Table of Contents for The Princeton Companion to Mathematics. Slightly off as the ToC available from the library of congress is different from the current editions. Also the number of files here is a case study in version control.

add-toc-to-pdf.sh

!#/usr/bin/env bash

function add-toc-to-pdf() {
  [[ ${#} -ne 3 ]] && echo -e 'usage: add-toc-to-pdf path/to/toc.txt /path/to/file.pdf /path/to/output.pdf' && return -1
  [[ $(file ${2} | cut -d ' ' -f 2) != "PDF" ]] && echo -e 'Error: second argument must be a pdf.' && return -1
  
  cpdf -add-bookmarks ${1} ${2} -o ${3}
}

final-cleaned-generated-toc.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

generate-toc-from-gist-for-cpdf.sh

!#/usr/bin/env bash

cat -- \
  <(echo '"') \
  <(curl https://gist.githubusercontent.com/colejhudson/b402192006835d194ec94a069bb49989/raw/d03d6d7592237e9a5d1bcb3f7f515afad04e372c/raw-toc.txt \
      | tr -d "'" \
      | tr -d '"'
   ) \
  <(echo '".split("\n").map {|s| p "1 #{s}"}') \
  | ruby -- \
  | copy # This is an alias defined as 'alias copy="xsel -ib"'

generate-toc.sh

!#/usr/bin/env bash

cat -- \
  <(echo '"') \
  <(curl https://gist.githubusercontent.com/colejhudson/b402192006835d194ec94a069bb49989/raw/237fa3fc851e88be507add13d6c6925e4ca9a8ed/raw-generated-toc.txt | tr -d "'" | tr -d '"') \
  <(echo '".split("\n").map {|s| STDOUT.write("1 \"#{s.split(" ")[1..-2].join(" ")}\" #{s.split(" ").last.to_i - 3}\n") }') \
  | ruby --

