Preface -- Contributors --

pt. 1. Introduction --

1.1.

What is mathematics about? --

1.2. The

language and grammar of mathematics --

1.3.

Some fundamental mathematical definitions --

1.4. The

general goals of mathematical research --

pt. 2. The

origins of modern mathematics --

2.1.

From numbers to number systems --

2.2.

Geometry --

2.3. The

development of abstract algebra --

2.4.

Algorithms --

2.5. The

development of rigor in mathematical analysis --

2.6. The

development of the idea of proof --

2.7. The

crisis in the foundations of mathematics --

pt. 3.

Mathematical concepts --

3.1. The

axiom of choice --

3.2. The

axiom of determinacy --

3.3.

Bayesian analysis --

3.4.

Braid groups --

3.5.

Buildings --

3.6.

Calabi-Yau manifolds --

3.7.

Cardinals --

3.8.

Categories --

3.9.

Compactness and compactification --

3.10.

Computational complexity classes --

3.11.

Countable and uncountable sets --

3.12.

C* - algebras --

3.13.

Curvature --

3.14.

Designs --

3.15.

Determinants --

3.15.

Differential forms and integration --

3.17.

Dimension --

3.18.

Distributions --

3.19.

Duality --

3.20.

Dynamical systems and chaos --

3.21.

Elliptic curves --

3.22. The

Euclidean algorithm and continued fractions --

3.23. The

Euler and Navier-Strokes equations --

3.24.

Expanders --

3.25. The

exponential and logarithmic functions --

3.26. The

fast Fourier transform --

3.27. The

Fourier transform --

3.28.

Fuchsian groups --

3.29.

Function spaces --

3.30.

Galois groups --

3.31. The

gamma function --

3.32.

Generating functions --

3.33.

Genus --

3.34.

Graphs --

3.35.

Hamiltonians --

3.36. The

heat equation --

3.37.

Hilbert spaces --

3.38.

Homology and cohomology --

3.39.

Homology and cohomology --

3.40. The

ideal class group --

3.41.

Irrational and transcendental numbers --

3.42. The

Isling model --

3.43.

Jordan normal form --

3.44.

Knot polynomials --

3.45.

K-theory --

3.46. The

leech lattice --

3.47.

L-function --

3.48.

Lie theory --

3.49.

Linear and nonlinear waves and solitons --

3.50.

Linear operators and their properties --

3.51.

Local and global in number theory --

3.52. The

Mandelbrot set --

3.53.

Manifolds --

3.54.

Matroids --

3.55.

Measures --

3.56.

Metric spaces --

3.57.

Models of set theory --

3.58.

Modular arithmetic --

3.59.

Modular forms --

3.60.

Moduli spaces --

3.61. The

monster group --

3.62.

Normed spaces and banach spaces --

3.63.

Number fields --

3.64.

Optimization and Lagrange multipliers --

3.65.

Orbifolds --

3.66.

Ordinals --

3.67. The

Peano axioms --

3.68.

Permutation groups --

3.69.

Phase transitions --

3.70.

[pi] --

3.71.

Probability distributions --

3.72.

Projective space --

3.73.

Quadratic forms --

3.74.

Quantum computation --

3.75.

Quantum computation --

3.76.

Quaternions, octonions, and normed division algebras --

3.77.

Representations --

3.78.

Ricci flow --

3.79.

Riemann surfaces --

3.80. The

Riemann zeta function --

3.81.

Rings, ideals, and modules --

3.82.

Schemes --

3.83. The

Schrödinger equation --

3.84. The

simplex algorithm --

3.85.

Special functions --

3.86. The

spectrum --

3.87.

Spherical harmonics --

3.88.

Symplectic manifolds --

3.89.

Tensor products --

3.90.

Topological spaces --

3.91.

Transforms --

3.92.

Trigonometric functions --

3.93.

Universal covers --

3.94.

Variational methods --

3.95.

Varieties --

3.96.

Vector bundles --

3.97.

