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1960

M Suzuki discovers new infinite families of finite simple groups.

1961

Edward Lorenz discovers a simple mathematical system with chaotic behaviour. It leads to the new mathematics of chaos theory which is widely applicable.

1961

Smale proves the higher dimensional Poincaré conjecture for *n* > 4, namely that any closed *n*-dimensional manifold which is homotopy equivalent to the *n*-sphere must be the *n*-sphere.

1962

Jacobson publishes his classic text *Lie algebras*.

1962

Sobolev publishes *Applications of Functional Analysis in Mathematical Physics*.

1963

John Thompson and Feit publish *Solvability of Groups of Odd Order* which proves that all nonabelian finite simple groups are of even order. Their paper requires 250 pages to prove the theorem.

1963

Cohen proves the independence of the axiom of choice and of the continuum hypothesis.

1964

Hironaka solves a major problem concerning the resolution of singularities on an algebraic variety.

1965

Sergi Novikov's work on differential topology, in particular in calculating stable homotopy groups and classifying smooth simply-connected manifolds, leads him to make the "Novikov Conjecture".

1965

Bombieri uses his improved large sieve method to prove what is now called "Bombieri's mean value theorem", which concerns the distribution of primes in arithmetic progressions.

1965

Tukey and Cooley publish a paper introducing the "Fast Fourier Transform" algorithm.

1965

Selten publishes important work on distinguishing between reasonable and unreasonable decisions in predicting the outcome of games. It will lead to the award of a Nobel Prize in 1994.

1966

Grothendieck receives a Fields Medal for his work on geometry, number theory, topology and complex analysis. His theory of schemes allows certain of Weil's number theory conjectures to be solved. His theory of topoi is highly relevant to mathematical logic, he had given an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve.

1966

Lander and Parkin use a computer to find a counterexample to Euler's Conjecture. They find 27^{5} + 84^{5} + 110^{5} + 133^{5} = 144^{5}.

1966

Alan Baker proves "Gelfond's Conjecture" about the linear independence of algebraic numbers over the rational numbers.

1967

Atiyah publishes *K-theory* which details his work on *K*-theory and the index theorem which led to the award of a Fields Medal in 1966.

1968

Novikov and Adian jointly publish a proof that the Burnside group *B*(*d*, *n*) is infinite for every *d* > 1 and every *n* > 4380.

1969

Conway publishes details of his discovery of new sporadic finite simple groups.

1970

Alan Baker is awarded a Fields Medal for his work on Diophantine equations.

1970

Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.

List of mathematicians alive in 1960.

List of mathematicians alive in 1970.

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