The Princeton Companion to Mathematics

Contents

Part I Introduction

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 II The Origins of Modern Mathematics

II.1 From Numbers to Number Systems

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 Compactification

III.10 Computational Complexity Classes

III.11 Countable and Uncountable Sets

III.12 C*-Algebras

III.13 Curvature

III.15 Determinants

III.16 Differential Forms and Integration

III.17 Dimension

III.18 Distributions

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.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.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.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 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.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 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 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-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 VII The Influence of Mathematics

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 VIII Final Perspectives

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