Richard Streit Hamilton (January 10, 1943 – September 29, 2024) was an American mathematician who served as the Davies Professor of Mathematics at Columbia University.

Hamilton is known for contributions to geometric analysis and partial differential equations, and particularly for developing the theory of Ricci flow. Hamilton introduced the Ricci flow in 1982 and, over the next decades, he developed a network of results and ideas for using it to prove the Poincaré conjecture and geometrization conjecture from the field of geometric topology.

Hamilton's work on the Ricci flow was recognized with an Oswald Veblen Prize, a Clay Research Award, a Leroy P. Steele Prize for Seminal Contribution to Research and a Shaw Prize. Grigori Perelman built upon Hamilton's research program, proving the Poincaré and geometrization conjectures in 2003. Perelman was awarded a Millennium Prize for resolving the Poincaré conjecture but declined it, regarding his contribution as no greater than Hamilton's.

Life

Hamilton was born in Cincinnati, Ohio, on January 10, 1943. He received his B.A. in 1963 from Yale University and PhD in 1966 from Princeton University. Robert Gunning supervised his thesis. In 2022, Hamilton additionally joined University of Hawaiʻi at Mānoa as an adjunct professor.

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 developing a research program aimed at the proof of William Thurston's geometrization conjecture, which contains the well-known Poincaré conjecture as a special case. In 2003, Grigori Perelman introduced new ideas into Hamilton's research program and completed a proof of the geometrization conjecture. 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.

In 1996, Hamilton was awarded the Oswald Veblen Prize in Geometry "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." In 2003 he received the Clay Research Award for "his introduction of the Ricci flow equation and his development of it into one of the most powerful tools in geometry and topology". He was elected to the National Academy of Sciences in 1999 and the American Academy of Arts and Sciences in 2003. In 2009, he received the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society for his "profoundly original" breakthrough article Three-manifolds with positive Ricci curvature, in which he first introduced and analyzed the Ricci flow. In 2011, the million-dollar Shaw Prize was 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 2024, he and Andrew Wiles received the Basic Science Lifetime Award in Mathematics at the International Congress of Basic Science.

Hamilton died at a hospital in Manhattan, New York City on September 29, 2024, at the age of 81.

Mathematical work

Hamilton was the author of forty-six research articles, the majority of which were in the field of geometric flows.

Harnack inequalities for heat equations

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. Their results take the form of asserting the nonnegativity of certain combinations of partial derivatives of a positive solution of the heat equation. These inequalities, known as differential Harnack inequalities or Li–Yau inequalities, are useful since they can be integrated along paths to compare the values of the solution at any two spacetime points. In 1993, Hamilton showed that the computations of Li and Yau could be extended, showing that their differential Harnack inequality was a consequence of a stronger inequality which asserts the nonnegativity of a matrix-valued function. His result required the stronger assumption that the underlying closed Riemannian manifold has nonnegative sectional curvature and parallel Ricci tensor (such as the flat torus or the Fubini–Study metric on complex projective space). Such matrix inequalities are sometimes known as Li–Yau–Hamilton inequalities.

Hamilton also found that Li and Yau's calculations were directly transferable to derive Harnack inequalities for the scalar curvature along a positively-curved Ricci flow on a two-dimensional closed manifold. With more effort, he was able to formulate an analogue of his matrix estimate in the case of the Riemann curvature tensor along a Ricci flow in general dimensions, provided that the curvature operator is nonnegative. As an important algebraic corollary, the values of the scalar curvature at two different spacetime points can be compared. 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. 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.

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. Although Dennis DeTurck gave a simpler proof in the particular case of the Ricci flow, Hamilton's result has been used for some other geometric flows for which DeTurck's method is inaccessible.

By a limiting procedure, Richard Schoen and Shing-Tung Yau used Hamilton's theorem to prove that any finite-energy map from a complete Riemannian manifold to a closed Riemannian manifold of nonpositive curvature can be deformed into a finite-energy harmonic map. With the use of such maps, they were able to derive a number of purely geometric corollaries, such as restrictions on the topology of precompact open subsets with simply-connected boundary inside complete Riemannian manifolds of nonnegative Ricci curvature.

Gage and Hamilton's main result is that, given any smoothly embedded circle in the plane which is convex, the corresponding mean curvature flow exists for a finite amount of time, and as the time approaches its maximal value, the curves asymptotically become increasingly small and circular. They made use of previous results of Gage, as well as a few special results for curves, such as Bonnesen's inequality. In combination with Gage and Hamilton's result, one has essentially a complete description of the asymptotic behavior of the mean curvature flow of embedded circles in the plane. This result, sometimes known as the Gage–Hamilton–Grayson theorem, says that the curve shortening flow gives systematic and geometrically defined means of deforming an arbitrary embedded circle in the Euclidean plane into a round circle.

Ricci flow

Hamilton extended the maximum principle for parabolic partial differential equations to the setting of symmetric 2-tensors which satisfy a parabolic partial differential equation. He also put this into the general setting of a parameter-dependent section of a vector bundle over a closed manifold which satisfies a heat equation, giving both strong and weak formulations. The essential outstanding issue was to carry out an analogous classification, for the small-scale geometry around high-curvature points on Ricci flows on three-dimensional manifolds, without any curvature restriction; the Hamilton–Ivey curvature estimate is the analogue to the condition of positive isotropic curvature. This was resolved by Grigori Perelman in his renowned canonical neighborhoods theorem. He also made contributions to the prescribed Ricci curvature problem.

Major publications

The collection

contains twelve of Hamilton's articles on Ricci flow, in addition to ten related articles by other authors.

References

  • Richard Hamilton – faculty bio at the homepage of the Department of Mathematics of Columbia University
  • Richard Hamilton – brief bio at the homepage of the Clay Mathematics Institute
  • 1996 Veblen Prize citation
  • Lecture by Hamilton on Ricci flow