[ToDo] The Princeton Companion to Mathematics

Dated Jan 1, 2008; last modified on Mon, 05 Sep 2022

Notes

Part I Introduction:

  • ✅ What Is Mathematics About?
  • ✅ The Language and Grammar of Mathematics
  • Some Fundamental Mathematical Definitions
  • The General Goals of Mathematical Research

Part II 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

Part III 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
  • Differential Forms and Integration
  • Dimension
  • Distributions
  • Duality
  • Dynamical Systems and Chaos
  • Elliptic Curves
  • The Euclidean Algorithm and Continued Fractions
  • The Euler and Navier-Stokes Equations
  • The Exponential and Logarithmic Functions
  • Expanders
  • 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
  • Holomorphic Functions
  • Homology and Cohomology
  • Homotopy Groups
  • The Hyperbolic Plane
  • 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
  • Optimization and Lagrange Multipliers
  • 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
  • Orbifolds
  • Ordinals
  • The Peano Axioms
  • Permutation Groups
  • Phase Transitions
  • ?
  • Probability Distributions
  • Projective Space
  • Quadratic Forms
  • Quantum Computation
  • Quantum Groups
  • Quaternions, Octonions, and Normed Division Algebras
  • Representations
  • Ricci Flow
  • Riemannian Metrics
  • Riemann Surfaces
  • Rings, Ideals, and Modules
  • Schemes
  • The Schrodinger Equation
  • The Simplex Algorithm
  • Special Functions
  • The Spectrum
  • Spherical Harmonics
  • Symplectic Manifolds
  • Tensor Products
  • Topological Spaces
  • Transforms
  • Trigonometric Functions
  • Variational Methods
  • Varieties
  • Vector Bundles
  • Von Neumann Algebras
  • Wavelets
  • Zeta Functions
  • The Zermelo-Fraenkel Axioms

Part IV Branches of Mathematics:

  • Set Theory
  • Logic and Model Theory
  • Algebraic Numbers
  • Analytic Number Theory
  • Computational Number Theory
  • Arithmetic Geometry
  • Algebraic Geometry
  • Moduli Spaces
  • Differential Topology
  • Algebraic Topology
  • Geometric and Combinatorial Group Theory
  • Representation Theory
  • Vertex Operator Algebras
  • Mirror Symmetry
  • Dynamics
  • Partial Differential Equations
  • General Relativity and the Einstein Equations
  • Harmonic Analysis
  • Operator Algebras
  • Numerical Analysis
  • Computational Complexity
  • Enumerative and Algebraic Combinatorics
  • Extremal and Probabilistic Combinatorics
  • High-Dimensional Geometry and Its Probabilistic Analogues
  • Stochastic Processes
  • Probabilistic Models of Critical Phenomena

Part V Theorems and Problems:

  • The ABC Conjecture
  • The Atiyah-Singer Index Theorem
  • The Banach-Tarski Paradox
  • The Birch-Swinnerton-Dyer Conjecture
  • Carleson’s Theorem
  • Cauchy’s Theorem
  • The Central Limit Theorem
  • The Classification of Finite Simple Groups
  • Dirichlet’s Theorem
  • Dvoretzky’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
  • The Fundamental Theorem of Calculus
  • Godel’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
  • Rational Points on Curves and the Mordell Conjecture
  • Mostow’s Strong Rigidity Theorem
  • The P = NP Problem
  • The Poincare Conjecture
  • Problems and Results in Additive Number Theory
  • From Quadratic Reciprocity to Class Field Theory
  • The Resolution of Singularities
  • The Riemann Hypothesis
  • The Riemann-Roch Theorem
  • The Robertson-Seymour Theorem
  • The Three-Body Problem
  • The Uniformization Theorem
  • The Weil Conjectures

Part VI Mathematicians:

  • Pythagoras
  • Euclid
  • Archimedes
  • Apollonius
  • Leonardo of Pisa (known as Fibonacci)
  • Girolamo Cardano
  • Rafael Bombelli
  • Francois Viete
  • Simon Stevin
  • Rene Descartes
  • Pierre Fermat
  • Blaise Pascal
  • Isaac Newton
  • Gottfried Wilhelm Leibniz
  • The Bernoullis
  • Brooke Taylor
  • Christian Goldbach
  • Leonhard Euler
  • Jean Le Rond d’Alembert
  • Edward Waring
  • Joseph Louis Lagrange
  • Pierre-Simon Laplace
  • Adrien-Marie Legendre
  • Jean-Baptiste Joseph Fourier
  • Carl Friedrich Gauss
  • Simeon-Denis Poisson
  • Bernard Bolzano
  • Augustin-Louis Cauchy
  • August Ferdinand Mobius
  • Nicolai Ivanovich Lobachevskii
  • George Green
  • Niels Henrik Abel
  • Janos Bolyai
  • Carl Gustav Jacob Jacobi
  • Peter Gustav Lejeune Dirichlet
  • William Rowan Hamilton
  • Augustus De Morgan
  • Joseph Liouville
  • Eduard Kummer
  • Evariste Galois
  • James Joseph Sylvester
  • George Boole
  • Karl Weierstrass
  • Pafnuty Chebyshev
  • Arthur Cayley
  • Charles Hermite
  • Leopold Kronecker
  • Georg Bernhard Friedrich Riemann
  • Julius Wilhelm Richard Dedekind
  • Emile Leonard Mathieu
  • Camille Jordan
  • Sophus Lie
  • Georg Cantor
  • William Kingdon Clifford
  • Gottlob Frege
  • Christian Felix Klein
  • Ferdinand Georg Frobenius
  • Sonya Kovalevskaya
  • William Burnside
  • Jules Henri Poincare
  • Giuseppe Peano
  • David Hilbert
  • Hermann Minkowski
  • Jacques Hadamard
  • Ivar Fredholm
  • Charles-Jean de la Vallee Poussin
  • Felix Hausdorff
  • Elie Joseph Cartan
  • Emile Borel
  • Bertrand Arthur William Russell
  • Henri Lebesgue
  • Godfrey Harold Hardy
  • Frigyes (Frederic) Riesz
  • Luitzen Egbertus Jan Brouwer
  • Emmy Noether
  • Waclaw Sierpi?nski
  • George Birkhoff
  • John Edensor Littlewood
  • Hermann Weyl
  • Thoralf Skolem
  • Srinivasa Ramanujan
  • Richard Courant
  • Stefan Banach
  • Norbert Wiener
  • Emil Artin
  • Alfred Tarski
  • Andrei Nikolaevich Kolmogorov
  • William Vallance Douglas Hodge
  • John von Neumann
  • Kurt Godel
  • Andre Weil
  • Alan Turing
  • Abraham Robinson
  • Nicolas Bourbaki

Part VII The Influence of Mathematics:

  • Mathematics and Chemistry
  • Mathematical Biology
  • Wavelets and Applications
  • The Mathematics of Traffic 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

Part VIII 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 Major Mathematical Events
  1. The Princeton Companion to Mathematics. Gowers, Timothy; Barrow-Green, June; Leader, Imre. 2008.
  2. Advice to a Young Mathematician. Sir Michael Atiyah; Béla Bollobás; Alain Connes; Dusa McDuff; Peter Sarnak. assets.press.princeton.edu . 2008.