Gowers Timothy (ed.) The Princeton Companion to Mathematics

Princeton University Press, 2008. — 1056 p. — ISBN 978-0-691-11880-2.The Princeton Companion to Mathematics by T. Gowers, J. Barrow-Green, I. Leader. This is an unusual book targeting a broad audience ranging from (even young) fans curious about the present state of mathematics to the professional mathematicians seeking to have a glimpse at the areas of mathematics not directly related to their special interests. The book consists of eight (unequal) parts: Introduction, The Origins of Modern Mathematics, Mathematical Concepts, Branches of Mathematics, Theorems and Problems, Mathematicians, The Influence of Mathematics, Final Perspectives. The book concentrates on what is commonly thought of as pure mathematics, although the fourth part - Branches of Mathematics - includes chapters on such borderline theories as, for example, General Relativity, Dynamics, Computational Complexity, Numerical Analysis, while the seventh part - The Influence of Mathematics - covers applications to biology, chemistry, networks, cryptography, medicine, music and art, among others. Such a selection of material left open a possibility of a similar book about applied mathematics and theoretical physics while simultaneously supplying hooks to the new volume.What Is Mathematics About?
The Language and Grammar of Mathematics
Some Fundamental Mathematical De.nitions
The General Goals of Mathematical Research
The Origins of Modern Mathematics
From Numbers to Number Systems
Geometry
The Development of Abstract Algebra
Algorithms
The Development of Rigor in Mathematical Analysis
The Development of the Idea of Proof
The Crisis in the Foundations of Mathematics
Mathematical Concepts
The Axiom of Choice
The Axiom of Determinacy
Bayesian Analysis
Braid Groups
Buildings
Calabi–Yau Manifolds
Cardinals
Categories
Compactness and Compactification
Computational Complexity Classes
Countable and Uncountable Sets
C.-Algebras
Curvature
Designs
Determinants
Di.erential Forms and Integration
Dimension
Distributions
Duality
Dynamical Systems and Chaos
Elliptic Curves
The Euclidean Algorithm and Continued Fractions
The Euler and Navier–Stokes Equations
Expanders
The Exponential and Logarithmic Functions
The Fast Fourier Transform
The Fourier Transform
Fuchsian Groups
Function Spaces
Galois Groups
The Gamma Function
Generating Functions
Genus
Graphs
Hamiltonians
The Heat Equation
Hilbert Spaces
Homology and Cohomology
Homotopy Groups
The Ideal Class Group
Irrational and Transcendental Numbers
The Ising Model
Jordan Normal Form
Knot Polynomials
K-Theory
The Leech Lattice
L-Functions
Lie Theory
Linear and Nonlinear Waves and Solitons
Linear Operators and Their Properties
Local and Global in Number Theory
The Mandelbrot Set
Manifolds
Matroids
Measures
Metric Spaces
Models of Set Theory
Modular Arithmetic
Modular Forms
Moduli Spaces
The Monster Group
Normed Spaces and Banach Spaces
Number Fields
Optimization and Lagrange Multipliers
Orbifolds
Ordinals
The Peano Axioms
Permutation Groups
Phase Transitions
p
Probability Distributions
Projective Space
Quadratic Forms
Quantum Computation
Quantum Groups
Quaternions, Octonions, and Normed Division Algebras
Representations
Ricci Flow
Riemann Surfaces
The Riemann Zeta Function
Rings, Ideals, and Modules
Schemes
The Schrödinger Equation
The Simplex Algorithm
Special Functions
The Spectrum
Spherical Harmonics
Symplectic Manifolds
Tensor Products
Topological Spaces
Transforms
Trigonometric Functions
Universal Covers
Variational Methods
Varieties
Vector Bundles
Von Neumann Algebras
Wavelets
The Zermelo–Fraenkel Axioms
Branches of Mathematics
Algebraic Numbers
Analytic Number Theory
Computational Number Theory
Algebraic Geometry
Arithmetic Geometry
Algebraic Topology
Di.erential Topology
Moduli Spaces
Representation Theory
Geometric and Combinatorial Group Theory
Harmonic Analysis
Partial Differential Equations
General Relativity and the Einstein Equations
Dynamics
Operator Algebras
Mirror Symmetry
Vertex Operator Algebras
Enumerative and Algebraic Combinatorics
Extremal and Probabilistic Combinatorics
Computational Complexity
Numerical Analysis
Set Theory
Logic and Model Theory
Stochastic Processes
Probabilistic Models of Critical Phenomena
High-Dimensional Geometry and Its Probabilistic Analogues
Theorems and Problems
The ABC Conjecture
The Atiyah–Singer Index Theorem
The Banach–Tarski Paradox
The Birch–Swinnerton-Dyer Conjecture
Carleson’s Theorem
The Central Limit Theorem
The Classification of Finite Simple Groups
Dirichlet’s Theorem
Ergodic Theorems
Fermat’s Last Theorem
Fixed Point Theorems
The Four-Color Theorem
The Fundamental Theorem of Algebra
The Fundamental Theorem of Arithmetic
Gödel’s Theorem
Gromov’s Polynomial-Growth Theorem
Hilbert’s Nullstellensatz
The Independence of the Continuum Hypothesis
Inequalities
The Insolubility of the Halting Problem
The Insolubility of the Quintic
Liouville’s Theorem and Roth’s Theorem
Mostow’s Strong Rigidity Theorem
The P versus NP Problem
The Poincaré Conjecture
The Prime Number Theorem and the Riemann Hypothesis
Problems and Results in Additive Number Theory
From Quadratic Reciprocity to Class Field Theory
Rational Points on Curves and the Mordell Conjecture
The Resolution of Singularities
The Riemann–Roch Theorem
The Robertson–Seymour Theorem
The Three-Body Problem
The Uniformization Theorem
The Weil Conjectures
Mathematicians
Pythagoras (ca. 569 b.c.e.–ca. 494 b.c.e.)
