The Princeton Companion to Mathematics -- Contents -- Preface -- Contributors -- Part I Introduction -- I.1 What Is Mathematics About? -- I.2 The Language and Grammar of Mathematics -- I.3 Some Fundamental Mathematical De.nitions -- I.4 The General Goals of Mathematical Research -- Part II The Origins of Modern Mathematics -- II.1 From Numbers to Number Systems -- II.2 Geometry -- II.3 The Development of Abstract Algebra -- II.4 Algorithms -- II.5 The Development of Rigor in Mathematical Analysis -- II.6 The Development of the Idea of Proof -- II.7 The Crisis in the Foundations of Mathematics -- Part III Mathematical Concepts -- III.1 The Axiom of Choice -- III.2 The Axiom of Determinacy -- III.3 Bayesian Analysis -- III.4 Braid Groups -- III.5 Buildings -- III.6 Calabi-Yau Manifolds -- III.7 Cardinals -- III.8 Categories -- III.9 Compactness and Compacti.cation -- III.10 Computational Complexity Classes -- III.11 Countable and Uncountable Sets -- III.12 C.-Algebras -- III.13 Curvature -- III.14 Designs -- III.15 Determinants -- III.16 Di.erential Forms and Integration -- III.17 Dimension -- III.18 Distributions -- III.19 Duality -- III.20 Dynamical Systems and Chaos -- III.21 Elliptic Curves -- III.22 The Euclidean Algorithm and Continued Fractions -- III.23 The Euler and Navier-Stokes Equations -- III.24 Expanders -- III.25 The Exponential and Logarithmic Functions -- III.26 The Fast Fourier Transform -- III.27 The Fourier Transform -- III.28 Fuchsian Groups -- III.29 Function Spaces -- III.30 Galois Groups -- III.31 The Gamma Function -- III.32 Generating Functions -- III.33 Genus -- III.34 Graphs -- III.35 Hamiltonians -- III.36 The Heat Equation -- III.37 Hilbert Spaces -- III.38 Homology and Cohomology -- III.39 Homotopy Groups -- III.40 The Ideal Class Group -- III.41 Irrational and Transcendental Numbers -- III.42 The Ising Model.

III.43 Jordan Normal Form -- III.44 Knot Polynomials -- III.45 K-Theory -- III.46 The Leech Lattice -- III.47 L-Functions -- III.48 Lie Theory -- III.49 Linear and Nonlinear Waves and Solitons -- III.50 Linear Operators and Their Properties -- III.51 Local and Global in Number Theory -- III.52 The Mandelbrot Set -- III.53 Manifolds -- III.54 Matroids -- III.55 Measures -- III.56 Metric Spaces -- III.57 Models of Set Theory -- III.58 Modular Arithmetic -- III.59 Modular Forms -- III.60 Moduli Spaces -- III.61 The Monster Group -- III.62 Normed Spaces and Banach Spaces -- III.63 Number Fields -- III.64 Optimization and Lagrange Multipliers -- III.65 Orbifolds -- III.66 Ordinals -- III.67 The Peano Axioms -- III.68 Permutation Groups -- III.69 Phase Transitions -- III.70 π -- III.71 Probability Distributions -- III.72 Projective Space -- III.73 Quadratic Forms -- III.74 Quantum Computation -- III.75 Quantum Groups -- III.76 Quaternions, Octonions, and Normed Division Algebras -- III.77 Representations -- III.78 Ricci Flow -- III.79 Riemann Surfaces -- III.80 The Riemann Zeta Function -- III.81 Rings, Ideals, and Modules -- III.82 Schemes -- III.83 The Schrödinger Equation -- III.84 The Simplex Algorithm -- III.85 Special Functions -- III.86 The Spectrum -- III.87 Spherical Harmonics -- III.88 Symplectic Manifolds -- III.89 Tensor Products -- III.90 Topological Spaces -- III.91 Transforms -- III.92 Trigonometric Functions -- III.93 Universal Covers -- III.94 Variational Methods -- III.95 Varieties -- III.96 Vector Bundles -- III.97 Von Neumann Algebras -- III.98 Wavelets -- III.99 The Zermelo-Fraenkel Axioms -- Part IV Branches of Mathematics -- IV.1 Algebraic Numbers -- IV.2 Analytic Number Theory -- IV.3 Computational Number Theory -- IV.4 Algebraic Geometry -- IV.5 Arithmetic Geometry -- IV.6 Algebraic Topology -- IV.7 Di.erential Topology.

