "IV.18 Harmonic Analysis" 550
1 "IV.19 Operator Algebras" 557
1 "IV.20 Numerical Analysis" 570
1 "IV.21 Computational Complexity" 582
1 "IV.22 Enumerative and Algebraic Combinatorics" 610
1 "IV.23 Extremal and Probabilistic Combinatorics" 622
1 "IV.24 High-Dimensional Geometry and Its Probabilistic Analogues" 636
1 "IV.25 Stochastic Processes" 646
1 "IV.26 Probabilistic Models of Critical Phenomena" 657
1 "Part V Theorems and" -3
1 "V.1 The ABC Conjecture" 670
1 "V.2 The Atiyah-Singer Index Theorem" 670
1 "V.3 The Banach-Tarski Paradox" 673
1 "V.4 The Birch-Swinnerton-Dyer Conjecture" 674
1 "V.5 Carlesons Theorem" 675
1 "V.6 Cauchys Theorem" 676
1 "V.7 The Central Limit Theorem" 676
1 "V.8 The Classification of Finite Simple Groups" 676
1 "V.9 Dirichlets Theorem" 678
1 "V.10 Dvoretzkys Theorem" 678
1 "V.11 Ergodic Theorems" 679
1 "V.12 Fermats Last Theorem" 681
1 "V.13 Fixed-Point Theorems" 683
1 "V.14 The Four-Color Theorem" 686
1 "V.15 The Fundamental Theorem of Algebra" 688
1 "V.16 The Fundamental Theorem of Arithmetic" 689
1 "V.17 The Fundamental Theorem of Calculus" 690
1 "V.18 Godels Theorem" 690
1 "V.19 Gromovs Polynomial-Growth Theorem" 692
1 "V.20 Hilberts Nullstellensatz" 693
1 "V.21 The Independence of the Continuum Hypothesis" 693
1 "V.22 Inequalities" 693
1 "V.23 The Insolubility of the Halting Problem" 696
1 "V.24 The Insolubility of the Quintic" 698
1 "V.25 Liouvilles Theorem and Roths Theorem" 700
1 "V.26 Rational Points on Curves and the Mordell Conjecture" 701
1 "V.27 Mostows Strong Rigidity Theorem" 703
1 "V.28 The P = NP Problem" 705
1 "V.29 The Poincare Conjecture" 706
1 "V.30 Problems and Results in Additive Number Theory" 706
1 "V.31 From Quadratic Reciprocity to Class Field Theory" 709
1 "V.32 The Resolution of Singularities" 712
1 "V.33 The Riemann Hypothesis" 712
1 "V.34 The Riemann-Roch Theorem" 713
1 "V.35 The Robertson-Seymour Theorem" 715
1 "V.36 The Three-Body Problem" 716
1 "V.37 The Uniformization Theorem" 718
1 "V.38 The Weil Conjectures" 719
1 "Part VI" 724
1 "VI.2 Euclid" 725
1 "VI.3 Archimedes" 725
1 "VI.4 Apollonius" 726
1 "VI.5 Leonardo of Pisa (known as Fibonacci)" 727
1 "VI.6 Girolamo Cardano" 727
1 "VI.7 Rafael Bombelli" 727
1 "VI.8 Francois Viete" 728
1 "VI.9 Simon Stevin" 729
1 "VI.10 Rene Descartes" 729
1 "VI.11 Pierre Fermat" 731
1 "VI.12 Blaise Pascal" 732
1 "VI.13 Isaac Newton" 732
1 "VI.14 Gottfried Wilhelm Leibniz" 734
1 "VI.15 The Bernoullis" 735
1 "VI.16 Brooke Taylor" 737
1 "VI.17 Christian Goldbach" 737
1 "VI.18 Leonhard Euler" 737
1 "VI.19 Jean Le Rond dAlembert" 739
1 "VI.20 Edward Waring" 741
1 "VI.21 Joseph Louis Lagrange" 741
1 "VI.22 Pierre-Simon Laplace" 742
1 "VI.23 Adrien-Marie Legendre" 744
1 "VI.24 Jean-Baptiste Joseph Fourier" 745
1 "VI.25 Carl Friedrich Gauss" 746
1 "VI.26 Simeon-Denis Poisson" 747
1 "VI.27 Bernard Bolzano" 747
1 "VI.28 Augustin-Louis Cauchy" 748
1 "VI.29 August Ferdinand Mobius" 749
1 "VI.30 Nicolai Ivanovich Lobachevskii" 749
1 "VI.31 George Green" 750
