Richard S. Hamilton
Richard Hamilton | |
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Robert Gunning | |
Doctoral students | Martin Lo |
Richard Streit Hamilton (born 10 January 1943) is an American mathematician who serves as the Davies Professor of
Biography
Hamilton received his
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.
For his work on the Ricci flow, Hamilton was awarded the
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.[1] 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.[2]
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."[3][4]
In 2022, Hamilton joined University of Hawaiʻi at Mānoa as an adjunct professor.[5]
Mathematical work
As of 2022, Hamilton has been the author of forty-six research articles, around forty of which are 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.[6] 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.[H93a] 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.[H88] 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.[H93b] 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.[H95c] 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.[7][8]
Nash–Moser theorem
In 1956,
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
Harmonic map heat flow
In 1964,
By taking limits of Hamilton's solutions of the boundary value problem for increasingly large boundaries, Richard Schoen and Shing-Tung Yau observed that a finite-energy map from a complete Riemannian manifold to a closed Riemannian manifold of nonpositive curvature could be deformed into a harmonic map of finite energy.[10] By proving extension of Eells and Sampson's vanishing theorem in various geometric settings, they were able to draw striking geometric conclusions, such as that if (M, g) is a complete Riemannian manifold of nonnegative Ricci curvature, then for any precompact open set D with smooth and simply-connected boundary, there cannot exist a nontrivial homomorphism from the fundamental group of D into any group which is the fundamental group of a closed Riemannian manifold of nonpositive curvature.
Mean curvature flow
In 1986, Hamilton and
Gage and Hamilton's main result is that, given any smooth embedding S1 → ℝ2 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.[GH86] They made use of previous results of Gage, as well as a few special results for curves, such as Bonnesen's inequality.
In 1987, Matthew Grayson proved a complementary result, showing that for any smooth embedding S1 → ℝ2, the corresponding mean curvature flow eventually becomes convex.
The modern understanding of the results of Gage–Hamilton and of Grayson usually treat both settings at once, without the need for showing that arbitrary curves become convex and separately studying the behavior of convex curves. Their results can also be extended to settings other than the mean curvature flow.[12]
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.[H82b] 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.[H86]
Partly due to these foundational technical developments, Hamilton was able to give an essentially complete understanding of how Ricci flow behaves on three-dimensional closed Riemannian manifolds of positive Ricci curvature
In 1995, Hamilton extended Jeff Cheeger's compactness theory for Riemannian manifolds to give a compactness theorem for sequences of Ricci flows.[H95a] Given a Ricci flow on a closed manifold with a finite-time singularity, Hamilton developed methods of rescaling around the singularity to produce a sequence of Ricci flows; the compactness theory ensures the existence of a limiting Ricci flow, which models the small-scale geometry of a Ricci flow around a singular point.[H95b] Hamilton used his maximum principles to prove that, for any Ricci flow on a closed three-dimensional manifold, the smallest value of the sectional curvature is small compared to its largest value. This is known as the Hamilton–Ivey estimate; it is extremely significant as a curvature inequality which holds with no conditional assumptions beyond three-dimensionality. An important consequence is that, in three dimensions, a limiting Ricci flow as produced by the compactness theory automatically has nonnegative curvature.[H95b] As such, Hamilton's Harnack inequality is applicable to the limiting Ricci flow. These methods were extended by Grigori Perelman, who due to his "noncollapsing theorem" was able to apply Hamilton's compactness theory in a number of extended contexts.
In 1997, Hamilton was able to combine the methods he had developed to define "Ricci flow with surgery" for four-dimensional Riemannian manifolds of positive isotropic curvature.[H97] For Ricci flows with initial data in this class, he was able to classify the possibilities for the small-scale geometry around points with large curvature, and hence to systematically modify the geometry so as to continue the Ricci flow. As a consequence, he obtained a result which classifies the smooth four-dimensional manifolds which support Riemannian metrics of positive isotropic curvature. Shing-Tung Yau has described this article as the "most important event" in geometric analysis in the period after 1993, marking it as the point at which it became clear that it could be possible to prove Thurston's geometrization conjecture by Ricci flow methods. 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." Building off of this result, Perelman modified the form of Hamilton's surgery procedure to define a "Ricci flow with surgery" given an arbitrary smooth Riemannian metric on a closed three-dimensional manifold. This led to the resolution of the geometrization conjecture in 2003.