IV.8 Moduli Spaces -- IV.9 Representation Theory -- IV.10 Geometric and Combinatorial Group Theory -- IV.11 Harmonic Analysis -- IV.12 Partial Di.erential Equations -- IV.13 General Relativity and the Einstein Equations -- IV.14 Dynamics -- IV.15 Operator Algebras -- IV.16 Mirror Symmetry -- IV.17 Vertex Operator Algebras -- IV.18 Enumerative and Algebraic Combinatorics -- IV.19 Extremal and Probabilistic Combinatorics -- IV.20 Computational Complexity -- IV.21 Numerical Analysis -- IV.22 Set Theory -- IV.23 Logic and Model Theory -- IV.24 Stochastic Processes -- IV.25 Probabilistic Models of Critical Phenomena -- IV.26 High-Dimensional Geometry and Its Probabilistic Analogues -- Part V Theorems and Problems -- V.1 The ABC Conjecture -- V.2 The Atiyah-Singer Index Theorem -- V.3 The Banach-Tarski Paradox -- V.4 The Birch-Swinnerton-Dyer Conjecture -- V.5 Carleson's Theorem -- V.6 The Central Limit Theorem -- V.7 The Classi.cation of Finite Simple Groups -- V.8 Dirichlet's Theorem -- V.9 Ergodic Theorems -- V.10 Fermat's Last Theorem -- V.11 Fixed Point Theorems -- V.12 The Four-Color Theorem -- V.13 The Fundamental Theorem of Algebra -- V.14 The Fundamental Theorem of Arithmetic -- V.15 Gödel's Theorem -- V.16 Gromov's Polynomial-Growth Theorem -- V.17 Hilbert's Nullstellensatz -- V.18 The Independence of the Continuum Hypothesis -- V.19 Inequalities -- V.20 The Insolubility of the Halting Problem -- V.21 The Insolubility of the Quintic -- V.22 Liouville's Theorem and Roth's Theorem -- V.23 Mostow's Strong Rigidity Theorem -- V.24 The P versus NP Problem -- V.25 The Poincaré Conjecture -- V.26 The Prime Number Theorem and the Riemann Hypothesis -- V.27 Problems and Results in Additive Number Theory -- V.28 From Quadratic Reciprocity to Class Field Theory -- V.29 Rational Points on Curves and the Mordell Conjecture.

V.30 The Resolution of Singularities -- V.31 The Riemann-Roch Theorem -- V.32 The Robertson-Seymour Theorem -- V.33 The Three-Body Problem -- V.34 The Uniformization Theorem -- V.35 The Weil Conjectures -- Part VI Mathematicians -- VI.1 Pythagoras (ca. 569 b.c.e.-ca. 494 b.c.e.) -- VI.2 Euclid (ca. 325 b.c.e.-ca. 265 b.c.e.) -- VI.3 Archimedes (ca. 287 b.c.e.-212 b.c.e.) -- VI.4 Apollonius (ca. 262 b.c.e.-ca. 190 b.c.e.) -- VI.5 Abu Ja'far Muhammad ibn Mūsā al-Khwārizmı̄ (800-847) -- VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170-ca. 1250) -- VI.7 Girolamo Cardano (1501-1576) -- VI.8 Rafael Bombelli (1526-after 1572) -- VI.9 François Viète (1540-1603) -- VI.10 Simon Stevin (1548-1620) -- VI.11 René Descartes (1596-1650) -- VI.12 Pierre Fermat (160?-1665) -- VI.13 Blaise Pascal (1623-1662) -- VI.14 Isaac Newton (1642-1727) -- VI.15 Gottfried Wilhelm Leibniz (1646-1716) -- VI.16 Brook Taylor (1685-1731) -- VI.17 Christian Goldbach (1690-1764) -- VI.18 The Bernoullis (.. 18th century) -- VI.19 Leonhard Euler (1707-1783) -- VI.20 Jean Le Rond d'Alembert (1717-1783) -- VI.21 Edward Waring (ca. 1735-1798) -- VI.22 Joseph Louis Lagrange (1736-1813) -- VI.23 Pierre-Simon Laplace (1749-1827) -- VI.24 Adrien-Marie Legendre (1752-1833) -- VI.25 Jean-Baptiste Joseph Fourier (1768-1830) -- VI.26 Carl Friedrich Gauss (1777-1855) -- VI.27 Siméon-Denis Poisson (1781-1840) -- VI.28 Bernard Bolzano (1781-1848) -- VI.29 Augustin-Louis Cauchy (1789-1857) -- VI.30 August Ferdinand Möbius (1790-1868) -- VI.31 Nicolai Ivanovich Lobachevskii (1792-1856) -- VI.32 George Green (1793-1841) -- VI.33 Niels Henrik Abel (1802-1829) -- VI.34 János Bolyai (1802-1860) -- VI.35 Carl Gustav Jacob Jacobi (1804-1851) -- VI.36 Peter Gustav Lejeune Dirichlet (1805-1859) -- VI.37 William Rowan Hamilton (1805-1865) -- VI.38 Augustus De Morgan (1806-1871).

