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1980

The classification of finite simple groups is complete.

1982

Mandelbrot publishes *The fractal geometry of nature* which develops his theory of fractal geometry more fully than his work of 1975.

1982

Freedman proves that any closed 4-dimensional manifold which is homotopy equivalent to the 4-sphere must be the 4-sphere. This proves a further case of the higher dimensional Poincaré conjecture following Smale's work in 1961.

1982

Shing-Tung Yau is awarded a Fields Medal for his contributions to partial differential equations, to the "Calabi conjecture" in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampère equations.

1983

Donaldson publishes *Self-dual connections and the topology of smooth 4-manifolds* which leads to totally new ideas concerning the geometry of 4-manifolds.

1983

Faltings proves the "Mordell conjecture". He makes a major contribution to Fermat's Last Theorem showing that for every *n* there are at most a finite number of coprime integers *x*, *y*, *z* satisfying *x*^{n} + *y*^{n} = *z*^{n}. (See this History Topic.)

1984

Louis de Brange solves the Bieberbach Conjecture.

1984

Vaughan Jones discovers a new polynomial invariant for knots and links in 3-space.

1984

Witten publishes *Supersymmetry and Morse theory* containing ideas that have become of central importance in the study of differential geometry.

1986

Margulis proves the "Oppenheim conjecture" on the values of indefinite irrational quadratic forms at integer points.

1987

Zelmanov proves an important conjecture about when an infinite dimensional Lie algebra is nilpotent.

1988

Langlands is the first recipient of the National Academy of Sciences Award in Mathematics. He receives it for "extraordinary vision that has brought the theory of group representations into a revolutionary new relationship with the theory of automorphic forms and number theory."

1988

Elkies finds a counterexample to Euler's Conjecture with *n* = 4, namely 2682440^{4} + 15365639^{4} + 18796760^{4} = 20615673^{4}.

Later in the year Frye finds the smallest counter-example: 95800^{4} + 217519^{4} + 414560^{4} = 422481^{4}.

1989

Bourgain, using analytic and probabilistic methods, solves the *L*(*p*) problem which had been a longstanding one in "Banach space" theory and harmonic analysis.

1990

Drinfeld is awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan for his work on quantum groups and for his work in number theory.

List of mathematicians alive in 1980.

List of mathematicians alive in 1990.

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