1 "VI.32 Niels Henrik Abel" 750
1 "VI.33 Janos Bolyai" 752
1 "VI.34 Carl Gustav Jacob Jacobi" 752
1 "VI.35 Peter Gustav Lejeune Dirichlet" 754
1 "VI.36 William Rowan Hamilton" 755
1 "VI.37 Augustus De Morgan" 756
1 "VI.38 Joseph Liouville" 756
1 "VI.39 Eduard Kummer" 757
1 "VI.40 Evariste Galois" 757
1 "VI.41 James Joseph Sylvester" 758
1 "VI.42 George Boole" 759
1 "VI.43 Karl Weierstrass" 760
1 "VI.44 Pafnuty Chebyshev" 761
1 "VI.45 Arthur Cayley" 762
1 "VI.46 Charles Hermite" 763
1 "VI.47 Leopold Kronecker" 763
1 "VI.48 Georg Bernhard Friedrich Riemann" 764
1 "VI.49 Julius Wilhelm Richard Dedekind" 766
1 "VI.50 Emile Leonard Mathieu" 766
1 "VI.51 Camille Jordan" 767
1 "VI.52 Sophus Lie" 767
1 "VI.53 Georg Cantor" 768
1 "VI.54 William Kingdon Clifford" 770
1 "VI.55 Gottlob Frege" 770
1 "VI.56 Christian Felix Klein" 772
1 "VI.57 Ferdinand Georg Frobenius" 773
1 "VI.58 Sonya Kovalevskaya" 774
1 "VI.59 William Burnside" 775
1 "VI.60 Jules Henri Poincare" 775
1 "VI.61 Giuseppe Peano" 777
1 "VI.62 David Hilbert" 778
1 "VI.63 Hermann Minkowski" 779
1 "VI.64 Jacques Hadamard" 780
1 "VI.65 Ivar Fredholm" 781
1 "VI.66 Charles-Jean de la Vallee Poussin" 781
1 "VI.67 Felix Hausdorff" 782
1 "VI.68 Elie Joseph Cartan" 783
1 "VI.69 Emile Borel" 784
1 "VI.70 Bertrand Arthur William Russell" 785
1 "VI.71 Henri Lebesgue" 785
1 "VI.72 Godfrey Harold Hardy" 787
1 "VI.73 Frigyes (Frederic) Riesz" 788
1 "VI.74 Luitzen Egbertus Jan Brouwer" 789
1 "VI.75 Emmy Noether" 790
1 "VI.76 Waclaw Sierpi?nski" 791
1 "VI.77 George Birkhoff" 792
1 "VI.78 John Edensor Littlewood" 793
1 "VI.79 Hermann Weyl" 795
1 "VI.80 Thoralf Skolem" 796
1 "VI.81 Srinivasa Ramanujan" 797
1 "VI.82 Richard Courant" 798
1 "VI.83 Stefan Banach" 799
1 "VI.84 Norbert Wiener" 801
1 "VI.85 Emil Artin" 802
1 "VI.86 Alfred Tarski" 803
1 "VI.87 Andrei Nikolaevich Kolmogorov" 804
1 "VI.88 William Vallance Douglas Hodge" 806
1 "VI.89 John von Neumann" 807
1 "VI.90 Kurt Godel" 808
1 "VI.91 Andre Weil" 809
1 "VI.92 Alan Turing" 810
1 "VI.93 Abraham Robinson" 811
1 "VI.94 Nicolas Bourbaki" 813
1 "Part VII The Influence of" 816
1 "VII.1 Mathematics and Chemistry" 816
1 "VII.2 Mathematical Biology" 827
1 "VII.3 Wavelets and Applications" 837
1 "VII.4 The Mathematics of Traffic in Networks" 852
1 "VII.5 The Mathematics of Algorithm Design" 860
1 "VII.6 Reliable Transmission of Information" 868
1 "VII.7 Mathematics and Cryptography" 876
1 "VII.8 Mathematics and Economic Reasoning" 885
1 "VII.9 The Mathematics of Money" 900
1 "VII.10 Mathematical Statistics" 906
1 "VII.11 Mathematics and Medical Statistics" 911
1 "VII.12 Analysis, Mathematical and Philosophical" 918
1 "VII.13 Mathematics and Music" 925
1 "VII.14 Mathematics and Art" 934
1 "Part VIII Final" 946
1 "VIII.1 The Art of Problem Solving" 946
1 "VIII.2 Why Mathematics? You Might Ask" 957
1 "VIII.3 The Ubiquity of Mathematics" 968
1 "VIII.4 Numeracy" 974
1 "VIII.5 Mathematics: An Experimental Science" 982
1 "VIII.6 Advice to a Young Mathematician" 991