raw-generated-toc.txt

"1 Contents"
"1 Preface ix"
"1 List of Contributors xvii"
"1 Part I Introduction"
"1 I.1 What Is Mathematics About? 1"
"1 I.2 The Language and Grammar of Mathematics 8"
"1 I.3 Some Fundamental Mathematical Definitions 16"
"1 I.4 The General Goals of Mathematical Research 46"
"1 Part II The Origins of Modern Mathematics"
"1 II.1 From Numbers to Number Systems 77"
"1 II.2 Geometry 83"
"1 II.3 The Development of Abstract Algebra 96"
"1 II.4 Algorithms 107"
"1 II.5 The Development of Rigor in Mathematical Analysis 118"
"1 II.6 The Development of the Idea of Proof 130"
"1 II.7 The Crisis in the Foundations of Mathematics 143"
"1 Part III Mathematical Concepts"
"1 III.1 The Axiom of Choice 159"
"1 III.2 The Axiom of Determinacy 161"
"1 III.3 Bayesian Analysis 161"
"1 III.4 Braid Groups 162"
"1 III.5 Buildings 163"
"1 III.6 Calabi-Yau Manifolds 164"
"1 III.7 Cardinals 167"
"1 III.8 Categories 167"
"1 III.9 Compactness and Compactification 169"
"1 III.10 Computational Complexity Classes 171"
"1 III.11 Countable and Uncountable Sets 172"
"1 III.12 C?-Algebras 173"
"1 III.13 Curvature 174"
"1 III.14 Designs 174"
"1 III.15 Determinants 175"
"1 III.16 Differential Forms and Integration 176"
"1 III.17 Dimension 182"
"1 III.18 Distributions 186"
"1 III.19 Duality 189"
"1 III.20 Dynamical Systems and Chaos 191"
"1 III.21 Elliptic Curves 192"
"1 III.22 The Euclidean Algorithm and Continued Fractions 193"
"1 III.23 The Euler and Navier-Stokes Equations 195"
"1 III.24 The Exponential and Logarithmic Functions 198"
"1 III.25 Expanders 201"
"1 III.26 The Fast Fourier Transform 204"
"1 III.27 The Fourier Transform 206"
"1 III.28 Fuchsian Groups 209"
"1 III.29 Function Spaces 211"
"1 III.30 Galois Groups 214"
"1 III.31 The Gamma Function 215"
"1 III.32 Generating Functions 216"
"1 III.33 Genus 216"
"1 III.34 Graphs 217"
"1 III.35 Hamiltonians 217"
"1 III.36 The Heat Equation 218"
"1 III.37 Hilbert Spaces 221"
"1 III.38 Holomorphic Functions 222"
"1 III.39 Homology and Cohomology 222"
"1 III.40 Homotopy Groups 223"
"1 III.41 The Hyperbolic Plane 223"
"1 III.42 The Ideal Class Group 223"
"1 III.43 Irrational and Transcendental Numbers 224"
"1 III.44 The Ising Model 225"
"1 III.45 Jordan Normal Form 225"
"1 III.46 Knot Polynomials 227"
"1 III.47 K-Theory 229"
"1 III.48 The Leech Lattice 229"
"1 III.49 L-Functions 230"
"1 III.50 Lie Theory 231"
"1 III.51 Linear and Nonlinear Waves and Solitons 236"
"1 III.52 Linear Operators and Their Properties 241"
"1 III.53 Local and Global in Number Theory 243"
"1 III.54 Optimization and Lagrange Multipliers 246"
"1 III.55 The Mandelbrot Set 248"
"1 III.56 Manifolds 248"
"1 III.57 Matroids 249"
"1 III.58 Measures 250"
"1 III.59 Metric Spaces 252"
"1 III.60 Models of Set Theory 253"
"1 III.61 Modular Arithmetic 253"
"1 III.62 Modular Forms 255"
"1 III.63 Moduli Spaces 256"
"1 III.64 The Monster Group 256"
"1 III.65 Normed Spaces and Banach Spaces 257"
"1 III.66 Number Fields 258"
"1 III.67 Orbifolds 259"
"1 III.68 Ordinals 260"
"1 III.69 The Peano Axioms 260"
"1 III.70 Permutation Groups 261"
"1 III.71 Phase Transitions 263"
"1 III.72 ? 263"
"1 III.73 Probability Distributions 264"
"1 III.74 Projective Space 269"
"1 III.75 Quadratic Forms 269"
"1 III.76 Quantum Computation 270"
"1 III.77 Quantum Groups 273"
"1 III.78 Quaternions, Octonions, and Normed Division Algebras 277"
"1 III.79 Representations 280"
"1 III.80 Ricci Flow 281"
"1 III.81 Riemannian Metrics 283"
"1 III.82 Riemann Surfaces 284"
"1 III.83 Rings, Ideals, and Modules 286"
"1 III.84 Schemes 287"
"1 III.85 The Schrodinger Equation 288"
"1 III.86 The Simplex Algorithm 290"
