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of positive or nonnegative curvature operator, and two-dimensional closed
Riemannian manifolds of nonpositive Euler characteristic or of positive curvature. In each case, after appropriate normalizations, the Ricci flow deforms the given Riemannian metric to one of constant curvature. This has strikingly simple immediate corollaries, such as the fact that any closed smooth 3-manifold which admits a Riemannian metric of positive curvature also admits a Riemannian metric of constant positive sectional curvature. Such results are notable in highly restricting the topology of such manifolds; the
953:'s compactness theory for Riemannian manifolds to give a compactness theorem for sequences of Ricci flows. 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. Hamilton used his maximum principles to prove that, for any Ricci flow on a closed three-dimensional manifold, the smallest value of the
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732:. 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
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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 and nonnegative Ricci curvature, four-dimensional closed Riemannian manifolds
426:, 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.
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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
HamiltonâIvey curvature estimate is the analogue to the
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is small compared to its largest value. This is known as the
HamiltonâIvey 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
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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.
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of positive curvature are largely understood. There are other corollaries, such as the fact that the topological space of
Riemannian metrics of positive Ricci curvature on a closed smooth 3-manifold is path-connected. These "convergence theorems" of Hamilton have been extended by later authors, in
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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
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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. By proving extension of Eells and Sampson's vanishing theorem in various geometric settings, they were able to draw
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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. For Ricci flows with initial data in this class, he was able to classify the possibilities for the small-scale geometry
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The modern understanding of the results of GageâHamilton 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
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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
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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.
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as applied to the distance between two points on a curve, they proved that if the initial immersion is an embedding, then all future immersions in the mean curvature flow are embeddings as well. Furthermore, convexity of the curves is preserved into the future.
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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
896:, the corresponding mean curvature flow eventually becomes convex. 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
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688:, 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
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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
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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
1048:. American Mathematical Society summer institute held at the University of California (Berkeley, CA) July 1â26, 1968. Proceedings of Symposia in Pure Mathematics. Vol. 16. Providence, RI:
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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, 333â341.
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by the compactness theory automatically has nonnegative curvature. As such, Hamilton's Harnack inequality is applicable to the limiting Ricci flow. These methods were extended by
437:"for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology."
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Gerhard Huisken and Carlo Sinestrari. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations 8 (1999), no. 1, 1â14.
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Blair, David E. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. BirkhÀuser Boston, Ltd., Boston, MA, 2010.
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Gerhard Huisken and Carlo Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183 (1999), no. 1, 45â70.
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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.
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cannot be trivially carried out, due to the fact that size of the gradient at the boundary is not automatically controlled by the boundary conditions.
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713:, 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
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Matthew A. Grayson. The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26 (1987), no. 2, 285â314.
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904:. It is somewhat surprising that there is such a systematic and geometrically defined means of deforming an arbitrary loop in
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satisfies a complicated inequality, formally analogous to his matrix extension of the LiâYau inequality, in the case that the
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962:, who due to his "noncollapsing theorem" was able to apply Hamilton's compactness theory in a number of extended contexts.
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Peter Li and Shing-Tung Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), no. 3-4, 153â201.
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Besse, Arthur L. Einstein manifolds. Reprint of the 1987 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008.
1967:
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applied Hamilton's NashâMoser theorem and well-posedness result for parabolic equations to prove the well-posedness for
784:. The analytic nature of the problem is more delicate in this setting, since Eells and Sampson's key application of the
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661:. In the case of two-dimensional manifolds, he found that the computation of Li and Yau can be directly adapted to the
650:), in the absence of which he obtained with a slightly weaker result. Such matrix inequalities are sometimes known as
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inequalities" or "LiâYau inequalities," are useful since they can be integrated along paths to compare the values of
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1692:. Surveys in Differential Geometry. Vol. II. Cambridge, MA: International Press. pp. 7â136. Reprinted in
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As of 2022, Hamilton has been the author of forty-six research articles, around forty of which are in the field of
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for his work. However, Perelman declined the award, regarding Hamilton's contribution as being equal to his own.
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gave a simpler proof in the particular case of the Ricci flow, Hamilton's result has been used for some other
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contains twelve of Hamilton's articles on Ricci flow, in addition to ten related articles by other authors.
