#### 1980

#### 1982

#### 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

#### 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.