Since Riemann’s invention of a geometry to describe higher dimensional curved spaces and Einstein’s introduction of his equations to describe gravity, the theory of the associated nonlinear partial differential equations has been a central one. These equations are elegant but in general they are notoriously difficult to study. One of the key issues is whether the solutions develop singularities.

Demetrios Christodoulou has made fundamental contributions to mathematical physics and especially in general relativity. His recent striking dynamical proof of the existence of trapped surfaces in the setting of Einstein’s equations in a vacuum demonstrates that black holes can be formed solely by the interaction of gravitational waves. Prior to that he made a deep study of this phenomenon in symmetrically reduced cases showing that unexpected naked singularities can occur but that they are unstable. In joint work with Klainerman he established the nonlinear stability of the Minkowski spacetime. His work is characterized by a profound understanding of the physics connected with these equations and brilliant mathematical technique.

Richard S Hamilton introduced the Ricci flow in Riemannian geometry. This is a differential equation which evolves the geometry of a space according to how it is curved. He used it to establish striking results about the shape (topology) of positively curved three and four dimensional spaces. During the last three decades he has developed a host of original and powerful techniques to study his flow; for example a technique called surgery allowing for the continuation of the evolution should singularities form. A primary goal of his theory was to classify all shapes in dimension three and in particular to resolve the Poincare Conjecture. Hamilton’s program was completed in the brilliant work of Perelman. With his Ricci flow, Hamilton has provided one of the most powerful tools in modern geometry.

Mathematical Sciences Selection Committee
The Shaw Prize