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to regularize functions was abstracted by Hamilton to the setting of exponentially decreasing sequences in
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By taking limits of Hamilton's solutions of the boundary value problem for increasingly large boundaries,
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In 1982, Hamilton published his formulation of Nash's reasoning, casting the theorem into the setting of
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Ben Andrews. Evolving convex curves. Calc. Var. Partial Differential Equations 7 (1998), no. 4, 315â371.
1282:. Series in Geometry and Topology. Vol. 37. Somerville, MA: International Press. pp. 521â524.
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into any group which is the fundamental group of a closed Riemannian manifold of nonpositive curvature.
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Mathematics and general relativity: proceedings of a summer research conference held June 22â28, 1986
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Proceedings of the conference on geometry and topology held at Harvard University, April 23â25, 1993
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into a smooth Riemannian manifold. Then, they specialized to the case of immersions of the circle
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with smooth and simply-connected boundary, there cannot exist a nontrivial homomorphism from the
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John Nash. The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63 (1956), 20â63.
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In 1987, Matthew Grayson proved a complementary result, showing that for any smooth embedding
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604:{\displaystyle {\frac {\partial u}{\partial t}}+{\frac {u}{2t}}+2du(v)+u|v|_{g}^{2}\geq 0}
273:(born 10 January 1943) is an American mathematician who serves as the Davies Professor of
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and the development of a corresponding program of techniques and ideas for resolving the
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This article is about the American mathematician. For other people named similarly, see
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Hamilton, Richard S. (1995). "A compactness property for solutions of the Ricci flow".
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1919:. Series in Geometry and Topology. Vol. 37. Somerville, MA: International Press.
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2153:â faculty bio at the homepage of the Department of Mathematics of Columbia University
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1036:; Hamilton, Richard S. (1970). "A fixed point theorem for holomorphic mappings". In
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Hamilton later adapted his LiâYau estimate for the Ricci flow to the setting of the
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Hamilton, Richard S. (2003). "Three-orbifolds with positive Ricci curvature". In
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at any two spacetime points. They also directly give pointwise information about
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Hamilton also discovered that the LiâYau methodology could be adapted to the
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289:. Hamilton is best known for foundational contributions to the theory of the
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946:, which had been a major conjecture in Riemannian geometry since the 1960s.
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which satisfies a heat equation, giving both strong and weak formulations.
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Ricci flow with surgery in four dimensions for positive isotropic curvature
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for this flow, proving an analogous result to Eells and Sampson's for the
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built upon Hamilton's results to prove the conjectures, and was awarded a
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996:. In unpublished lecture notes from the 1980s, Hamilton introduced the
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480:. Among other results, they showed that if one has a positive solution
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and starting a research program that ultimately led to the proof, by
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Gage and Hamilton's main result is that, given any smooth embedding
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Hamilton's mathematical contributions are primarily in the field of
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1989:
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to a closed manifold of nonpositive curvature can be deformed to a
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684:, which is slightly simpler since the geometry is governed by the
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condition of positive isotropic curvature. This was resolved by
665:
along the Ricci flow. In general dimensions, he showed that the
2129:
1680:(1995). "The formation of singularities in the Ricci flow". In
227:
1435:
Hamilton, Richard S. (1988). "The Ricci flow on surfaces". In
1854:"Non-singular solutions of the Ricci flow on three-manifolds"
673:
is nonnegative. As an immediate algebraic consequence, the
1443:. Contemporary Mathematics. Vol. 71. Providence, RI:
1000:
and proved its long-time existence. In collaboration with
459:
409:
Leroy P. Steele Prize for Seminal Contribution to Research
3113:
Members of the United States National Academy of Sciences
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and parallel Ricci tensor (such as the flat torus or the
391:
For his work on the Ricci flow, Hamilton was awarded the
102:
2011:$ 500,000 for mathematician who laid Poincaré groundwork
429:
In June 2011, it was announced that the million-dollar
415:, in which he introduced and analyzed the Ricci flow.
504:
2033:
World-renowned mathematician joins UH MÄnoa faculty.
422:, having listed the Poincaré conjecture among their
1796:"Four-manifolds with positive isotropic curvature"
988:In one of his earliest works, Hamilton proved the
603:
1503:"A matrix Harnack estimate for the heat equation"
1377:"Four-manifolds with positive curvature operator"
1321:"The heat equation shrinking convex plane curves"
816:is a complete Riemannian manifold of nonnegative
772:. In 1975, Hamilton considered the corresponding
740:follows from Hamilton's general result. Although
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1168:"The inverse function theorem of Nash and Moser"
804:striking geometric conclusions, such as that if
2035:University of Hawaiʻi News (February 28, 2022).