"1 III.87 Special Functions 292"
"1 III.88 The Spectrum 296"
"1 III.89 Spherical Harmonics 297"
"1 III.90 Symplectic Manifolds 299"
"1 III.91 Tensor Products 303"
"1 III.92 Topological Spaces 303"
"1 III.93 Transforms 305"
"1 III.94 Trigonometric Functions 309"
"1 III.95 Variational Methods 311"
"1 III.96 Varieties 313"
"1 III.97 Vector Bundles 314"
"1 III.98 Von Neumann Algebras 314"
"1 III.99 Wavelets 314"
"1 III.100 Zeta Functions 315"
"1 III.101 The Zermelo-Fraenkel Axioms 315"
"1 Part IV Branches of Mathematics"
"1 IV.1 Set Theory 317"
"1 IV.2 Logic and Model Theory 336"
"1 IV.3 Algebraic Numbers 348"
"1 IV.4 Analytic Number Theory 365"
"1 IV.5 Computational Number Theory 382"
"1 IV.6 Arithmetic Geometry 396"
"1 IV.7 Algebraic Geometry 407"
"1 IV.8 Moduli Spaces 416"
"1 IV.9 Differential Topology 428"
"1 IV.10 Algebraic Topology 440"
"1 IV.11 Geometric and Combinatorial Group Theory 453"
"1 IV.12 Representation Theory 470"
"1 IV.13 Vertex Operator Algebras 482"
"1 IV.14 Mirror Symmetry 493"
"1 IV.15 Dynamics 509"
"1 IV.16 Partial Differential Equations 526"
"1 IV.17 General Relativity and the Einstein Equations 553"
"1 IV.18 Harmonic Analysis 553"
"1 IV.19 Operator Algebras 560"
"1 IV.20 Numerical Analysis 573"
"1 IV.21 Computational Complexity 585"
"1 IV.22 Enumerative and Algebraic Combinatorics 613"
"1 IV.23 Extremal and Probabilistic Combinatorics 625"
"1 IV.24 High-Dimensional Geometry and Its Probabilistic Analogues 639"
"1 IV.25 Stochastic Processes 649"
"1 IV.26 Probabilistic Models of Critical Phenomena 660"
"1 Part V Theorems and Problems"
"1 V.1 The ABC Conjecture 673"
"1 V.2 The Atiyah-Singer Index Theorem 673"
"1 V.3 The Banach-Tarski Paradox 676"
"1 V.4 The Birch-Swinnerton-Dyer Conjecture 677"
"1 V.5 Carlesons Theorem 678"
"1 V.6 Cauchys Theorem 679"
"1 V.7 The Central Limit Theorem 679"
"1 V.8 The Classification of Finite Simple Groups 679"
"1 V.9 Dirichlets Theorem 681"
"1 V.10 Dvoretzkys Theorem 681"
"1 V.11 Ergodic Theorems 682"
"1 V.12 Fermats Last Theorem 684"
"1 V.13 Fixed-Point Theorems 686"
"1 V.14 The Four-Color Theorem 689"
"1 V.15 The Fundamental Theorem of Algebra 691"
"1 V.16 The Fundamental Theorem of Arithmetic 692"
"1 V.17 The Fundamental Theorem of Calculus 693"
"1 V.18 Godels Theorem 693"
"1 V.19 Gromovs Polynomial-Growth Theorem 695"
"1 V.20 Hilberts Nullstellensatz 696"
"1 V.21 The Independence of the Continuum Hypothesis 696"
"1 V.22 Inequalities 696"
"1 V.23 The Insolubility of the Halting Problem 699"
"1 V.24 The Insolubility of the Quintic 701"
"1 V.25 Liouvilles Theorem and Roths Theorem 703"
"1 V.26 Rational Points on Curves and the Mordell Conjecture 704"
"1 V.27 Mostows Strong Rigidity Theorem 706"
"1 V.28 The P = NP Problem 708"
"1 V.29 The Poincare Conjecture 709"
"1 V.30 Problems and Results in Additive Number Theory 709"
"1 V.31 From Quadratic Reciprocity to Class Field Theory 712"
"1 V.32 The Resolution of Singularities 715"
"1 V.33 The Riemann Hypothesis 715"
"1 V.34 The Riemann-Roch Theorem 716"
"1 V.35 The Robertson-Seymour Theorem 718"
"1 V.36 The Three-Body Problem 719"
"1 V.37 The Uniformization Theorem 721"
"1 V.38 The Weil Conjectures 722"
"1 Part VI Mathematicians"
"1 VI.1 Pythagoras 727"
"1 VI.2 Euclid 728"
"1 VI.3 Archimedes 728"
"1 VI.4 Apollonius 729"
"1 VI.5 Leonardo of Pisa (known as Fibonacci) 730"
"1 VI.6 Girolamo Cardano 730"
"1 VI.7 Rafael Bombelli 730"
"1 VI.8 Francois Viete 731"
"1 VI.9 Simon Stevin 732"
"1 VI.10 Rene Descartes 732"
"1 VI.11 Pierre Fermat 734"
"1 VI.12 Blaise Pascal 735"
"1 VI.13 Isaac Newton 735"
"1 VI.14 Gottfried Wilhelm Leibniz 737"
"1 VI.15 The Bernoullis 738"
