Richard S. Hamilton

Richard Streit Hamilton (born 10 January 1943) is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University.

He is known for contributions to geometric analysis and partial differential equations.

Hamilton is best known for foundational contributions to the theory of the Ricci flow and the development of a corresponding program of techniques and ideas for resolving the Poincaré conjecture and geometrization conjecture in the field of geometric topology.

Grigori Perelman built upon Hamilton's results to prove the conjectures, and was awarded a Millennium Prize for his work.

However, Perelman declined the award, regarding Hamilton's contribution as being equal to his own.

In 1956, John Nash resolved the problem of smoothly isometrically embedding Riemannian manifolds in Euclidean space.

The core of his proof was a novel "small perturbation" result, showing that if a Riemannian metric could be isometrically embedded in a certain way, then any nearby Riemannian metric could be isometrically embedded as well.

Such a result is highly reminiscent of an implicit function theorem, and many authors have attempted to put the logic of the proof into the setting of a general theorem.

Such theorems are now known as Nash–Moser theorems.

Hamilton received his B.A. in 1963 from Yale University and Ph.D. in 1966 from Princeton University.

Robert Gunning supervised his thesis.

He has taught at University of California, Irvine, University of California, San Diego, Cornell University, and Columbia University.

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 starting a research program that ultimately led to the proof, by Grigori Perelman, of William Thurston's geometrization conjecture and the Poincaré conjecture.

In 1982, Hamilton published his formulation of Nash's reasoning, casting the theorem into the setting of tame Fréchet spaces; Nash's fundamental use of restricting the Fourier transform to regularize functions was abstracted by Hamilton to the setting of exponentially decreasing sequences in Banach spaces.

His formulation has been widely quoted and used in the subsequent time.

He used it himself to prove a general existence and uniqueness theorem for geometric evolution equations; the standard implicit function theorem does not often apply in such settings due to the degeneracies introduced by invariance under the action of the diffeomorphism group.

In particular, the well-posedness of the Ricci flow follows from Hamilton's general result.

In 1986, Peter Li and Shing-Tung Yau discovered a new method for applying the maximum principle to control the solutions of the heat equation.

Among other results, they showed that if one has a positive solution u of the heat equation on a closed Riemannian manifold of nonnegative Ricci curvature, then one has

for any tangent vector v. Such inequalities, known as "differential Harnack inequalities" or "Li–Yau inequalities," are useful since they can be integrated along paths to compare the values of u at any two spacetime points.

They also directly give pointwise information about u, by taking v to be zero.

In 1993, Hamilton showed that the computations of Li and Yau could be extended to show that their differential Harnack inequality was a consequence of a stronger matrix inequality.

His result required the closed Riemannian manifold to have nonnegative sectional curvature and parallel Ricci tensor (such as the flat torus or the Fubini–Study metric on complex projective space), in the absence of which he obtained with a slightly weaker result.

Such matrix inequalities are sometimes known as Li–Yau–Hamilton inequalities.

Hamilton also discovered that the Li–Yau methodology could be adapted to the Ricci flow.

In the case of two-dimensional manifolds, he found that the computation of Li and Yau can be directly adapted to the scalar curvature along the Ricci flow.

In general dimensions, he showed that the Riemann curvature tensor satisfies a complicated inequality, formally analogous to his matrix extension of the Li–Yau inequality, in the case that the curvature operator is nonnegative.

As an immediate algebraic consequence, the scalar curvature satisfies an inequality which is almost identical to that of Li and Yau.

This fact is used extensively in Hamilton and Perelman's further study of Ricci flow.

Hamilton later adapted his Li–Yau estimate for the Ricci flow to the setting of the mean curvature flow, which is slightly simpler since the geometry is governed by the second fundamental form, which has a simpler structure than the Riemann curvature tensor.

Hamilton's theorem, which requires strict convexity, is naturally applicable to certain singularities of mean curvature flow due to the convexity estimates of Gerhard Huisken and Carlo Sinestrari.

For his work on the Ricci flow, 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 the American Academy of Arts and Sciences in 2003.

He also received the AMS Leroy P. Steele Prize for Seminal Contribution to Research in 2009, for his 1982 article Three-manifolds with positive Ricci curvature, in which he introduced and analyzed the Ricci flow.

In March 2010, the Clay Mathematics Institute, having listed the Poincaré conjecture among their Millennium Prize Problems, awarded Perelman with one million USD for his 2003 proof of the conjecture.

In July 2010, Perelman turned down the award and prize money, saying that he believed his contribution in proving the Poincaré conjecture was no greater than that of Hamilton, who had developed the program for the solution.

In June 2011, it was announced that the million-dollar Shaw Prize would be split equally between Hamilton and Demetrios Christodoulou "for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology."

In 2022, Hamilton joined University of Hawaiʻi at Mānoa as an adjunct professor.

As of 2022, Hamilton has been the author of forty-six research articles, around forty of which are in the field of geometric flows.