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1224:"Three-manifolds with positive Ricci curvature"
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1024:
1740:"Harnack estimate for the mean curvature flow"
368:. He is best known for having discovered the
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1173:Bulletin of the American Mathematical Society
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413:Three-manifolds with positive Ricci curvature
1315:
748:for which DeTurck's method is inaccessible.
724:; Nash's fundamental use of restricting the
618:. Such inequalities, known as "differential
433:would be split equally between Hamilton and
3123:University of California, San Diego faculty
736:. In particular, the well-posedness of the
233:Variation of structure on Riemann surfaces
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760:and Joseph Sampson initiated the study of
709:Such a result is highly reminiscent of an
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1557:"The Harnack estimate for the Ricci flow"
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858:into the two-dimensional Euclidean space
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472:discovered a new method for applying the
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2179:. The Shaw Prize Foundation. 2011-09-28.
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1106:Harmonic Maps of Manifolds with Boundary
1103:
900:. This result is sometimes known as the
341:supervised his thesis. He has taught at
144:for Ricci flow and other geometric flows
2022:Shaw Prize in Mathematical Studies 2011
1858:Communications in Analysis and Geometry
1800:Communications in Analysis and Geometry
1507:Communications in Analysis and Geometry
460:Harnack inequalities for heat equations
148:Maximum principle for parabolic systems
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835:
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2187:
2177:"Autobiography of Richard S Hamilton"
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405:American Academy of Arts and Sciences
3078:21st-century American mathematicians
3073:20th-century American mathematicians
1012:. He also made contributions to the
862:, which is the simplest context for
447:
2159:â brief bio at the homepage of the
980:geometrization conjecture in 2003.
820:, then for any precompact open set
347:University of California, San Diego
281:. He is known for contributions to
213:University of California, San Diego
13:
1112:. Vol. 471. BerlinâNew York:
1014:prescribed Ricci curvature problem
990:EarleâHamilton fixed point theorem
942:the 2000s, to give a proof of the
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508:
407:in 2003. He also received the AMS
130:EarleâHamilton fixed-point theorem
16:American mathematician (born 1943)
14:
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2171:Lecture by Hamilton on Ricci flow
2123:
1986:"Maths genius declines top prize"
1447:. pp. 237â262. Reprinted in
2135:Richard Hamilton (mathematician)
2128:
1745:Journal of Differential Geometry
1562:Journal of Differential Geometry
1382:Journal of Differential Geometry
1326:Journal of Differential Geometry
1229:Journal of Differential Geometry
476:to control the solutions of the
343:University of California, Irvine
2860:Oswald Veblen Prize in Geometry
2110:
2101:
2092:
2083:
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1736:
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1618:American Journal of Mathematics
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1314:
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1187:10.1090/s0273-0979-1982-15004-2
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399:in 2003. He was elected to the
393:Oswald Veblen Prize in Geometry
3083:Clay Research Award recipients
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2047:
2038:
2026:
2015:
2004:
1978:
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1917:Collected papers on Ricci flow
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1373:
1280:Collected papers on Ricci flow
1102:
580:
571:
561:
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442:University of HawaiÊ»i at MÄnoa
411:in 2009, for his 1982 article
287:partial differential equations
221:University of HawaiÊ»i at MÄnoa
1:
2146:Mathematics Genealogy Project
1953:
1852:Hamilton, Richard S. (1999).
1794:Hamilton, Richard S. (1997).
1738:Hamilton, Richard S. (1995).
1555:Hamilton, Richard S. (1993).
1501:Hamilton, Richard S. (1993).
1445:American Mathematical Society
1375:Hamilton, Richard S. (1986).
1222:Hamilton, Richard S. (1982).
1166:Hamilton, Richard S. (1982).
1104:Hamilton, Richard S. (1975).
