1.1. The 1996 Prize.

The 1996 Oswald Veblen Prize in Geometry was awarded at the Joint Mathematics Meetings in Orlando in January 1996 to Richard Hamilton of the University of California, San Diego, and to Gang Tian of the Massachusetts Institute of Technology.

Oswald Veblen (1880-1960), who served as president of the Society in 1923 and 1924, was well known for his mathematical work in geometry and topology. In 1961 the trustees of the Society established a fund in memory of Professor Veblen, contributed originally by former students and colleagues and later doubled by his widow. Since 1964 the fund has been used for the award of the Oswald Veblen Prize in Geometry. Subsequent awards were made at five-year intervals.

At present, the award is supplemented from the Steele Prize Fund, bringing the value of the Veblen Prize to $4,000, divided equally between this year's recipients.

The 1996 Veblen Prize was awarded by the AMS Council on the basis of a recommendation by a selection committee consisting of Jeff Cheeger, Peter Li, and Clifford Taubes (chair).

1.2. Citation for Richard Hamilton.

Richard Hamilton is cited for his continuing study of the Ricci flow and related parabolic equations for a Riemannian metric and he is cited in particular for his analysis of the singularities which develop along these flows.

The Ricci flow equations were introduced to geometers by Hamilton in 1982 ("Three manifolds with positive Ricci curvature" (1982)). 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.)

Over the subsequent years, Hamilton has continued his study of the Ricci flow equations and related equations, delving ever deeper to 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, three-dimensional 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 characterisation 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, analysed 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 characterisation 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.)

The Oswald Veblen Prize in Geometry is awarded to Richard Hamilton in recognition of his recent and continuing work to uncover the geometric and analytic properties of singularities of the Ricci flow equation and related systems of differential equations.

1.3. Response by Richard Hamilton.

It is a great honour to receive the Oswald Veblen Prize from the AMS. This award recognises the tremendous growth of the whole field of nonlinear parabolic partial differential equations in geometry, of which my own work is but a small part. Especial thanks are due to my parents, Dr and Mrs Selden Hamilton, who provided me with every conceivable head start in education; my high school geometry teacher, Mrs Becker, for an enduring love of three-dimensional geometry; my mentor, James Eells, Jr., whose work with Joseph Sampson on the Harmonic Map Heat Flow originated and inspired the field; and my colleagues S-T Yau and Richard Schoen, who suggested the neck-pinching phenomenon and encouraged me to study the formation of singularities.

It is a pleasure to share the prize with Gang Tian, whose work on Kähler manifolds is outstanding.

1.4. Brief biography.

Professor Hamilton was born in Cincinnati, Ohio, in 1943. He received his B.A. from Yale University in 1963 and his Ph.D. from Princeton University in 1966 under the direction of Robert Gunning. He has held professorships at Cornell University and the University of California at Berkeley and visiting positions at the University of Warwick, the Courant Institute, the Institute for Advanced Study in Princeton, and the University of Hawaii. He is currently professor of mathematics at the University of California, San Diego.

2.1. The Clay Research Awards.

The Clay Research Awards, presented annually at the Clay Research Conference, celebrate the outstanding achievements of the world's most gifted mathematicians. Although perhaps less well known outside the mathematical world than the Clay Millennium Prize Problems, the Clay Research Award is widely appreciated within it.

2.2. Citation for Richard Hamilton.

The 2003 Clay Research Award was made to Richard Hamilton for his discovery of the Ricci Flow Equation and its development into one of the most powerful tools of geometric analysis. Hamilton conceived of his work as a way to approach both the Poincaré Conjecture and the Thurston Geometrization Conjecture.

2.3. Report in Clay Institute Year 2003.

Each year Clay Mathematics Institute recognises outstanding achievements in mathematics with the Clay Research Award. In 2003, the award went to Richard Hamilton for his visionary work on Ricci flow and to Terence Tao for his fundamental contributions to analysis and other fields.

