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The Princeton companion to mathematics
Publisher
Princeton University Press
Publication Date
c2008
Language
English
Summary
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Table of Contents
From the Book
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 Schreodinger 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. Geodel'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 Poincarae 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 Ja?far Muhammad ibn M*us*a al-Khw*arizm?i --
6.6. Leonardo of Pisa (known as Fibonacci) --
6.7. Girolamo Cardano --
6.8. Rafael Bombelli --
6.9. Fran?cois Viaete --
6.10. Simon Stevin --
6.11. Renae 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. Simaeon-Denis Poisson --
6.28. Bernard Bolzano --
6.29. Augustin-Louis Cauchy --
6.30. August Ferdinand Meobius --
6.31. Nicolai Ivanovich Lobachevskii --
6.32. George Green --
6.33. Niels Henrik Abel --
6.34. Jaanos 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. aEvariste 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. aEmile Laeonard 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 Poincarae --
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 Vallaee Poussin --
6.68. Felix Hausdorff --
6.69. aElie Joseph Cartan --
6.70. Emile Borel --
6.71. Bertrand Arthur William Russell --
6.72. Henri Lebesgue --
6.73. Godfrey Harold Hardy --
6.74. Frigyes (Fraedaeric) Riesz --
6.75. Luitzen Egbertus Jan Brouwer --
6.76. Emmy Noether --
6.77. Wac?aw Sierpi*nski --
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 Geodel --
6.93. Andrae 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.
From the eBook
Part I. Introduction
Part II. The origins of modern mathematics
Part III. Mathematical concepts
Part IV. Branches of mathematics
Part V. Theorems and problems
Part VI. Mathematicians
Part VII. Final perspectives.
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ISBN
9780691118802
9781400830398
9781849726955
9781400830398
9781849726955
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