Euclid (ca. 325 b.c.e.–ca. 265 b.c.e.)
Archimedes (ca. 287 b.c.e.–212 b.c.e.)
Apollonius (ca. 262 b.c.e.–ca. 190 b.c.e.)
Abu Ja’far Muhammad ibn Musa al-Khwarizmi (800–847)
Leonardo of Pisa (known as Fibonacci) (ca. 1170–ca. 1250)
Girolamo Cardano (1501–1576)
Rafael Bombelli (1526–after 1572)
François Viète (1540–1603)
Simon Stevin (1548–1620)
René Descartes (1596–1650)
Pierre Fermat (160?–1665)
Blaise Pascal (1623–1662)
Isaac Newton (1642–1727)
Gottfried Wilhelm Leibniz (1646–1716)
Brook Taylor (1685–1731)
Christian Goldbach (1690–1764)
The Bernoullis (.. 18th century)
Leonhard Euler (1707–1783)
Jean Le Rond d’Alembert (1717–1783)
Edward Waring (ca. 1735–1798)
Joseph Louis Lagrange (1736–1813)
Pierre-Simon Laplace (1749–1827)
Adrien-Marie Legendre (1752–1833)
Jean-Baptiste Joseph Fourier (1768–1830)
Carl Friedrich Gauss (1777–1855)
Siméon-Denis Poisson (1781–1840)
Bernard Bolzano (1781–1848)
Augustin-Louis Cauchy (1789–1857)
August Ferdinand Möbius (1790–1868)
Nicolai Ivanovich Lobachevskii (1792–1856)
George Green (1793–1841)
Niels Henrik Abel (1802–1829)
János Bolyai (1802–1860)
Carl Gustav Jacob Jacobi (1804–1851)
Peter Gustav Lejeune Dirichlet (1805–1859)
William Rowan Hamilton (1805–1865)
Augustus De Morgan (1806–1871)
Joseph Liouville (1809–1882)
Ernst Eduard Kummer (1810–1893)
Évariste Galois (1811–1832)
James Joseph Sylvester (1814–1897)
George Boole (1815–1864)
Karl Weierstrass (1815–1897)
Pafnuty Chebyshev (1821–1894)
Arthur Cayley (1821–1895)
Charles Hermite (1822–1901)
Leopold Kronecker (1823–1891)
Georg Friedrich Bernhard Riemann (1826–1866)
Julius Wilhelm Richard Dedekind (1831–1916)
Émile Léonard Mathieu (1835–1890)
Camille Jordan (1838–1922)
Sophus Lie (1842–1899)
Georg Cantor (1845–1918)
William Kingdon Clifford (1845–1879)
Gottlob Frege (1848–1925)
Christian Felix Klein (1849–1925)
Ferdinand Georg Frobenius (1849–1917)
Sofya (Sonya) Kovalevskaya (1850–1891)
William Burnside (1852–1927)
Jules Henri Poincaré (1854–1912)
Giuseppe Peano (1858–1932)
David Hilbert (1862–1943)
Hermann Minkowski (1864–1909)
Jacques Hadamard (1865–1963)
Ivar Fredholm (1866–1927)
Charles-Jean de la Vallée Poussin (1866–1962)
Felix Hausdorff (1868–1942)
Élie Joseph Cartan (1869–1951)
Emile Borel (1871–1956)
Bertrand Arthur William Russell (1872–1970)
Henri Lebesgue (1875–1941)
Godfrey Harold Hardy (1877–1947)
Frigyes (Frédéric) Riesz (1880–1956)
Luitzen Egbertus Jan Brouwer (1881–1966)
Emmy Noether (1882–1935)
Waclaw Sierpi´nski (1882–1969)
George Birkhoff (1884–1944)
John Edensor Littlewood (1885–1977)
Hermann Weyl (1885–1955)
Thoralf Skolem (1887–1963)
Srinivasa Ramanujan (1887–1920)
Richard Courant (1888–1972)
Stefan Banach (1892–1945)
Norbert Wiener (1894–1964)
Emil Artin (1898–1962)
Alfred Tarski (1901–1983)
Andrei Nikolaevich Kolmogorov (1903–1987)
Alonzo Church (1903–1995)
William Vallance Douglas Hodge (1903–1975)
John von Neumann (1903–1957)
Kurt Gödel (1906–1978)
André Weil (1906–1998)
Alan Turing (1912–1954)
Abraham Robinson (1918–1974)
Nicolas Bourbaki (1935–)
The Influence of Mathematics
Mathematics and Chemistry
Mathematical Biology
Wavelets and Applications
The Mathematics of Tra.c in Networks
The Mathematics of Algorithm Design
Reliable Transmission of Information
Mathematics and Cryptography
Mathematics and Economic Reasoning
The Mathematics of Money
Mathematical Statistics
Mathematics and Medical Statistics
Analysis, Mathematical and Philosophical
Mathematics and Music
Mathematics and Art
Final Perspectives
The Art of Problem Solving
“Why Mathematics?” You Might Ask
The Ubiquity of Mathematics
Numeracy
Mathematics: An Experimental Science
Advice to a Young Mathematician
A Chronology of Mathematical Events