Richard Hamilton of Columbia University was recognised for his introduction of the Ricci flow equation and his development of it into one of the most powerful tools in geometry and topology. The Ricci flow equation is a system of nonlinear partial differential equations somewhat analogous to the classical (scalar) heat equation. However, the quantity that evolves is not temperature, but rather the metric on a manifold, that is, its geometry. In his seminal 1982 paper, Hamilton showed that a compact manifold with positive Ricci curvature evolves toward a state of constant curvature. This theorem is the basis of his visionary program to prove William Thurston's geometrisation conjecture, of which the celebrated Poincaré conjecture is a special case. Recent work by Grigori Perelman of St Petersburg has spectacularly advanced these ideas and brought us much closer to an understanding of the conjectures.

The awards were presented at the Clay Mathematics Institute Annual Meeting on 14 November 2003. The Massachusetts Institute of Technology hosted the event, which was held at its Media Lab.

3.1. The Leroy P Steele Prize for Seminal Contribution to research.

The Steele Prize for Seminal Contribution to Research is awarded for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research.

These prizes were established in 1970 in honour of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein, and are endowed under the terms of a bequest from Leroy P Steele. From 1970 to 1976 one or more prizes were awarded each year for outstanding published mathematical research; most favourable consideration was given to papers distinguished for their exposition and covering broad areas of mathematics. In 1977 the Council of the American Mathematical Society modified the terms under which the prizes are awarded. In 1993, the Council formalised the three categories of the prize by naming each of them: (1) The Leroy P Steele Prize for Lifetime Achievement; (2) The Leroy P Steele Prize for Mathematical Exposition; and (3) The Leroy P Steele Prize for Seminal Contribution to Research.

3.2. The 2009 Steele Prizes.

The 2009 AMS Leroy P Steele Prizes were presented at the 115th Annual Meeting of the American Mathematical Society in Washington, DC, in January 2009. The Steele Prizes were awarded to I G Macdonald for Mathematical Exposition, to Richard Hamilton for a Seminal Contribution to Research, and to Luis Caffarelli for Lifetime Achievement.

3.3. Citation for Seminal Contribution to Research: Richard Hamilton.

The 2009 Leroy P Steele Prize for Seminal Contribution to Research is awarded to Richard Hamilton for his paper "Three-manifolds with positive Ricci curvature", Journal of Differential Geometry 17 (1982), 255-306.

Differential geometry includes the study of Riemannian metrics and their associated geometric entities. These include the curvature tensor, geodesic distance function, natural differential operators on functions, forms, and tensors as well as many others. A given smooth manifold has an infinite-dimensional space of Riemannian metrics whose geometric behaviour may vary dramatically. By its very nature geometry must be coordinate invariant, so two Riemannian metrics which are related by a diffeomorphism of the manifold must be considered equivalent. The question of choosing a natural metric from the infinite-dimensional family is nicely illustrated by the case of compact oriented two-dimensional surfaces. For surfaces of genus 0 there is a unique choice of equivalence class of metrics with curvature 1, while for genus 1 (respectively genus greater than 1) there is a finite-dimensional moduli space of inequivalent metrics with curvature 0 (respectively curvature -1).

The cited paper of Richard Hamilton introduced a profoundly original approach to the construction of natural metrics on manifolds. This approach is the Ricci flow, which is an evolution equation in the space of Riemannian metrics on a manifold. The stationary points (for the normalised flow) are the Einstein metrics (constant curvature in dimensions 2 and 3). The Ricci flow is a nonlinear diffusion equation which may be used to deform any chosen initial metric for a short time interval. In the cited paper Hamilton showed that, in dimension 3, if the initial metric has positive Ricci curvature, then the flow exists for all time and converges to a constant curvature metric. This implies the remarkable result that a three manifold of positive Ricci curvature is a spherical space form (a space of constant curvature). Over the next twenty years Hamilton laid the groundwork for understanding the long time evolution for an arbitrary initial metric on a three-manifold with an eye toward the topological classification problem. For this purpose he developed the idea of the Ricci flow with singularities in which the flow would be continued past singular times by performing surgeries in a controlled way. Finally, through the spectacular work of Grisha Perelman in 2002, the difficult issues remaining in the construction were resolved, and the program became successful.

