Title Copyright Contents Preface Contributors I.1 What Is Mathematics About? I.2 The Language and Grammar of Mathematics I.3 Some Fundamental Mathematical Definitions I.4 The General Goals of Mathematical Research Part I Introduction 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 II The Origins of Modern Mathematics 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 Compactification 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 Differential 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.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 III Mathematical Concepts 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 Differential Topology IV.8 Moduli Spaces IV.9 Representation Theory IV.10 Geometric and Combinatorial Group Theory IV.11 Harmonic Analysis IV.12 Partial Differential 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 IV Branches of Mathematics 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 Classification of Finite Simple Groups V.8 Dirichiet’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 Liouvflle’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 V Theorems and Problems 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-Khwārizmī (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 (f1.18th 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 Clifford (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 Hausdorff (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 Waclaw Sierpiński (1882–1969) VI.78 George Birkhoff (1884–1944) VI.79 John Edensor Littlewood (1885–1977) VI.80 Hermann Weyl (1885–1955) VI.81 Thoralf Skolem (1887–1963) VI.82 Srinwasa 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) VI.87 Alfred Tarski (1901–1983) VI.88 Andrei Nikolaevich Kolmogorov (1903–1987) VI.89 Alonzo Church (1903–1995) VI.90 William Valiance Douglas Hodge (1903–1975) VI.91 John von Neumann (1903–1957) VI.92 Kurt Gödel (1906–1978) VI.93 André Weil (1906–1998) VI.94 Alan Turing (1912–1954) VI.95 Abraham Robinson (1918–1974) VI.96 Nicolas Bourbaki (1935–) Part VI Mathematicians VII.1 Mathematics and Chemistry VII.2 Mathematical Biology VII.3 Wavelets and Applications VII.4 The Mathematics of Traffic in Networks VII.5 The Mathematics of Algorithm Design VII.6 Reliable Transmission of Information VII.7 Mathematics and Cryptography VII.8 Mathematics and Economic Reasoning VII.9 The Mathematics of Money VII.10 Mathematical Statistics VII.11 Mathematics and Medical Statistics VII.12 Analysis, Mathematical and Philosophical VII.13 Mathematics and Music VII.14 Mathematics and Art Part VII The Influence of Mathematics VIII.1 The Art of Problem Solving VIII.2 “Why Mathematics” You Might Ask VIII.3 The Ubiquity of Mathematics VIII.4 Numeracy VIII.5 Mathematics: An Experimental Science VIII.6 Advice to a Young Mathematician VIII.7 A Chronology of Mathematical Events Part VIII Final Perspectives Index