"1 VI.16 Brooke Taylor 740"
"1 VI.17 Christian Goldbach 740"
"1 VI.18 Leonhard Euler 740"
"1 VI.19 Jean Le Rond dAlembert 742"
"1 VI.20 Edward Waring 744"
"1 VI.21 Joseph Louis Lagrange 744"
"1 VI.22 Pierre-Simon Laplace 745"
"1 VI.23 Adrien-Marie Legendre 747"
"1 VI.24 Jean-Baptiste Joseph Fourier 748"
"1 VI.25 Carl Friedrich Gauss 749"
"1 VI.26 Simeon-Denis Poisson 750"
"1 VI.27 Bernard Bolzano 750"
"1 VI.28 Augustin-Louis Cauchy 751"
"1 VI.29 August Ferdinand Mobius 752"
"1 VI.30 Nicolai Ivanovich Lobachevskii 752"
"1 VI.31 George Green 753"
"1 VI.32 Niels Henrik Abel 753"
"1 VI.33 Janos Bolyai 755"
"1 VI.34 Carl Gustav Jacob Jacobi 755"
"1 VI.35 Peter Gustav Lejeune Dirichlet 757"
"1 VI.36 William Rowan Hamilton 758"
"1 VI.37 Augustus De Morgan 759"
"1 VI.38 Joseph Liouville 759"
"1 VI.39 Eduard Kummer 760"
"1 VI.40 Evariste Galois 760"
"1 VI.41 James Joseph Sylvester 761"
"1 VI.42 George Boole 762"
"1 VI.43 Karl Weierstrass 763"
"1 VI.44 Pafnuty Chebyshev 764"
"1 VI.45 Arthur Cayley 765"
"1 VI.46 Charles Hermite 766"
"1 VI.47 Leopold Kronecker 766"
"1 VI.48 Georg Bernhard Friedrich Riemann 767"
"1 VI.49 Julius Wilhelm Richard Dedekind 769"
"1 VI.50 Emile Leonard Mathieu 769"
"1 VI.51 Camille Jordan 770"
"1 VI.52 Sophus Lie 770"
"1 VI.53 Georg Cantor 771"
"1 VI.54 William Kingdon Clifford 773"
"1 VI.55 Gottlob Frege 773"
"1 VI.56 Christian Felix Klein 775"
"1 VI.57 Ferdinand Georg Frobenius 776"
"1 VI.58 Sonya Kovalevskaya 777"
"1 VI.59 William Burnside 778"
"1 VI.60 Jules Henri Poincare 778"
"1 VI.61 Giuseppe Peano 780"
"1 VI.62 David Hilbert 781"
"1 VI.63 Hermann Minkowski 782"
"1 VI.64 Jacques Hadamard 783"
"1 VI.65 Ivar Fredholm 784"
"1 VI.66 Charles-Jean de la Vallee Poussin 784"
"1 VI.67 Felix Hausdorff 785"
"1 VI.68 Elie Joseph Cartan 786"
"1 VI.69 Emile Borel 787"
"1 VI.70 Bertrand Arthur William Russell 788"
"1 VI.71 Henri Lebesgue 788"
"1 VI.72 Godfrey Harold Hardy 790"
"1 VI.73 Frigyes (Frederic) Riesz 791"
"1 VI.74 Luitzen Egbertus Jan Brouwer 792"
"1 VI.75 Emmy Noether 793"
"1 VI.76 Waclaw Sierpi?nski 794"
"1 VI.77 George Birkhoff 795"
"1 VI.78 John Edensor Littlewood 796"
"1 VI.79 Hermann Weyl 798"
"1 VI.80 Thoralf Skolem 799"
"1 VI.81 Srinivasa Ramanujan 800"
"1 VI.82 Richard Courant 801"
"1 VI.83 Stefan Banach 802"
"1 VI.84 Norbert Wiener 804"
"1 VI.85 Emil Artin 805"
"1 VI.86 Alfred Tarski 806"
"1 VI.87 Andrei Nikolaevich Kolmogorov 807"
"1 VI.88 William Vallance Douglas Hodge 809"
"1 VI.89 John von Neumann 810"
"1 VI.90 Kurt Godel 811"
"1 VI.91 Andre Weil 812"
"1 VI.92 Alan Turing 813"
"1 VI.93 Abraham Robinson 814"
"1 VI.94 Nicolas Bourbaki 816"
"1 Part VII The Influence of Mathematics"
"1 VII.1 Mathematics and Chemistry 819"
"1 VII.2 Mathematical Biology 830"
"1 VII.3 Wavelets and Applications 840"
"1 VII.4 The Mathematics of Traffic in Networks 855"
"1 VII.5 The Mathematics of Algorithm Design 863"
"1 VII.6 Reliable Transmission of Information 871"
"1 VII.7 Mathematics and Cryptography 879"
"1 VII.8 Mathematics and Economic Reasoning 888"
"1 VII.9 The Mathematics of Money 903"
"1 VII.10 Mathematical Statistics 909"
"1 VII.11 Mathematics and Medical Statistics 914"
"1 VII.12 Analysis, Mathematical and Philosophical 921"
"1 VII.13 Mathematics and Music 928"
"1 VII.14 Mathematics and Art 937"
"1 Part VIII Final Perspectives"
"1 VIII.1 The Art of Problem Solving 949"
"1 VIII.2 Why Mathematics? You Might Ask 960"
"1 VIII.3 The Ubiquity of Mathematics 971"
"1 VIII.4 Numeracy 977"
"1 VIII.5 Mathematics: An Experimental Science 985"
"1 VIII.6 Advice to a Young Mathematician 994"
"1 VIII.7 A Chronology of Major Mathematical Events 1003"

