Aubin and Yau's proof

The work of the Calabi conjectureUhlenbeck, which involves the existence Taubes, Donaldson, and uniqueness of solutions to a complex Monge-Ampere equation has had others on Yang-Mills connections is a lot gorgeous application of impact in algebraic geometrynonlinear elliptic PDE theory.

Aubin and Yau's proof of the Calabi conjecture, which involves the existence and uniqueness of solutions to a complex Monge-Ampere equation has had a lot of impact in algebraic geometry.