Von Neumann algebras --

3.98.

Wavelets --

3.99. The

Zermelo-Fraenkel axioms --

pt. 4.

Branches of mathematics --

4.1.

Algebraic numbers --

4.2.

Analytic number theory --

4.3.

Computational number theory --

4.4.

Algebraic geometry --

4.5.

Arithmetic geometry --

4.6.

Algebraic topology --

4.7.

Differential topology --

4.8.

Moduli spaces --

4.9.

Representation theory --

4.10.

Geometric and combinatorial group theory --

4.11.

Harmonic analysis --

4.12.

Partial differential equations --

4.13.

General relativity and the Einstein equations --

4.14.

Dynamics --

4.15.

Operator algebras --

4.16.

Mirror symmetry --

4.17.

Vertex operator algebras --

4.18.

Enumerative and algebraic combinatorics --

4.19.

Extremal and probabilistic combinatorics --

4.20.

Computational complexity --

4.21.

Numerical analysis --

4.22.

Set theory --

4.23.

Logic and model theory --

4.24.

Stochastic processes --

4.25.

Probabilistic models of critical phenomena --

4.26.

High-dimensional geometry and its probabilistic analogues --

pt. 5.

Theorems and problems --

5.1. The

ABC conjecture --

5.2. The

Atiyah-Singer index theorem --

5.3. The

Banach-Tarski paradox --

5.4. The

Birch-Swinnerton-Dyer conjecture --

5.5.

Carleson's theorem --

5.6. The

central limit theorem --

5.7. The

classification of finite simple groups --

5.8.

Dirichlet's theorem --

5.9.

Ergodic theorems --

5.10.

Fermat's last theorem --

5.11.

Fixed point theorems --

5.12. The

four-color theorem --

5.13. The

fundamental theorem of algebra --

5.14. The

fundamental theorem of arithmetic --

5.15.

Gödel's theorem --

5.16.

Gromov's polynomial-growth theorem --

5.17.

Hilbert's nullstellensatz --

5.18. The

independence of the continuum hypothesis --

5.19.

Inequalities --

5.20. The

insolubility of the halting problem --

5.21. The

insolubility of the quintic --

5.22.

Liousville's theorem and Roth's theorem --

5.23.

Mostow's strong rigidity theorem --

5.24. The

p versus NP problem --

5.25. The

Poincaré conjecture --

5.26. The

prime number theorem and the Riemann hypothesis --

5.27.

Problems and results in additive number theory --

5.28.

From quadratic reciprocity to class field theory --

5.29.

Rational points on curves and the Mordell conjecture --

5.30. The

resolution of singularities --

5.31. The

Riemann-Roch theorem --

5.32. The

Robertson-Seymour theorem --

5.33. The

three-body problem --

5.34. The

uniformization theorem --

5.35. The

Weil conjecture -- pt. 6. Mathematicians --

6.1.

Pythagoras --

6.2.

Euclid --

6.3.

Archimedes --

6.4.

Apollonius --

6.5.

Abu Jaʼfar Muhammad ibn Mūsā al-Khwārizmī --

6.6.

Leonardo of Pisa (known as Fibonacci) --

6.7.

Girolamo Cardano --

6.8.

Rafael Bombelli --

6.9.

François Viète --

6.10.

Simon Stevin --

6.11.

René Descartes --

6.12.

Pierre Fermat --

6.13.

Blaise Pascal --

6.14.

Isaac Newton --

6.15.

Gottfried Wilhelm Leibnitz --

6.16.

Brook Taylor --

6.17.

Christian Goldbach --

6.18. The

Bernoullis --

6.19.

Leonhard Euler --

6.20.

Jean Le Rond d'Alembert --

6.21.

Edward Waring --

6.22.

Joseph Louis Lagrange --

6.23.

Pierre-Simon Laplace --

6.24.

Adrien-Marie Legendre --

6.25.

Jean-Baptiste Joseph Fourier --

6.26.