raw-toc.txt

Contents
Preface ix
List of 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 46
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 96
II.4 Algorithms 107
II.5 The Development of Rigor in
Mathematical Analysis 118
II.6 The Development of the Idea of Proof 130
II.7 The Crisis in the Foundations of Mathematics 143
Part III Mathematical Concepts
III.1 The Axiom of Choice 159
III.2 The Axiom of Determinacy 161
III.3 Bayesian Analysis 161
III.4 Braid Groups 162
III.5 Buildings 163
III.6 Calabi-Yau Manifolds 164
III.7 Cardinals 167
III.8 Categories 167
III.9 Compactness and Compactification 169
III.10 Computational Complexity Classes 171
III.11 Countable and Uncountable Sets 172
III.12 C?-Algebras 173
III.13 Curvature 174
III.14 Designs 174
III.15 Determinants 175
III.16 Differential Forms and Integration 176
III.17 Dimension 182
III.18 Distributions 186
III.19 Duality 189
III.20 Dynamical Systems and Chaos 191
III.21 Elliptic Curves 192
III.22 The Euclidean Algorithm and
Continued Fractions 193
III.23 The Euler and Navier-Stokes Equations 195
III.24 The Exponential and Logarithmic Functions 198
III.25 Expanders 201
III.26 The Fast Fourier Transform 204
III.27 The Fourier Transform 206
III.28 Fuchsian Groups 209
III.29 Function Spaces 211
III.30 Galois Groups 214
III.31 The Gamma Function 215
III.32 Generating Functions 216
III.33 Genus 216
III.34 Graphs 217
III.35 Hamiltonians 217
III.36 The Heat Equation 218
III.37 Hilbert Spaces 221
III.38 Holomorphic Functions 222
III.39 Homology and Cohomology 222
III.40 Homotopy Groups 223
III.41 The Hyperbolic Plane 223
III.42 The Ideal Class Group 223
III.43 Irrational and Transcendental Numbers 224
III.44 The Ising Model 225
III.45 Jordan Normal Form 225
III.46 Knot Polynomials 227
III.47 K-Theory 229
III.48 The Leech Lattice 229
III.49 L-Functions 230
III.50 Lie Theory 231
III.51 Linear and Nonlinear Waves and Solitons 236
III.52 Linear Operators and Their Properties 241
III.53 Local and Global in Number Theory 243
III.54 Optimization and Lagrange Multipliers 246
_
vi Contents
III.55 The Mandelbrot Set 248
III.56 Manifolds 248
III.57 Matroids 249
III.58 Measures 250
III.59 Metric Spaces 252
III.60 Models of Set Theory 253
III.61 Modular Arithmetic 253
III.62 Modular Forms 255
III.63 Moduli Spaces 256
III.64 The Monster Group 256
III.65 Normed Spaces and Banach Spaces 257
III.66 Number Fields 258
III.67 Orbifolds 259
III.68 Ordinals 260
III.69 The Peano Axioms 260
III.70 Permutation Groups 261
III.71 Phase Transitions 263
III.72 ? 263
III.73 Probability Distributions 264
III.74 Projective Space 269
III.75 Quadratic Forms 269
III.76 Quantum Computation 270
III.77 Quantum Groups 273
III.78 Quaternions, Octonions, and Normed
Division Algebras 277
III.79 Representations 280
III.80 Ricci Flow 281
III.81 Riemannian Metrics 283
III.82 Riemann Surfaces 284
III.83 Rings, Ideals, and Modules 286
III.84 Schemes 287
III.85 The Schrodinger Equation 288
III.86 The Simplex Algorithm 290
III.87 Special Functions 292
III.88 The Spectrum 296
III.89 Spherical Harmonics 297
III.90 Symplectic Manifolds 299
III.91 Tensor Products 303
III.92 Topological Spaces 303
