The mathematical century: the 30 greatest problems of the last 100 years

Ch. 1. Foundations

1.1. 1920s: Sets

1.2. 1940s: Structures

1.3. 1960s: Categories

1.4. 1980s: Functions

Ch. 2. Pure Mathematics

2.1. Mathematical Analysis: Lebesgue Measure (1902)

2.2. Algebra: Steinitz Classification of Fields (1910)

2.3. Topology: Brouwer's Fixed-Point Theorem (1910)

2.4. Number Theory: Gelfand Transcendental Numbers (1929)

2.5. Logic: Godel's Incompleteness Theorem (1931)

2.6. Calculus of Variations: Douglas's Minimal Surfaces (1931)

2.7. Mathematical Analysis: Schwartz's Theory of Distributions (1945)

2.8. Differential Topology: Milnor's Exotic Structures (1956)

2.9. Model Theory: Robinson's Hyperreal Numbers (1961)

2.10. Set Theory: Cohen's Independence Theorem (1963)

2.11. Singularity Theory: Thom's Classification of Catastrophes (1964)

2.12. Algebra: Gorenstein's Classification of Finite Groups (1972)

2.13. Topology: Thurston's Classification of 3-Dimensional Surfaces (1982)

2.14. Number Theory: Wiles's Proof of Fermat's Last Theorem (1995)

2.15. Discrete Geometry: Hales's Solution of Kepler's Problem (1998)

Ch. 3. Applied Mathematics

3.1. Crystallography: Bieberbach's Symmetry Groups (1910)

3.2. Tensor Calculus: Einstein's General Theory of Relativity (1915)

3.3. Game Theory: Von Neumann's Minimas Theorem (1928)

3.4. Functional Analysis: Von Neumann's Axiomatization of Quantum Mechanics (1932)

3.5. Probability Theory: Kolmogorov's Axiomatization (1933)

3.6. Optimization Theory: Dantzig's Simplex Method (1947)

3.7. General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954)

3.8. Theory of Formal Languages: Chomsky's Classification (1957)

3.9. Dynamical Systems Theory: The KAM Theorem (1962)

3.10. Knot Theory: Jones Invariants (1984)

Ch. 4. Mathematics and the Computer

4.1. Theory of Algorithms: Turing's Characterization (1936)

4.2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950)

4.3. Chaos Theory: Lorenz's Strange Attractor (1963)

4.4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976)

4.5. Fractals: The Mandelbrot Set (1980)

Ch. 5. Open Problems

5.1. Arithmetic: The Perfect Numbers Problem (300 B.C.)

5.2. Complex Analysis: The Riemann Hypothesis (1859)

5.3. Algebraic Topology: The Poincare Conjecture (1904)

5.4. Complexity Theory: The P = NP Problem (1972).