Demetrios Christodoulou & Richard Hamilton.

Demetrios Christodoulou one of the Shaw Prize winners has one single article from 2008 on arXive, a whole book on 'The Formation of Black Holes in General Relativity', a book where a result which complements the stability result is proved. Namely, that a sufficiently strong flux of incoming gravitational waves leads to the formation of a black hole. Prologue 34 p, sensible for everyone. Read it!

The Formation of Black Holes in General Relativity, (monograph, 589 pp.), EMS Monographs in Mathematics, EMS Publishing House (ISBN 978-3-03719-068-5), 2009.

Research field: Partial di fferential equations, geometric analysis, general relativity, fluid mechanics. His publications started from1970 with black holes, which also was his theme for the thesis 1971. Investigations in Gravitational Collapse and the Physics of Black Holes. So I guess he is qualified enough. Mathematical Problems of General Relativity I, 2008, The formation of shocks in 3-dimensional fluids, The Euler equations of compressible fluid flow, 2007, Recent developments in nonlinear hyperbolic PDE, 2001 etc.

On wikipedia: well known in the field of general relativity for his proof, together with Sergiu Klainerman, of the nonlinear stability of the Minkowski spacetime of special relativity in the framework of general relativity. The extraordinarily difficult proof of the stability result is laid out in detail.

• Christodoulou, Demetrios & Klainerman, Sergiu (1993). The global nonlinear stability of the Minkowski space. Princeton: Princeton University Press. ISBN 0-691-08777-6.

• Christodoulou, Demetrios (2000). The action principle and partial differential equations. Princeton: Princeton University Press. ISBN 0-691-04957-2.

Richard Hamilton's mathematical contributions are primarily in the field of differential geometry and more specifically geometric analysis. He is best known for having discovered the Ricci flow and suggesting the research program that ultimately led to the proof, by Grigori Perelman, of the Thurston geometrization conjecture and the solution of the Poincaré conjecture. Research field: Partial differential equations, differential geometry. (Peter Woit is also at the same institution. Algebraic Geometry, Mathematical Physics and Number Theory are other fields.) Woits blog. Not many words for the 1mill prize! The second Nobel?

Several stages of Ricci flow on a 2D manifold. Wikipedia. Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

Colombia Universitys release. Analytic number theory is the study of the distribution of prime numbers. One of the most important unsolved problems in mathematics is the Riemann hypothesis about the zeros of the Riemann zeta function, which gives a square root type error term for the number of primes in a large interval. One of the greatest applications of Grothendieck's theory of schemes is Deligne's proof of the Riemann hypothesis for L-functions for varieties over finite fields (which was first formulated by Weil). Thanks to the profound insight of Langlands, now embodied in the Langlands program: there is a sweeping vision of connections between automorphic L-functions on the one hand, and motivic L-functions, on the other. This vision encompasses the Artin and Shimura-Taniyama conjectures, both of which played a key role in Wiles' proof of Fermat last theorem. The main technique of Wiles, the deformation of Galois representations, is a new direction, now quite extensively developed

Wiles' proof of his modularity lifting theorems is a perfect illustration of p-adic techniques in number theory where the basic objects are deformation of Galois representations, congruences between modular forms, and their deep connections with special values of L-functions. Another spectacular illustration of the p-adic techniques for automorphic forms attached to higher rank reductive groups is the recent proof of the Sato-Tate conjecture. Mazur's theory of deformations of Galois representations used in Wiles' proof has been inspired by the theory of p-adic families of automorphic forms developed originally by Hida. This theory has been developing for reductive groups of higher rank and has many powerful applications for the understanding of the connections between L-functions (or p-adic L-functions) and Galois representations which are at the heart of modern research in algebraic number theory and arithmetic geometry. The theory of p-adic families has also inspired some of the new developments of p-adic Hodge theory and the so-called p-adic Langlands program which establishes a conjectural connection between p-adic Galois representations of a local field of residual characteristic p and certain p-adic representations of p-adic reductive groups. These subjects where the notion of p-adic variation is involved are advancing very quickly and a substantial breakthrough is expected in the near future.

• "Three-manifolds with positive Ricci curvature"

• Lecture by Hamilton on Ricci flow

• Joint seminars see Relative p-adic Hodge theory or THE FUNDAMENTAL CURVE OF p-ADIC HODGE THEORY

From 1996 Oswald Veblen Prize motivations.
The Ricci flow equations were introduced to geometers by Hamilton in 1982 (“Three manifolds with positive Ricci curvature”, J. Differential Geometry 17 (1982), 255–306). These equations form a very nonlinear system of differential equations (of essentially parabolic type) for the time evolution of a Riemannian metric on a smooth manifold. The equations assert simply that the time derivative of the metric is equal to minus twice the Ricci curvature tensor. (The Ricci curvature tensor is a symmetric, rank two tensor which is obtained by a natural average of the sectional curvatures.) This flow equation can be thought of as a nonlinear heat equation for the Riemannian metric. After an appropriate, time-dependent rescaling, the static solutions are simply the Einstein metrics. In introducing the Ricci flow equations, Hamilton proved that compact, three-dimensional manifolds with positive definite Ricci curvature are diffeomorphic to spherical space forms. (These are quotients of the three-dimensional sphere by free, finite
group actions.)
... understand the nature of the singularities which arise under the flow. (Hamilton proved that singularities do not arise in three dimensions when the Ricci curvature starts out positive.)
Hamilton has come to understand the geometric constraints on the singularities which arise under the Ricci flow on a compact, threedimensional Riemannian manifold and under a related flow equation (for the “isotropic curvature tensor”) on a compact, four-dimensional manifold. This understanding has allowed him, in many cases, to classify all possible singularities of the flow. In the four-dimensional case, Hamilton was recently able to give a topological characterization of the possible singularities which arise from the isotropic curvature tensor flow if the starting metric has positive isotropic curvature tensor. The conclusion is as follows: If a singularity arises, then it can be described as a lengthening neck in the manifold whose cross-section is an embedded spherical space form with injective fundamental group. Hamilton deduced from this fact that simply connected manifolds with positive isotropic curvature are diffeomorphic to the four-dimensional sphere.
For the compact 3-manifold case, Hamilton, in a recent paper, analyzed the development of singularities in the Ricci flow by studying the evolution of stable, closed geodesics and stable, minimal surfaces under their own, compatible, geometric flows. This analysis of the flows of stable geodesics and minimal surfaces leads to a characterization of the developing singularities in terms of Ricci soliton solutions to the flow equations along degenerating, geometric subsets of the original manifold. (A Ricci soliton is a solution whose motion in time is generated by a 1-parameter group of diffeomorphisms of the underlying manifold.)
etc.
He shared the prize with Gang Tian: The basic Kähler-Einstein problem is to find necessary and sufficient conditions for the existence of a Kähler metric on a given complex manifold whose Ricci curvature is a constant multiple of the metric itself. The sign of the constant is determined by the degree of the manifold’s first Chern class. The case where the sign is negative was solved independently by Aubin and Yau, while the sign zero case (where the first Chern class vanishes) was solved by Yau in his celebrated solution to the Calabi Conjecture.