1050:American Mathematical Society
983:
944:differentiable sphere theorem
915:
902:GageâHamiltonâGrayson theorem
136:GageâHamiltonâGrayson theorem
1110:Lecture Notes in Mathematics
652:LiâYauâHamilton inequalities
401:National Academy of Sciences
316:
7:
3118:Princeton University alumni
3088:Columbia University faculty
1864:(4): 695â729. Reprinted in
1625:(3): 545â572. Reprinted in
1569:(1): 225â243. Reprinted in
1389:(2): 153â179. Reprinted in
1236:(2): 255â306. Reprinted in
1004:, Hamilton studied certain
949:In 1995, Hamilton extended
10:
3144:
3093:Cornell University faculty
2872:Christos Papakyriakopoulos
2166:1996 Veblen Prize citation
2161:Clay Mathematics Institute
1319:; Hamilton, R. S. (1986).
1008:for Riemannian metrics in
484:of the heat equation on a
420:Clay Mathematics Institute
125:and harmonic map heat flow
18:
3103:Educators from Cincinnati
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2642:
2423:
2222:
1968:"The Poincaré Conjecture"
1875:10.4310/CAG.1999.v7.n4.a2
1846:
1817:10.4310/CAG.1997.v5.n1.a1
1788:
1732:
1703:10.4310/SDG.1993.v2.n1.a2
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1520:10.4310/CAG.1993.v1.n1.a6
1495:
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1216:
1160:
1098:
1027:
972:geometrization conjecture
790:parabolic Bochner formula
711:implicit function theorem
444:as an adjunct professor.
440:In 2022, Hamilton joined
424:Millennium Prize Problems
382:geometrization conjecture
299:geometrization conjecture
264:
254:
242:
226:
204:
194:
187:
161:
114:Convergence theorems for
109:
85:
75:
47:
35:
28:
3108:Mathematicians from Ohio
1911:; Chow, B.; Chu, S. C.;
1806:(1): 1â92. Reprinted in
1274:; Chow, B.; Chu, S. C.;
667:Riemann curvature tensor
648:complex projective space
2710:Demetrios Christodoulou
2551:Everett Peter Greenberg
686:second fundamental form
614:for any tangent vector
435:Demetrios Christodoulou
271:Richard Streit Hamilton
3128:Yale University alumni
3098:Differential geometers
1759:10.4310/jdg/1214456010
1580:10.4310/jdg/1214453430
1400:10.4310/jdg/1214440433
1340:10.4310/jdg/1214439902
1247:10.4310/jdg/1214436922
992:in collaboration with
920:Hamilton extended the
840:In 1986, Hamilton and
774:boundary value problem
762:harmonic map heat flow
752:Harmonic map heat flow
692:and Carlo Sinestrari.
605:
364:and more specifically
321:Hamilton received his
121:Dirichlet problem for
3024:Fernando CodĂĄ Marques
2414:Shrinivas R. Kulkarni
2137:at Wikimedia Commons
908:into a round circle.
884:Bonnesen's inequality
864:curve shortening flow
606:
362:differential geometry
176:Leroy P. Steele Prize
2473:Keith H. S. Campbell
2433:Stanley Norman Cohen
1678:Hamilton, Richard S.
1006:variational problems
734:diffeomorphism group
502:
335:Princeton University
99:Princeton University
2952:Richard S. Hamilton
2918:James Harris Simons
2776:Alexander Beilinson
2714:Richard S. Hamilton
2384:Chryssa Kouveliotou
955:sectional curvature
846:mean curvature flow
836:Mean curvature flow
778:Dirichlet condition
722:tame Fréchet spaces
715:NashâMoser theorems
682:mean curvature flow
644:FubiniâStudy metric
640:sectional curvature
594:
489:Riemannian manifold
418:In March 2010, the
397:Clay Research Award
386:Poincaré conjecture
355:Columbia University
295:Poincaré conjecture
279:Columbia University
217:Columbia University
171:Clay Research Award
142:LiâYau inequalities
2970:Michael J. Hopkins
2858:Recipients of the
2786:Jean-Michel Bismut
2513:Franz-Ulrich Hartl
2336:William J. Borucki
1437:Isenberg, James A.
1122:10.1007/BFb0087227
1052:. pp. 61â65.
1038:Chern, Shiing-Shen
1034:Earle, Clifford J.