Other work
In one of his earliest works, Hamilton proved the
Major publications
EH70. | Zbl 0205.14702 . |
H75. | Hamilton, Richard S. (1975). Harmonic Maps of Manifolds with Boundary.
Zbl 0308.35003 . |
H82a. | Hamilton, Richard S. (1982). "The inverse function theorem of Nash and Moser".
Zbl 0499.58003 . |
H82b. | Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature".
MR 2143256 . |
GH86. | Zbl 0621.53001 . |
H86. | Hamilton, Richard S. (1986). "Four-manifolds with positive curvature operator".
Zbl 0628.53042 . |
H88. | Hamilton, Richard S. (1988). "The Ricci flow on surfaces". In
Zbl 0663.53031 . |
H93a. | Hamilton, Richard S. (1993). "A matrix Harnack estimate for the heat equation". Communications in Analysis and Geometry. 1 (1): 113–126.
Zbl 0799.53048 . |
H93b. | Hamilton, Richard S. (1993). "The Harnack estimate for the Ricci flow".
Zbl 0804.53023 . |
H95a. | Hamilton, Richard S. (1995). "A compactness property for solutions of the Ricci flow".
Zbl 0840.53029 . |
H95b. | Hamilton, Richard S. (1995). "The formation of singularities in the Ricci flow". In
Zbl 0867.53030 . |
H95c. | Hamilton, Richard S. (1995). "Harnack estimate for the mean curvature flow".
Zbl 0827.53006 . |
H97. | Hamilton, Richard S. (1997). "Four-manifolds with positive isotropic curvature". Communications in Analysis and Geometry. 5 (1): 1–92. Reprinted in
Zbl 0892.53018 . |
H99. | Hamilton, Richard S. (1999). "Non-singular solutions of the Ricci flow on three-manifolds". Communications in Analysis and Geometry. 7 (4): 695–729. Reprinted in
Zbl 0939.53024 . |
The collection
- Zbl 1108.53002.
contains twelve of Hamilton's articles on Ricci flow, in addition to ten related articles by other authors.
References
- ^ "The Poincaré Conjecture". Archived from the original on 2013-07-27.
- ^ "Maths genius declines top prize". BBC News. 22 August 2006. Retrieved 16 June 2011.
- ^ $500,000 for mathematician who laid Poincaré groundwork
- ^ Shaw Prize in Mathematical Studies 2011
- ^ World-renowned mathematician joins UH Mānoa faculty. University of Hawaiʻi News (February 28, 2022).
- ^ Peter Li and Shing-Tung Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), no. 3-4, 153–201.
- ^ Gerhard Huisken and Carlo Sinestrari. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations 8 (1999), no. 1, 1–14.
- ^ Gerhard Huisken and Carlo Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183 (1999), no. 1, 45–70.
- ^ John Nash. The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63 (1956), 20–63.
- ^ Richard Schoen and Shing Tung Yau. Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51 (1976), no. 3, 333–341.
- ^ Matthew A. Grayson. The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26 (1987), no. 2, 285–314.
- ^ Ben Andrews. Evolving convex curves. Calc. Var. Partial Differential Equations 7 (1998), no. 4, 315–371.
- ^ Blair, David E. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhäuser Boston, Ltd., Boston, MA, 2010.
- ^ Besse, Arthur L. Einstein manifolds. Reprint of the 1987 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008.
External links
Media related to Richard Hamilton (mathematician) at Wikimedia Commons
- Richard Hamilton at the Mathematics Genealogy Project
- 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
- "Autobiography of Richard S Hamilton". The Shaw Prize Foundation. 2011-09-28.