III.93 Transforms 305
III.94 Trigonometric Functions 309
III.95 Variational Methods 311
III.96 Varieties 313
III.97 Vector Bundles 314
III.98 Von Neumann Algebras 314
III.99 Wavelets 314
III.100 Zeta Functions 315
III.101 The Zermelo-Fraenkel Axioms 315
Part IV Branches of Mathematics
IV.1 Set Theory 317
IV.2 Logic and Model Theory 336
IV.3 Algebraic Numbers 348
IV.4 Analytic Number Theory 365
IV.5 Computational Number Theory 382
IV.6 Arithmetic Geometry 396
IV.7 Algebraic Geometry 407
IV.8 Moduli Spaces 416
IV.9 Differential Topology 428
IV.10 Algebraic Topology 440
IV.11 Geometric and Combinatorial Group Theory 453
IV.12 Representation Theory 470
IV.13 Vertex Operator Algebras 482
IV.14 Mirror Symmetry 493
IV.15 Dynamics 509
IV.16 Partial Differential Equations 526
IV.17 General Relativity and the Einstein Equations 553
IV.18 Harmonic Analysis 553
IV.19 Operator Algebras 560
IV.20 Numerical Analysis 573
IV.21 Computational Complexity 585
IV.22 Enumerative and Algebraic Combinatorics 613
IV.23 Extremal and Probabilistic Combinatorics 625
IV.24 High-Dimensional Geometry and Its
Probabilistic Analogues 639
IV.25 Stochastic Processes 649
IV.26 Probabilistic Models of Critical Phenomena 660
Part V Theorems and Problems
V.1 The ABC Conjecture 673
V.2 The Atiyah-Singer Index Theorem 673
V.3 The Banach-Tarski Paradox 676
V.4 The Birch-Swinnerton-Dyer Conjecture 677
V.5 Carleson's Theorem 678
V.6 Cauchy's Theorem 679
V.7 The Central Limit Theorem 679
V.8 The Classification of Finite Simple Groups 679
V.9 Dirichlet's Theorem 681
V.10 Dvoretzky's Theorem 681
V.11 Ergodic Theorems 682
V.12 Fermat's Last Theorem 684
V.13 Fixed-Point Theorems 686
V.14 The Four-Color Theorem 689
V.15 The Fundamental Theorem of Algebra 691
V.16 The Fundamental Theorem of Arithmetic 692
V.17 The Fundamental Theorem of Calculus 693
V.18 Godel's Theorem 693
V.19 Gromov's Polynomial-Growth Theorem 695
V.20 Hilbert's Nullstellensatz 696
_
Contents vii
V.21 The Independence of the
Continuum Hypothesis 696
V.22 Inequalities 696
V.23 The Insolubility of the Halting Problem 699
V.24 The Insolubility of the Quintic 701
V.25 Liouville's Theorem and Roth's Theorem 703
V.26 Rational Points on Curves and
the Mordell Conjecture 704
V.27 Mostow's Strong Rigidity Theorem 706
V.28 The P = NP Problem 708
V.29 The Poincare Conjecture 709
V.30 Problems and Results in
Additive Number Theory 709
V.31 From Quadratic Reciprocity to
Class Field Theory 712
V.32 The Resolution of Singularities 715
V.33 The Riemann Hypothesis 715
V.34 The Riemann-Roch Theorem 716
V.35 The Robertson-Seymour Theorem 718
V.36 The Three-Body Problem 719
V.37 The Uniformization Theorem 721
V.38 The Weil Conjectures 722
Part VI Mathematicians
VI.1 Pythagoras 727
VI.2 Euclid 728
VI.3 Archimedes 728
VI.4 Apollonius 729
VI.5 Leonardo of Pisa (known as Fibonacci) 730
VI.6 Girolamo Cardano 730
VI.7 Rafael Bombelli 730
VI.8 Francois Viete 731
VI.9 Simon Stevin 732
VI.10 Rene Descartes 732
VI.11 Pierre Fermat 734
VI.12 Blaise Pascal 735
VI.13 Isaac Newton 735
VI.14 Gottfried Wilhelm Leibniz 737
VI.15 The Bernoullis 738
VI.16 Brooke Taylor 740
VI.17 Christian Goldbach 740
VI.18 Leonhard Euler 740
