Rick Schoen is known for his solution of the Yamabe conjecture (also solved by Aubin), and for his solution of the positive mass conjecture together with Yau. He also has important works about minimal surfaces and super-rigidity of rank 1 symmetric spaces.
More recently, and I suspect largely why he is giving a plenary lecture at the ICM, he and Simon Brendle proved that $1/4$-pinched Riemannian manifolds are space forms. If the manifold is simply-connected, then previous sphere theorems imply that $1/4$-pinched manifolds are homeomorphic to spheres (Berger, Klingenberg). Together with the Poincare conjecture, this implies that they are homeomorphic to spheres. However, there are spheres which are homeomorphic but not diffeomorphic to the $n$-sphere (Milnor), so the question was left open as to whether these manifolds are diffeomorphic to the $n$-sphere. This is the question that Brendle and Schoen resolved, using Ricci flow.
