Introduction: What is mathematics about? ; The language and grammar of mathematics ; Some fundamental mathematical definitions ; 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 ; Differential forms and integration ; Dimension ; Distributions.

Mathematical concepts (continued): 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-function ; 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.

Mathematical concepts (continued): 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 ; [pi] ; 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 ; Differential 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 conjecture.

Mathematicians: Pythagoras ; Euclid ; Archimedes ; Apollonius ; Abu Jaʼfar Muhammad ibn Mūsā al-Khwārizmī ; Leonardo of Pisa (known as Fibonacci) ; Girolamo Cardano ; Rafael Bombelli ; François Viète ; Simon Stevin ; René Descartes ; Pierre Fermat ; Blaise Pascal ; Isaac Newton ; Gottfried Wilhelm Leibniz ; Brook Taylor ; Christian Goldbach ; The Bernoullis ; 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 ; Siméon-Denis Poisson ; Bernard Bolzano ; Augustin-Louis Cauchy ; August Ferdinand Möbius ; Nicolai Ivanovich Lobachevskii ; George Green ; Niels Henrik Abel ; János Bolyai ; Carl Gustav Jacob Jacobi ; Peter Gustav Lejeune Dirichlet ; William Rowan Hamilton ; Augustus De Morgan ; Joseph Liouville ; Eduard Kumme ; Évariste Galois ; James Joseph Sylvester ; George Boole ; Karl Weierstrass ; Pafnuty Chebyshev ; Arthur Cayley ; Charles Hermite ; Leopold Kronecker.

Mathematicians (continued): Georg Friedrich Bernhard Riemann ; Julius Wilhelm Richard Dedekind ; Émile Léonard Mathieu ; Camille Jordan ; Sophus Lie ; Georg Cantor ; William Kingdon Clifford ; Gottlob Frege ; Christian Felix Klein ; Ferdinand Georg Frobenius ; Sofya (Sonya) Kovalevskaya ; William Burnside ; Jules Henri Poincaré ; Giuseppe Peano ; David Hilbert ; Hermann Minkowski ; Jacques Hadamard ; Ivar Fredholm ; Charles-Jean de la Vallée Poussin ; Felix Hausdorff ; Élie Joseph Cartan ; Emile Borel ; Bertrand Arthur William Russell ; Henri Lebesgue ; Godfrey Harold Hardy ; Frigyes (Frédéric) Riesz

Luitzen Egbertus Jan Brouwer ;

Emmy Noether ; Wacław Sierpiński ; George Birkhoff ; John Edensor Littlewood ; Hermann Weyl ; Thoralf Skolem ; Srinivasa Ramanujan ; Richard Courant ; Stefan Banach ; Norbert Wiener ; Emil Artin ; Alfred Tarski ; Andrei Nikolaevich Kolmogorov ; Alonzo Church ; William Vallance Douglas Hodge ; John von Neumann ; Kurt Gödel ; André Weil ; Alan Turing ; Abraham Robinson ; Nicolas Bourbaki.

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 --

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.