Maths in 100 Key Breakthroughs

Dr Richard Elwes is a writer, teacher and researcher in Mathematics and a Visiting Fellow at the University of Leeds. He contributes to New Scientist and Plus Magazine and publishes research on model theory. Dr Elwes is a committed popularizer of mathematics, on which he speaks regularly at public lectures and on radio. He lives in Leeds.

Introduction. The evolution of counting. Tallies. Place-value notation. Area and volume. Pythagoras' theorem. Irrational numbers. Zeno's paraodoxes. The Platonic solids. Logic. Euclidean geometry. Prime numbers. The area of a circle. Conic sections. Trigonometry. Perfect numbers. Diophantine equations. Hindu-Arabic numerals. Modular arithmetic. Negative numbers. Algebra. Combinatorics. The Fibonacci sequence. The harmonic series. Cubic and quartic equations. The complex numbers. Logarithms. Polyhedra. Tessellations. Kepler's laws. Projective geometry. Coordinates. Calculus. Differential geometry. Polar coordinates. Normal distribution. Graph theory. Exponentiation. Euler characteristic. Conditional probability. Fundamental theorem of algebra. Fourier analysis. The real numbers. The unsolvability of the quintic. The Navier-Stokes equations. Curvature. Hyperbolic geometry. Constructible numbers. Transcendental numbers. Polytopes. Riemann's zeta function. Jordan curve theorem. Classification of surfaces. Cardinal numbers. Wallpaper groups. Digital geometry. Russell's paradox. Special relativity. The three-body problem. Waring's problem. Markov's processes. General relativity. Fractals. Abstract algebra. Knot polynomials. Quantum mechanics. Quantum field theory. Ramsey theory. Godel's incompleteness theorem. Turing machines. Numerical analysis. Information theory. Arrow's impossibility theorem. Game theory. Exotic spheres. Randomness. The continuum hypothesis. Singularity theory. Quasicrystals. Friendship theorem. Non-standard analysis. Hilbert's tenth problem. The game of Life. Complexity theory. The travelling salesman problem. Chaos theory. Four colour theorem. Public key cryptography. Elliptic curves. Weaire-Phelan foam. Quantum computing. Fermat's Last Theorem. Kepler's conjecture. Catalan's conjecture. Poincare's conjecture. Constellations of primes. The classification of finite simple groups. Langlands program. Reverse mathematics. Partitions. Sudoku. Glossary. Index.