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 --
2.3. The development of abstract algebra --
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 --
3.6. Calabi-Yau manifolds --
3.9. Compactness and compactification --
3.10. Computational complexity classes --
3.11. Countable and uncountable sets --
3.15. Differential forms and integration --
3.20. Dynamical systems and chaos --
3.22. The Euclidean algorithm and continued fractions --
3.23. The Euler and Navier-Stokes equations --
3.25. The exponential and logarithmic functions --
3.26. The fast Fourier transform --
3.27. The Fourier transform --
3.31. The gamma function --
3.32. Generating functions --
3.36. The heat equation --
3.38. Homology and cohomology --
3.40. The ideal class group --
3.41. Irrational and transcendental numbers --
3.43. Jordan normal form --
3.44. Knot polynomials --
3.46. The leech lattice --
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 --
3.57. Models of set theory --
3.58. Modular arithmetic --
3.61. The monster group --
3.62. Normed spaces and banach spaces --
3.64. Optimization and Lagrange multipliers --
3.67. The Peano axioms --
3.68. Permutation groups --
3.69. Phase transitions --
3.71. Probability distributions --
3.72. Projective space --
3.74. Quantum computation --
3.76. Quaternions, octonions, and normed division algebras --
3.79. Riemann surfaces --
3.80. The Riemann zeta function --
3.81. Rings, ideals, and modules --
3.83. The Schrödinger equation --
3.84. The simplex algorithm --
3.85. Special functions --
3.87. Spherical harmonics --
3.88. Symplectic manifolds --
3.90. Topological spaces --
3.92. Trigonometric functions --
3.93. Universal covers --
3.94. Variational methods --
3.97. Von Neumann algebras --
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 --
4.9. 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 --
4.15. Operator algebras --
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 --
4.23. 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 --
5.10. 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 --
5.16. Gromov's polynomial-growth theorem --
5.17. Hilbert's nullstellensatz --
5.18. The independence of the continuum hypothesis --
5.20. The insolubility of the halting problem --
5.21. The insolubility of the quintic --
5.22. Liouville'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.
6.5. Abu Jaʼfar Muhammad ibn Mūsā al-Khwārizmī --
6.6. Leonardo of Pisa (known as Fibonacci) --
6.15. Gottfried Wilhelm Leibniz --
6.17. Christian Goldbach --
6.20. Jean Le Rond d'Alembert --
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 --
6.29. Augustin-Louis Cauchy --
6.30. August Ferdinand Möbius --
6.31. Nicolai Ivanovich Lobachevskii --
6.33. Niels Henrik Abel --
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 --
6.42. James Joseph Sylvester --
6.44. Karl Weierstrass --
6.45. Pafnuty Chebyshev --
6.48. Leopold Kronecker --
6.49. Georg Friedrich Bernhard Riemann --
6.50. Julius Wilhelm Richard Dedekind --
6.51. Émile Léonard Mathieu --
6.55. William Kingdon Clifford --
6.57. Christian Felix Klein --
6.58. Ferdinand Georg Frobenius --
6.59. Sofya (Sonya) Kovalevskaya --
6.60. William Burnside --
6.61. Jules Henri Poincaré --
6.64. Hermann Minkowski --
6.65. Jacques Hadamard --
6.67. Charles-Jean de la Vallée Poussin --
6.69. Élie Joseph Cartan --
6.71. Bertrand Arthur William Russell --
6.73. Godfrey Harold Hardy --
6.74. Frigyes (Frédéric) Riesz.
6.75. Luitzen Egbertus Jan Brouwer --
6.77. Wacław Sierpiński --
6.79. John Edensor Littlewood --
6.82. Srinivasa Ramanujan --
6.88. Andrei Nikolaevich Kolmogorov --
6.90. William Vallance Douglas Hodge --
6.91. John von Neumann --
6.95. Abraham Robinson --
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 statistics --
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 --
8.5. Mathematics : an experimental science --
8.6. Advice to a young mathematician --
8.7. A chronology of mathematical events --