Richard S Hamilton

for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.

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.

After Newton’s introduction of calculus and in particular differential equations to describe the motion of the planets, classical physics and geometry developed with more complex phenomena naturally being formulated in terms of partial differential equations. The Einstein equations in general relativity and the Ricci Flow equation in Riemannian geometry are two celebrated geometric partial differential equations. The first describes the geometry of four dimensional spacetime and it relates gravitation to curvature. The second gives an evolution of Riemannian geometries in which the flow at a given time is dictated by the curvature of the space at that time. Both of these equations are very elegant in their formulation. They are nonlinear partial differential equations in several unknown quantities which in turn depend on several variables. While they are of quite different characteristics in terms of the classification of such equations, they share the feature that they are notoriously difficult to study rigorously (even on a computer). Central to the understanding of the solutions, is whether they form singularities or not, and if so what is their nature. In the spacetime setting, examples of singularities are black holes and more generally gravitational collapse. In the Ricci Flow, should singularities arise in the course of the evolution, then for certain applications they need to be resolved. Christodoulou, in the case of Einstein’s equations, and Hamilton in the case of the Ricci Flow, have made many of the fundamental breakthroughs in the theory of these geometric equations and especially in understanding their singularities. Their works have spectacular applications both to mathematics and to physics.

**Demetrios Christodoulou** was born in 1951 in Athens, Greece. He is currently Professor of Mathematics and Physics at the ETH, Zurich in Switzerland. He received his MA in Physics in 1970 and PhD in Physics in 1971 from Princeton University. He was Professor of Mathematics at Syracuse University (1985 – 1987), the Courant Institute (1988 – 1992) and the Princeton University (1992 – 2001). He is a member of the American Academy of Arts and Sciences and European Academy of Sciences.

**Richard S Hamilton** was born in 1943 in Ohio, USA. He is currently Davies Professor of Mathematics at the Columbia University, USA. He received his BA in 1963 from Yale University and PhD in 1966 from Princeton University. He has held positions at UC Irvine, UC San Diego and Cornell University. He is a member of the United States National Academy of Sciences and the American Academy of Arts and Sciences.