VI.19 Jean Le Rond d'Alembert 742
VI.20 Edward Waring 744
VI.21 Joseph Louis Lagrange 744
VI.22 Pierre-Simon Laplace 745
VI.23 Adrien-Marie Legendre 747
VI.24 Jean-Baptiste Joseph Fourier 748
VI.25 Carl Friedrich Gauss 749
VI.26 Simeon-Denis Poisson 750
VI.27 Bernard Bolzano 750
VI.28 Augustin-Louis Cauchy 751
VI.29 August Ferdinand Mobius 752
VI.30 Nicolai Ivanovich Lobachevskii 752
VI.31 George Green 753
VI.32 Niels Henrik Abel 753
VI.33 Janos Bolyai 755
VI.34 Carl Gustav Jacob Jacobi 755
VI.35 Peter Gustav Lejeune Dirichlet 757
VI.36 William Rowan Hamilton 758
VI.37 Augustus De Morgan 759
VI.38 Joseph Liouville 759
VI.39 Eduard Kummer 760
VI.40 Evariste Galois 760
VI.41 James Joseph Sylvester 761
VI.42 George Boole 762
VI.43 Karl Weierstrass 763
VI.44 Pafnuty Chebyshev 764
VI.45 Arthur Cayley 765
VI.46 Charles Hermite 766
VI.47 Leopold Kronecker 766
VI.48 Georg Bernhard Friedrich Riemann 767
VI.49 Julius Wilhelm Richard Dedekind 769
VI.50 Emile Leonard Mathieu 769
VI.51 Camille Jordan 770
VI.52 Sophus Lie 770
VI.53 Georg Cantor 771
VI.54 William Kingdon Clifford 773
VI.55 Gottlob Frege 773
VI.56 Christian Felix Klein 775
VI.57 Ferdinand Georg Frobenius 776
VI.58 Sonya Kovalevskaya 777
VI.59 William Burnside 778
VI.60 Jules Henri Poincare 778
VI.61 Giuseppe Peano 780
VI.62 David Hilbert 781
VI.63 Hermann Minkowski 782
VI.64 Jacques Hadamard 783
VI.65 Ivar Fredholm 784
VI.66 Charles-Jean de la Vallee Poussin 784
VI.67 Felix Hausdorff 785
VI.68 Elie Joseph Cartan 786
VI.69 Emile Borel 787
VI.70 Bertrand Arthur William Russell 788
VI.71 Henri Lebesgue 788
VI.72 Godfrey Harold Hardy 790
VI.73 Frigyes (Frederic) Riesz 791
VI.74 Luitzen Egbertus Jan Brouwer 792
VI.75 Emmy Noether 793
VI.76 Waclaw Sierpi?nski 794
VI.77 George Birkhoff 795
VI.78 John Edensor Littlewood 796
VI.79 Hermann Weyl 798
VI.80 Thoralf Skolem 799
_
viii Contents
VI.81 Srinivasa Ramanujan 800
VI.82 Richard Courant 801
VI.83 Stefan Banach 802
VI.84 Norbert Wiener 804
VI.85 Emil Artin 805
VI.86 Alfred Tarski 806
VI.87 Andrei Nikolaevich Kolmogorov 807
VI.88 William Vallance Douglas Hodge 809
VI.89 John von Neumann 810
VI.90 Kurt Godel 811
VI.91 Andre Weil 812
VI.92 Alan Turing 813
VI.93 Abraham Robinson 814
VI.94 Nicolas Bourbaki 816
Part VII The Influence of Mathematics
VII.1 Mathematics and Chemistry 819
VII.2 Mathematical Biology 830
VII.3 Wavelets and Applications 840
VII.4 The Mathematics of Traffic in Networks 855
VII.5 The Mathematics of Algorithm Design 863
VII.6 Reliable Transmission of Information 871
VII.7 Mathematics and Cryptography 879
VII.8 Mathematics and Economic Reasoning 888
VII.9 The Mathematics of Money 903
VII.10 Mathematical Statistics 909
VII.11 Mathematics and Medical Statistics 914
VII.12 Analysis, Mathematical and Philosophical 921
VII.13 Mathematics and Music 928
VII.14 Mathematics and Art 937
Part VIII Final Perspectives
VIII.1 The Art of Problem Solving 949
VIII.2 "Why Mathematics?" You Might Ask 960
VIII.3 The Ubiquity of Mathematics 971
VIII.4 Numeracy 977
VIII.5 Mathematics: An Experimental Science 985
VIII.6 Advice to a Young Mathematician 994
VIII.7 A Chronology of Major Mathematical Events 1003
Index