VI.39 Joseph Liouville (1809-1882) -- VI.40 Ernst Eduard Kummer (1810-1893) -- VI.41 Évariste Galois (1811-1832) -- VI.42 James Joseph Sylvester (1814-1897) -- VI.43 George Boole (1815-1864) -- VI.44 Karl Weierstrass (1815-1897) -- VI.45 Pafnuty Chebyshev (1821-1894) -- VI.46 Arthur Cayley (1821-1895) -- VI.47 Charles Hermite (1822-1901) -- VI.48 Leopold Kronecker (1823-1891) -- VI.49 Georg Friedrich Bernhard Riemann (1826-1866) -- VI.50 Julius Wilhelm Richard Dedekind (1831-1916) -- VI.51 Émile Léonard Mathieu (1835-1890) -- VI.52 Camille Jordan (1838-1922) -- VI.53 Sophus Lie (1842-1899) -- VI.54 Georg Cantor (1845-1918) -- VI.55 William Kingdon Cli.ord (1845-1879) -- VI.56 Gottlob Frege (1848-1925) -- VI.57 Christian Felix Klein (1849-1925) -- VI.58 Ferdinand Georg Frobenius (1849-1917) -- VI.59 Sofya (Sonya) Kovalevskaya (1850-1891) -- VI.60 William Burnside (1852-1927) -- VI.61 Jules Henri Poincaré (1854-1912) -- VI.62 Giuseppe Peano (1858-1932) -- VI.63 David Hilbert (1862-1943) -- VI.64 Hermann Minkowski (1864-1909) -- VI.65 Jacques Hadamard (1865-1963) -- VI.66 Ivar Fredholm (1866-1927) -- VI.67 Charles-Jean de la Vallée Poussin (1866-1962) -- VI.68 Felix Hausdor. (1868-1942) -- VI.69 Élie Joseph Cartan (1869-1951) -- VI.70 Emile Borel (1871-1956) -- VI.71 Bertrand Arthur William Russell (1872-1970) -- VI.72 Henri Lebesgue (1875-1941) -- VI.73 Godfrey Harold Hardy (1877-1947) -- VI.74 Frigyes (Frédéric) Riesz (1880-1956) -- VI.75 Luitzen Egbertus Jan Brouwer (1881-1966) -- VI.76 Emmy Noether (1882-1935) -- VI.77 Wacław Sierpiński (1882-1969) -- VI.78 George Birkho. (1884-1944) -- VI.79 John Edensor Littlewood (1885-1977) -- VI.80 Hermann Weyl (1885-1955) -- VI.81 Thoralf Skolem (1887-1963) -- VI.82 Srinivasa Ramanujan (1887-1920) -- VI.83 Richard Courant (1888-1972) -- VI.84 Stefan Banach (1892-1945) -- VI.85 Norbert Wiener (1894-1964).

VI.86 Emil Artin (1898-1962).