Carl Friedrich Gauss --

6.27.

Siméon-Denis Poisson --

6.28.

Bernard Bolzano --

6.29.

Augustin-Louis Cauchy --

6.30.

August Ferdinand Möbius --

6.31.

Nicolai Ivanovich Lobachevskii --

6.32.

George Green --

6.33.

Niels Henrik Abel --

6.34.

János Bolyai --

6.35.

Carl Gustav Jacob Jacobi --

6.36.

Peter Gustav Lejeune Dirichlet --

6.37.

William Rowan Hamilton --

6.38.

Augustus De Morgan --

6.39.

Joseph Liouville --

6.40.

Eduard Kummer --

6.41.

Évariste Galois --

6.42.

James Joseph Sylvester --

6.43.

George Boole --

6.44.

Karl Weierstrass --

6.45.

Pafnuty Chebyshev --

6.46.

Arthur Cayley --

6.47.

Charles Hermite --

6.48.

Leopold Kronecker --

6.49.

Georg Friedrich Bernhard Riemann --

6.50.

Julius Wilhelm Richard Dedekind --

6.51.

Émile Léonard Mathieu --

6.52.

Camille Jordan --

6.53.

Sophus Lie --

6.54.

Georg Cantor --

6.55.

William Kingdon Clifford --

6.56.

Gottlob Frege --

6.57.

Christian Felix Klein --

6.58.

Ferdinand Georg Frobenius --

6.59.

Sofya (Sonya) Kovalevskaya --

6.60.

William Burnside --

6.61.

Jules Henri Poincaré --

6.62.

Giuseppe Peano --

6.63.

David Hilbert --

6.64.

Hermann Minkowski --

6.65.

Jacques Hadamard --

6.66.

Ivar Fredholm --

6.67.

Charles-Jean de la Vallée Poussin --

6.68.

Felix Hausdorff --

6.69.

Élie Joseph Cartan --

6.70.

Emile Borel --

6.71.

Bertrand Arthur William Russell --

6.72.

Henri Lebesgue --

6.73.

Godfrey Harold Hardy --

6.74.

Frigyes (Frédéric) Riesz --

6.75.

Luitzen Egbertus Jan Brouwer --

6.76.

Emmy Noether --

6.77.

Wacław Sierpiński --

6.78.

George Birkhoff --

6.79.

John Edensor Littlewood --

6.80.

Hermann Weyl --

6.81.

Thoralf Skolem --

6.82.

Srinivasa Ramanujan --

6.83.

Richard Courant --

6.84.

Stefan Banach --

6.85.

Norbert Wiener --

6.86.

Emil Artin --

6.87.

Alfred Tarski --

6.88.

Andrei Nikolaevich Kolmogorov --

6.89.

Alonzo Church --

6.90.

William Vallance Douglas Hodge --

6.91.

John von Neumann --

6.92.

Kurt Gödel --

6.93.

André Weil --

6.94.

Alan Turing --

6.95.

Abraham Robinson --

6.96.

Nicolas Bourbaki --

pt. 7. The

influence of mathematics --

7.1.

Mathematics and chemistry --

7.2.

Mathematical biology --

7.3.

Wavelets and applications --

7.4. The

mathematics of traffic in networks --

7.5. The

mathematics of algorithm design --

7.6

Reliable transmission of information --

7.7.

Mathematics and cryptography --

7.8.

Mathematics and economic reasoning --

7.9. The

mathematics of money --

7.10.

Mathematical statistucs --

7.11.

Mathematics and medical statistics --

7.12.

Analysis, mathematical and philosophical --

7.13.

Mathematics and music --

7.14.

Mathematics and art --

pt. 8.

Final perspectives --

8.1. The

art of problem solving --

8.2.

"Why mathematics?" you might ask --

8.3. The

ubiquity of mathematics --

8.4.

Numeracy --

8.5.

Mathematics : an experimental science --

8.6.

Advice to a young mathematician --

8.7. A

chronology of mathematical events -- Index