The mathematical century

TABLE OF CONTENTS:

Foreword xi
Acknowledgments xvii
Introduction 1

CHAPTER 1: THE FOUNDATIONS 8
1.1. The 1920s: Sets 10
1.2. The 1940s: Structures 14
1.3. The 1960s: Categories 17
1.4. The 1980s: Functions 21

CHAPTER TWO: PURE MATHEMATICS 25
2.1. Mathematical Analysis: Lebesgue Measure (1902) 29
2.2. Algebra: Steinitz Classification of Fields (1910) 33
2.3. Topology: Brouwer´s Fixed-Point Theorem (1910) 37
2.4. Number Theory: Gelfand Transcendental Numbers (1929) 39
2.5. Logic: Gödel´s Incompleteness Theorem (1931) 43
2.6. The Calculus of Variations: Douglas´s Minimal Surfaces (1931) 47
2.7. Mathematical Analysis: Schwartz´s Theory of Distributions (1945) 52
2.8. Differential Topology: Milnor´s Exotic Structures (1956) 56
2.9. Model Theory: Robinson´s Hyperreal Numbers (1961) 59
2.10. Set Theory: Cohen´s Independence Theorem (1963) 63
2.11. Singularity Theory: Thom´s Classification of Catastrophes (1964) 66
2.12. Algebra: Gorenstein´s Classification of Finite Groups (1972) 71
2.13. Topology: Thurston´s Classification of 3-Dimensional Surfaces (1982) 78
2.14. Number Theory: Wiles´s Proof of Fermat´s Last Theorem (1995) 82
2.15. Discrete Geometry: Hales´s Solution of Kepler´s Problem (1998) 87

CHAPTER THREE: APPLIED MATHEMATICS 92
3.1. Crystallography: Bieberbach´s Symmetry Groups (1910) 98
3.2. Tensor Calculus: Einstein´s General Theory of Relativity (1915) 104
3.3. Game Theory: Von Neumann´s Minimax Theorem (1928) 108
3.4. Functional Analysis: Von Neumann´s Axiomatization of Quantum Mechanics (1932) 112
3.5. Probability Theory: Kolmogorov´s Axiomatization (1933) 116
3.6. Optimization Theory: Dantzig´s Simplex Method (1947) 120
3.7. General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954) 122
3.8. The Theory of Formal Languages: Chomsky´s Classification (1957) 125
3.9. Dynamical Systems Theory: The KAM Theorem (1962) 128
3.10. Knot Theory: Jones Invariants (1984) 132

CHAPTER FOUR: MATHEMATICS AND THE COMPUTER 139
4.1. The Theory of Algorithms: Turing´s Characterization (1936) 145
4.2. Artificial Intelligence: Shannon´s Analysis of the Game of Chess (1950) 148
4.3. Chaos Theory: Lorenz´s Strange Attractor (1963) 151
4.4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976) 154
4.5. Fractals: The Mandelbrot Set (1980) 159

CHAPTER FIVE: OPEN PROBLEMS 165
5.1. Arithmetic: The Perfect Numbers Problem (300 BC) 166
5.2. Complex Analysis: The Riemann Hypothesis (1859) 168
5.3. Algebraic Topology: The Poincare Conjecture (1904) 172
5.4. Complexity Theory: The P=NP Problem (1972) 176

Conclusion 181
References and Further Reading 187
Index 189