toc.md

The Princeton Companion to Mathematics

Contents

Preface ix

List of 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 46

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 96

II.4 Algorithms 107

II.5 The Development of Rigor in Mathematical Analysis 118

II.6 The Development of the Idea of Proof 130

II.7 The Crisis in the Foundations of Mathematics 143

Part III Mathematical Concepts

III.1 The Axiom of Choice 159

III.2 The Axiom of Determinacy 161

III.3 Bayesian Analysis 161

III.4 Braid Groups 162

III.5 Buildings 163

III.6 Calabi-Yau Manifolds 164

III.7 Cardinals 167

III.8 Categories 167

III.9 Compactness and Compactification 169

III.10 Computational Complexity Classes 171

III.11 Countable and Uncountable Sets 172

III.12 C?-Algebras 173

III.13 Curvature 174

III.14 Designs 174

III.15 Determinants 175

III.16 Differential Forms and Integration 176

III.17 Dimension 182

III.18 Distributions 186

III.19 Duality 189

III.20 Dynamical Systems and Chaos 191

III.21 Elliptic Curves 192

III.22 The Euclidean Algorithm and Continued Fractions 193

III.23 The Euler and Navier-Stokes Equations 195

III.24 The Exponential and Logarithmic Functions 198

III.25 Expanders 201

III.26 The Fast Fourier Transform 204

III.27 The Fourier Transform 206

III.28 Fuchsian Groups 209

III.29 Function Spaces 211

III.30 Galois Groups 214

III.31 The Gamma Function 215

III.32 Generating Functions 216

III.33 Genus 216

III.34 Graphs 217

III.35 Hamiltonians 217

III.36 The Heat Equation 218

III.37 Hilbert Spaces 221

III.38 Holomorphic Functions 222

III.39 Homology and Cohomology 222

III.40 Homotopy Groups 223

III.41 The Hyperbolic Plane 223

III.42 The Ideal Class Group 223

III.43 Irrational and Transcendental Numbers 224

III.44 The Ising Model 225

III.45 Jordan Normal Form 225

III.46 Knot Polynomials 227

III.47 K-Theory 229

III.48 The Leech Lattice 229

III.49 L-Functions 230

III.50 Lie Theory 231

III.51 Linear and Nonlinear Waves and Solitons 236

III.52 Linear Operators and Their Properties 241

III.53 Local and Global in Number Theory 243

III.54 Optimization and Lagrange Multipliers 246

III.55 The Mandelbrot Set 248

III.56 Manifolds 248

III.57 Matroids 249

III.58 Measures 250

III.59 Metric Spaces 252

III.60 Models of Set Theory 253

III.61 Modular Arithmetic 253

III.62 Modular Forms 255

III.63 Moduli Spaces 256

III.64 The Monster Group 256

III.65 Normed Spaces and Banach Spaces 257

III.66 Number Fields 258

III.67 Orbifolds 259

III.68 Ordinals 260

III.69 The Peano Axioms 260

III.70 Permutation Groups 261

III.71 Phase Transitions 263

III.72 ? 263

III.73 Probability Distributions 264

III.74 Projective Space 269

III.75 Quadratic Forms 269

III.76 Quantum Computation 270

III.77 Quantum Groups 273

III.78 Quaternions, Octonions, and Normed Division Algebras 277

III.79 Representations 280

III.80 Ricci Flow 281

III.81 Riemannian Metrics 283

III.82 Riemann Surfaces 284

III.83 Rings, Ideals, and Modules 286

III.84 Schemes 287

III.85 The Schrodinger Equation 288

III.86 The Simplex Algorithm 290

III.87 Special Functions 292

III.88 The Spectrum 296

III.89 Spherical Harmonics 297

III.90 Symplectic Manifolds 299

III.91 Tensor Products 303

III.92 Topological Spaces 303

III.93 Transforms 305

III.94 Trigonometric Functions 309

III.95 Variational Methods 311

III.96 Varieties 313

III.97 Vector Bundles 314

III.98 Von Neumann Algebras 314

III.99 Wavelets 314

III.100 Zeta Functions 315

III.101 The Zermelo-Fraenkel Axioms 315

Part IV Branches of Mathematics

IV.1 Set Theory 317

IV.2 Logic and Model Theory 336

IV.3 Algebraic Numbers 348

IV.4 Analytic Number Theory 365

IV.5 Computational Number Theory 382

IV.6 Arithmetic Geometry 396

IV.7 Algebraic Geometry 407

IV.8 Moduli Spaces 416

IV.9 Differential Topology 428

IV.10 Algebraic Topology 440

IV.11 Geometric and Combinatorial Group Theory 453

IV.12 Representation Theory 470

IV.13 Vertex Operator Algebras 482

IV.14 Mirror Symmetry 493

IV.15 Dynamics 509

IV.16 Partial Differential Equations 526

IV.17 General Relativity and the Einstein Equations 553

IV.18 Harmonic Analysis 553

IV.19 Operator Algebras 560

IV.20 Numerical Analysis 573

IV.21 Computational Complexity 585

IV.22 Enumerative and Algebraic Combinatorics 613

IV.23 Extremal and Probabilistic Combinatorics 625

IV.24 High-Dimensional Geometry and Its Probabilistic Analogues 639

IV.25 Stochastic Processes 649

IV.26 Probabilistic Models of Critical Phenomena 660

Part V Theorems and Problems

V.1 The ABC Conjecture 673

V.2 The Atiyah-Singer Index Theorem 673

V.3 The Banach-Tarski Paradox 676

V.4 The Birch-Swinnerton-Dyer Conjecture 677