This is still today very actual.

From the Columbia University research pages:
Topology is concerned with the intrinsic properties of shapes of spaces. One class of spaces which plays a central role in mathematics, and whose topology is extensively studied, are the n dimensional manifolds. These are spaces which locally look like Euclidean n-dimensional space.

Historically, topology has been a nexus point where algebraic geometry, differential geometry and partial differential equations meet and influence each other, influence topology, and are influencedby topology. More recently, topology and differential geometry have provided the language in which to formulate much of modern theoretical high energy physics. This interaction has brought topology, and mathematics more generally, a whole host of new questions and ideas. Because of its central place in a broad spectrum of mathematics there has always been a great deal of interaction between work in topology and work in these neighboring disciplines.

Ironically, in topology, the case of manifolds of dimensions 3 and 4, the physical dimensions in which we live, has eluded undestanding for the longest time. The case of manifolds of dimension n=1 is straightforward, and the case where n=2 was understood thoroughly in the 19^th century. Moreover, intense activity in the 1960's (including the pioneering work of Browder, Milnor, Novikov, and Smale) expresses the topology of manifolds of dimension n>4 in terms of an elaborate but purely algebraic description.

The study of manifolds of dimension n=3 and 4 is quite different from the higher-dimensional cases; and, though both cases n=3 and 4 are quite different in their overall character, both are generally referred to as low-dimensional topology.

Low-dimensional topology is currently a very active part of mathematics, benefiting greatly from its interactions with the fields of partial differential equations, differential geometry, algebraic geometry, modern physics, representation theory, number theory, and algebra.

The case of manifolds of dimension n=4 remains the most elusive. In view of the foundational results of Freedman, understanding manifolds up to their topological equivalence is a theory which is similar in character to the higher-dimensional manifold theory. However, the theory of differentiable four-manifolds is quite different. The subject was fundamentally transformed by the pioneering work of Simon Donaldson, who was studying moduli spaces of solutions to certain partial differential equations which came from mathematical physics. Studying algebro-topological properties of these moduli spaces, Donaldson came up with very interesting smooth invariants for four-manifolds which demonstrated the unique and elusive character of smooth four-manifold topology. In the case where the underlying manifold is Kähler, these moduli spaces also admit an interpretation in terms of stable bundles, and hence shed light on the differential topology of smooth algebraic surfaces. Since Donaldson's work, the physicists Seiberg and Witten introduced another smooth invariant of four-manifolds. Since then, the study of four-manifolds and their invariants has undergone several further exciting developments, tying them deeply with ideas from symplectic geometry and pseudo-holomorphic curves, and hence forming further bridges with algebraic and symplectic geometry, but also connecting them more closely with knot theory and three-manifold topology.

Geometry and analysis are vast fields, with myriad facets reflected differently in the leading mathematics departments worldwide. At Columbia, they are closely intertwined, with partial differential equations as the common unifying thread, and fundamental questions from several complex variables, algebraic geometry, topology, theoretical physics, probability, and applied mathematics as guiding goals.

The theory of partial differential equations, PDE, at Columbia is practically indistinguishable from its analytic, geometric, or physical contexts: the d-bar-equation from several complex variables and complex geometry, real and complex Monge-Ampère equations from differential geometry and applied mathematics, Schrodinger and Landau-Ginzburg equations from mathematical physics, and especially the powerful theory of geometric evolution equations from topology, algebraic geometry, general relativity, and gauge theories of elementary particle physics. Of particular interest are manifestations of non-linearity and curvature, long-time behavior and inherently non-perturbative aspects, formation of singularities, generalized and viscosity solutions, and global obstructions to the existence and regularity of solutions. Although real and complex differential geometry can be quite different in orientation - the latter having closer ties with algebraic geometry and number theory - both are strongly represented at Columbia.

Other less analytic aspects of the theory of partial differential equations also thrive at Columbia. Of particular importance is the theory of solitons and integrable models, with their hidden symmetries and deep geometric structures, and stochastic differential equations, with the ever growing manifestations of random phenomena.

From its PDE and differential geometry core, the group branches out for strong interactions with other groups in the department and the university, notably the groups in algebraic geometry, topology, number theory, string theory, and applied mathematics.
In arXive is so many articles, but look at this one from 2009!
arXiv:0905.4215 [ps, pdf, other]

Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

Stephen C. Anco, Esmaeel Asadi