1020:Major publications
696:NashâMoser theorem
671:curvature operator
601:
578:
366:geometric analysis
351:Cornell University
303:geometric topology
283:geometric analysis
209:Cornell University
152:NashâMoser theorem
3050:
3049:
3004:William Minicozzi
2825:
2824:
2806:Vladimir Drinfeld
2652:Shiing-Shen Chern
2609:Paul A. Negulescu
2517:Arthur L. Horwich
2390:Lennart Lindegren
2322:Daniel Eisenstein
2133:Media related to
1900:
1899:
1866:Cao et al. (2003)
1842:
1841:
1808:Cao et al. (2003)
1784:
1783:
1728:
1727:
1694:Cao et al. (2003)
1667:
1666:
1627:Cao et al. (2003)
1605:
1604:
1571:Cao et al. (2003)
1545:
1544:
1491:
1490:
1449:Cao et al. (2003)
1425:
1424:
1391:Cao et al. (2003)
1365:
1364:
1306:
1305:
1238:Cao et al. (2003)
1212:
1211:
1156:
1155:
1131:978-3-540-07185-3
1094:
1093:
1059:10.1090/pspum/016
1002:Shiing-Shen Chern
922:maximum principle
868:maximum principle
826:fundamental group
786:maximum principle
782:Neumann condition
726:Fourier transform
541:
523:
474:maximum principle
448:Mathematical work
268:
267:
255:Doctoral students
189:Scientific career
3135:
2982:Peter Kronheimer
2966:Yakov Eliashberg
2936:Michael Freedman
2912:William Thurston
2852:
2845:
2838:
2829:
2828:
2770:Michel Talagrand
2720:Maxim Kontsevich
2674:Robert Langlands
2613:Michael J. Welsh
2577:Mary-Claire King
2531:Michael W. Young
2503:Ruslan Medzhitov
2487:Jeffrey Friedman
2463:Robert Lefkowitz
2451:Michael Berridge
2408:Maura McLaughlin
2394:Michael Perryman
2208:
2201:
2194:
2185:
2184:
2180:
2157:Richard Hamilton
2151:Richard Hamilton
2142:Richard Hamilton
2132:
2117:
2114:
2108:
2105:
2099:
2096:
2090:
2087:
2081:
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2051:
2045:
2042:
2036:
2030:
2024:
2019:
2013:
2008:
2002:
2001:
1999:
1997:
1992:. 22 August 2006
1982:
1976:
1975:
1970:. Archived from
1964:
1946:
1903:The collection
1895:
1877:
1844:
1837:
1819:
1786:
1779:
1761:
1730:
1723:
1705:
1669:
1662:
1607:
1600:
1582:
1547:
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1522:
1493:
1486:
1457:10.1090/conm/071
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1420:
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1301:
1267:
1249:
1214:
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1158:
1151:
1096:
1089:
1061:
1025:
1010:contact geometry
977:Grigori Perelman
960:Grigori Perelman
912:curvature flow.
907:
899:
895:
880:
861:
857:
831:
823:
815:
675:scalar curvature
663:scalar curvature
633:
629:
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607:
602:
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588:
583:
574:
542:
540:
529:
524:
522:
514:
506:
483:
403:in 1999 and the
395:in 1996 and the
378:William Thurston
374:Grigori Perelman
311:Millennium Prize
307:Grigori Perelman
301:in the field of
244:Doctoral advisor
238:
65:Cincinnati, Ohio
61:
58:January 10, 1943
57:
55:
42:Hamilton in 1982
40:
30:Richard Hamilton
26:
25:
21:Richard Hamilton
3143:
3142:
3138:
3137:
3136:
3134:
3133:
3132:
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3052:
3051:
3046:
3038:Simon Donaldson
2946:Clifford Taubes
2906:Dennis Sullivan
2862:
2856:
2826:
2821:
2800:Ehud Hrushovski
2764:Luis Caffarelli
2698:Clifford Taubes
2694:Simon Donaldson
2684:Vladimir Arnold
2644:
2638:
2629:Stuart H. Orkin
2589:Gero Miesenböck
2527:Michael Rosbash
2523:Jeffrey C. Hall
2483:Douglas Coleman
2477:Shinya Yamanaka
2425:
2419:
2374:Roger Blandford
2368:Edward C. Stone
2362:Jean-Loup Puget
2330:John A. Peacock
2302:David C. Jewitt
2278:Charles Bennett
2266:Reinhard Genzel
2260:Peter Goldreich
2246:Saul Perlmutter
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2212:
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2043:
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2020:
2016:
2009:
2005:
1995:
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1984:
1983:
1979:
1966:
1965:
1961:
1956:
1927:
1915:, eds. (2003).