In addition to the applications to the topology of three-manifolds, the Ricci flow has had, and continues to have, a wide range of applications to Riemannian and Kähler geometry. The cited paper truly fits the definition of a seminal contribution; that is, "containing or contributing the seeds of later development".

3.4. Response from Richard Hamilton.

It is a great honour to receive the Steele Prize acknowledging the role of my 1982 paper in launching the Ricci flow, which has now succeeded even beyond my dreams. I am deeply grateful to the prize committee and the AMS.

When I first arrived at Cornell in 1966, James Eells Jr. introduced me to the idea of using a nonlinear parabolic partial differential equation to construct an ideal geometric object, lecturing on his brilliant 1964 paper with Joseph Sampson on harmonic maps, which was the origin of the field of geometric flows. This now encompasses the harmonic map flow, the mean curvature flow (used in physics to describe the motion of an interface, and also in image processing as well as isoperimetric estimates), the Gauss curvature flow (describing wear under random impact), the inverse mean curvature flow (used by Huisken and Ilmanen to prove the Penrose conjecture in relativity), and many others, including Ricci flow.

James Eells Jr. also first suggested I use analysis rather than topology to prove the Poincaré conjecture on the grounds that it is difficult for topologists to solve a problem where the hypothesis is the absence of topological invariants. And indeed as Lysander said, "Where the lion's skin will not reach, we must patch it out with the fox's." So I started thinking in the 1970s about how to use a parabolic flow to round out a general Riemannian metric to an Einstein metric by spreading the curvature evenly over the manifold. Now the Ricci curvature tensor is in a certain sense the Laplacian of the metric, so that zero Ricci curvature in the Riemannian case is really the elliptic equation for a harmonic metric, while in the Lorentzian case it is the hyperbolic wave equation for a metric, which is Einstein's theory of relativity. So it is only natural to guess that the parabolic heat equation for a metric is to evolve

It is often the case that the credit for a discovery goes not to the first person to stumble upon a thing, but to the first who sees how to use it. So the significance of my 1982 paper was that it proves a very nice result in geometry, that a three-dimensional manifold with a metric of positive Ricci curvature is always a quotient of the sphere. To prove this I developed a number of new techniques and estimates that opened up the field, in particular using the maximum principle on systems to obtain pinching estimates on curvature. Right afterward Shing-Tung Yau pointed out to me that the Ricci flow would pinch necks, performing a connected sum decomposition. I was very fortunate that shortly after I moved to University of California, San Diego, where I could collaborate with Shing-Tung Yau, Richard Schoen, and Gerhard Huisken, who coached me in the use of blow-ups to analyse singularities, making it possible to handle surgeries. It was also very important that Peter Li and Yau pointed out the fundamental importance of their seminal 1986 paper on Harnack inequalities, leading to my Harnack estimate for the Ricci flow, which is fundamental to the classification of singularities. And in 1997 I proved a surgical decomposition on four-manifolds with positive isotropic curvature. In 1999 I published a paper outlining a program for proving geometrisation of three-manifolds by performing surgeries on singularities and identifying incompressible hyperbolic pieces as time goes to infinity, only as I was still lacking control of the injectivity radius, I had to assume a curvature bound. This was supplied four years later in 2003 by the brilliant work of Grigory Perelman in his noncollapsing estimate, which led to his remarkable pointwise derivative estimates, allowing him to complete the program.

