| Description |
xx, 1034 pages : illustrations ; 26 cm |
| Bibliography |
Includes bibliographical references and index. |
| Contents |
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 -- |
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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-Stokes 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. Homotopy Groups -- 3.40. The ideal class group -- 3.41. Irrational and transcendental numbers -- 3.42. The Ising 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 -- |
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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 groups -- 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 -- |
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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 -- |
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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. 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 -- |
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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 Leibniz -- 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 -- |
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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 -- |
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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 -- |
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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 -- 8.4. Numeracy -- 8.5. Mathematics : an experimental science -- 8.6. Advice to a young mathematician -- 8.7. A chronology of mathematical events -- Index. |
| Subject |
Mathematics.
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| Added Author |
Gowers, Timothy.
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Barrow-Green, June, 1953-
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Leader, Imre.
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Princeton University.
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| ISBN |
9780691118802 hardcover alkaline paper |
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0691118809 hardcover alkaline paper |
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