1901:
1896:
1838:
1780:
1724:
1663:
1635:10.2307/2375080
1601:
1541:
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1467:
1421:
1361:
1302:
1290:
1268:
1208:
1152:
1132:
1114:Springer-Verlag
1090:
1070:
1046:Global analysis
1022:
986:
931:closed manifold
918:
905:
897:
890:
875:
859:
853:
850:closed manifold
838:
829:
821:
818:Ricci curvature
805:
766:closed manifold
754:
746:geometric flows
698:
690:Gerhard Huisken
631:
627:
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615:
589:
584:
579:
570:
533:
528:
515:
507:
505:
503:
500:
499:
495:, then one has
493:Ricci curvature
491:of nonnegative
481:
462:
454:geometric flows
450:
327:Yale University
319:
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90:Yale University
86:Alma mater
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2924:Mikhail Gromov
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2732:George Lusztig
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2678:Richard Taylor
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2593:Peter Hegemann
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2580:
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2567:Ian R. Gibbons
2564:
2554:
2547:Bonnie Bassler
2544:
2537:Kazutoshi Mori
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2499:Jules Hoffmann
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2404:Duncan Lorimer
2400:Matthew Bailes
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2380:Victoria Kaspi
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1176:. New Series.
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1042:Smale, Stephen
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994:Clifford Earle
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332:
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69:United States
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2962:Jeff Cheeger
2951:
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2890:Morton Brown
2816:Peter Sarnak
2790:Jeff Cheeger
2754:JĂĄnos KollĂĄr
2726:David Donoho
2713:
2658:Andrew Wiles
2643:Mathematical
2603:Scott D. Emr
2541:Peter Walter
2493:David Julius
2445:Richard Doll
2441:Yuet-Wai Kan
2426:and medicine
2424:Life science
2350:Rainer Weiss
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166:Veblen Prize
3063:1943 births
3028:André Neves
3018:Daniel Wise
3008:Paul Seidel
2976:David Gabai
2894:Barry Mazur
2623:Eva Nogales
2597:Georg Nagel
2583:Maria Jasin
2571:Ronald Vale
2561:Huda Zoghbi
2557:Adrian Bird
2356:Simon White
2230:Jim Peebles
998:Yamabe flow
939:space forms
758:James Eells
275:Mathematics
199:Mathematics
76:Nationality
3057:Categories
2878:Raoul Bott
2668:Wentsun Wu
2469:Ian Wilmut
2346:Kip Thorne
2326:Shaun Cole
2282:Lyman Page
2250:Adam Riess
2215:Shaw Prize
1954:References
1943:1108.53002
1913:Yau, S. T.
1909:Cao, H. D.
1892:0939.53024
1834:0892.53018
1776:0827.53006
1720:0867.53030
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1264:0504.53034
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1148:0308.35003
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984:Other work
916:Ricci flow
738:Ricci flow
659:Ricci flow
431:Shaw Prize
370:Ricci flow
291:Ricci flow
181:Shaw Prize
116:Ricci flow
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2956:Gang Tian
2796:Noga Alon
2272:Frank Shu
2223:Astronomy
2217:laureates
756:In 1964,
702:John Nash
700:In 1956,
596:≥
517:∂
509:∂
464:In 1986,
317:Biography
259:Martin Lo
3042:Song Sun
3014:Ian Agol
2306:Jane Luu
1990:BBC News
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1317:Gage, M.
1278:(eds.).
1044:(eds.).
466:Peter Li
384:and the
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1996:16 June
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706:problem
620:Harnack
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2812:(2023)
2802:(2022)
2792:(2021)
2782:(2020)
2772:(2019)
2766:(2018)
2760:(2017)
2750:(2016)
2744:(2015)
2734:(2014)
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2579:(2018)
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2519:(2012)
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2479:(2008)
2465:(2007)
2459:(2006)
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2447:(2004)
2416:(2024)
2410:(2023)
2396:(2022)
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2376:(2020)
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2364:(2018)
2358:(2017)
2352:(2016)
2338:(2015)
2332:(2014)
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2308:(2012)
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228:Thesis
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168:(1996)
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3022:2016
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