But the importance of Ricci flow is not confined to three dimensions. For example, we can hope to prove results on four-manifold topology, which are far more difficult. The Ricci flow on canonical Kähler manifolds is well advanced, based on the work of Huai-Dong Cao and Grigory Perelman, which might lead to a theorem in algebra. Ricci flow also is closely connected to the renormalisation group in string theory, and might be used to find stationary Lorentzian Einstein metrics in higher dimensions, giving applications to physics. And just recently we have the very lovely result of Richard Schoen and Simon Brendle using Ricci flow to prove the much stronger result in differential geometry of diffeomorphism rather than homeomorphism in the quarter-pinching theorem using the much weaker assumption of pointwise rather than global pinching. Now that many outstanding mathematicians are working on it, the story of the Ricci flow is just beginning. it by its Ricci curvature, which is the Ricci flow.

3.5. Brief Biographical Sketch.

Richard Streit Hamilton was born in Cincinnati, Ohio, in 1943. He graduated from Yale summa cum laude in 1963, and received his Ph.D. from Princeton in 1966, writing his thesis under Robert Gunning. He has taught at Cornell University, the University of California at San Diego, and UC Irvine, where he held a Bren Chair. He is currently Davies Professor of Mathematics at Columbia University in New York City, where he does research on geometric flows. In 1996 Richard Hamilton was awarded the Oswald Veblen Prize of the American Mathematical Society, and he is a Member of the American Academy of Arts and Sciences and the National Academy of Sciences.

4.1. The Shaw Prize.

The Shaw Prize was established under the vision and generosity of the late Mr Run Run Shaw in 2002. It is an international award to honour individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The award is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation.

The Shaw Prize consists of three disciplines, namely, Astronomy, Life Science and Medicine, and Mathematical Sciences and is presented annually by the Shaw Prize Foundation since 2004.

4.2. The 2011 Shaw Prize in Mathematical Sciences.

The 2011 Prize was awarded to Demetrios Christodoulou and 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.

4.3. Contribution of Richard Hamilton.

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.

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.

4.4. An Essay on the Prize.

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.

Hamilton's work is in geometric analysis. He has provided the theory and one of the most powerful tools to study the shapes (in other words the topology, where we allow deforming but no tearing) of low dimensional spaces. He introduced the Ricci Flow on the space geometries on a given shape as a means of deforming a given geometry into a regular one. An early success of his theory was his proof that any positively curved three dimensional space is the ordinary three dimensional sphere. This opened the door for him to attack the much more general problem of classifying all three dimensional shapes. This classification problem for two dimensional spaces has been understood for over 100 years. Thanks to the work of Thurston, one had, by 1980, a strong expectation of what the corresponding classification in three dimensions should be. It is formulated in Thurston's Geometrization Conjecture, which itself is a far reaching extension of the well-known Poincare Conjecture. During the past three decades, Hamilton developed a host of original techniques to study the long time behaviour of the geometry under his Ricci Flow. Among these is his technique of surgery allowing for a continuation of the flow should singularities form. In particular, his methods allowed him to classify four dimensional positively curved spaces. He has also pursued the application of his techniques in the study of related geometric flows, such as the curve shortening flow and the Gauss and the mean curvature flows. His ideas and techniques have been used by others to resolve a number of long standing problems in topology and geometry. In particular, Hamilton's program in three dimensions was completed in brilliant work of Perelman leading to a complete solution of the Conjectures of Poincare and Thurston. This classification of three dimensional shapes constitutes one of the finest achievements in mathematics.

4.5. Autobiography: Richard Hamilton.

I was born in Cincinnati, Ohio in 1943. My father was a surgeon; he had recently finished his residency at the Mayo Clinic when the Japanese bombed Pearl Harbor. He volunteered as a naval surgeon in the war, and was stationed in Portsmouth, England during my first two years of life, repairing wounded pilots. My mother lived with my grandmother, together with me and my brother Billie who was only one year older than I, until my father returned.

