Chronology

1980 - 1990

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.
• 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.
• 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.
• 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 LINK.)

1984

• Louis de Brange solves the Bieberbach Conjecture.
• Vaughan Jones discovers a new polynomial invariant for knots and links in 3-space.
• 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."
• Elkies finds a counterexample to Euler's Conjecture with $n = 4$, namely 26824404 + 153656394 + 187967604 = 206156734 .
  Later in the year Frye finds the smallest counter-example: 958004 + 2175194 + 4145604 = 4224814 .

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.