I attended Lotspeich elementary school, where I received an excellent education. In the fourth grade, out of curiosity I went to the library and took out a book on first year algebra. I taught myself first year algebra in a month, and went back for the second year algebra book. Then I attended Walnut Hills High School, one of the best in the nation, which was a public high school taking the brightest kids from the whole city. Skipping senior year, I went to Yale at 16, along with my brother. The most interesting classes were in ancient Greek, where we read the tragedies, the comedies and the great orators (in the original), and my philosophy classes with Brand Blanshard, a wonderful old scholar who had not changed his philosophical ideas since before WWI.

I did my graduate work at Princeton, writing my thesis at age 23 in 1966 on Riemann surfaces with Bob Gunning. During that time I was married (and divorced several years later) and my only son Andrew was born. My first academic position was at Cornell, where for several years I had the pleasure to work with Jim Eells Jr., who had just finished his ground-breaking paper with Joe Sampson on the harmonic map flow. This was the first example of using a nonlinear parabolic flow to solve an elliptic equation in geometry, and was my inspiration for creating the Ricci flow. My son Andrew would visit and we could snow ski in winter, and water ski and scuba dive in the summer.

By the mid seventies I had begun work on the Ricci flow, and published the first result in 1982 on the case of three dimensional manifolds with positive Ricci curvature. This got a lot of attention, and I was invited to visit the Mathematical Sciences Research Institute (MSRI), Berkeley, CA the first year it opened, along with S T Yau and Rick Schoen. The next year Yau, Rick and I all moved to the University of California at San Diego (UCSD), CA and Gerhard Huisken came to visit also. With all these excellent mathematicians around working on similar problems in geometric analysis, it was the ideal environment to develop the Ricci flow further. Yau had already pointed out that the Ricci flow could pinch along necks, and that this could provide the first step in the proof, by breaking the manifold into simpler pieces which could sustain constant curvature geometries.

Later, I moved back to the East coast to Columbia University, where I am now Davies Professor. In the Ricci flow I extended the results to four dimensions, derived the important Li-Yau type estimate for the curvature, developed the classification of singularities by ancient solutions and proved many properties of ancient solutions, showed that in three dimensions the curvature is pinched toward non-negative, developed the method of analytic surgery to bypass pinching necks and used it to classify four dimensional manifolds with positive isotropic curvature. I also showed how one could complete the proof of the three dimensional Poincare conjecture provided one could also do similar surgeries in three dimensions, and explained the program to Grigory Perelman, pointing out the importance of avoiding collapses. In a series of brilliant papers in 2003 Perelman did this with a very clever non-collapsing result derived from a novel Li-Yau type estimate for the adjoint heat equation.

Now I am looking at future possible developments in the Ricci flow, including possible applications to four dimensional topology, Kaehler geometry, and stationary solutions in Relativity.

4.6. Columbia News Report.

Richard Hamilton, Davies Professor of Mathematics, has won the 2011 Shaw Prize in Mathematical Sciences. The Shaw Prize is given annually in three areas: astronomy, life science and medicine, and mathematical sciences. This is the eighth year of the Shaw Prize; awardees will be honoured at a ceremony on Wednesday, Sept. 28.

The Shaw Prize is awarded to individuals who have made outstanding contributions and significant advances in their current field of study. The award is dedicated to "furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation," according to the Shaw Prize website. Professor Hamilton is receiving the award for his work with Ricci flow in Riemannian geometry. The $1 million award will be shared equally with fellow winner Demetrios Christodoulou, professor of mathematics and physics at the ETH, a science and technology university in Zurich, Switzerland.

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 geometrisation conjecture and the solution of the Poincaré conjecture.

Hamilton was awarded the Oswald Veblen Prize in Geometry in 1996 and the Clay Research Award in 2003. He was elected to the National Academy of Sciences in 1999 and to the American Academy of Arts and Sciences in 2003. He also received the AMS Leroy P Steele Prize for a Seminal Contribution to Research in 2009.