495:, are two-dimensional. The surface of a ball has trivial fundamental group, meaning that any loop drawn on the surface can be continuously deformed to a single point. By contrast, the surface of a torus has nontrivial fundamental group, as there are loops on the surface which cannot be so deformed. Both are topological manifolds which are closed (meaning that they have no boundary and take up a finite region of space) and connected (meaning that they consist of a single piece). Two closed manifolds are said to be homeomorphic when it is possible for the points of one to be reallocated to the other in a continuous way. Because the (non)triviality of the fundamental group is known to be invariant under homeomorphism, it follows that the two-dimensional sphere and torus are not homeomorphic.
1332:
1396:, which describes the way heat flows in a solid. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. A singularity in a manifold is a place where it is not differentiable: like a corner or a cusp or a pinching. The Ricci flow was only defined for smooth differentiable manifolds. Hamilton used the Ricci flow to prove that some compact manifolds were
1385:. Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery"), causing the separate pieces to form into ball-like shapes. Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed, and establishing that the surgery need not be repeated infinitely many times.
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that certain kinds of singularities do not occur. Perelman discovered the singularities were all very simple: consider that a cylinder is formed by 'stretching' a circle along a line in another dimension, repeating that process with spheres instead of circles essentially gives the form of the singularities. Perelman proved this using something called the "Reduced Volume", which is closely related to an
593:, which is a closed connected three-dimensional manifold which has the homology of the sphere but whose fundamental group has 120 elements. This example made it clear that homology is not powerful enough to characterize the topology of a manifold. In the closing remarks of the fifth supplement, Poincaré modified his erroneous theorem to use the fundamental group instead of homology:
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the torus, as above.) This analogue is known to be true via the classification of closed and connected two-dimensional topological manifolds, which was understood in various forms since the 1860s. In higher dimensions, the closed and connected topological manifolds do not have a straightforward classification, precluding an easy resolution of the
Poincaré conjecture.
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he is left with a collection of round three-dimensional spheres. Then, he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape, and sees that, despite all the initial confusion, the manifold was, in fact, homeomorphic to a sphere.
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If, instead, one only has an arbitrary
Riemannian metric, the Ricci flow equations must lead to more complicated singularities. Perelman's major achievement was to show that, if one takes a certain perspective, if they appear in finite time, these singularities can only look like shrinking spheres or
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smoothly converge to one of constant positive curvature. According to classical
Riemannian geometry, the only simply-connected compact manifold which can support a Riemannian metric of constant positive curvature is the sphere. So, in effect, Hamilton showed a special case of the Poincaré conjecture:
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on the unknown simply connected closed 3-manifold. The basic idea is to try to "improve" this metric; for example, if the metric can be improved enough so that it has constant positive curvature, then according to classical results in
Riemannian geometry, it must be the 3-sphere. Hamilton prescribed
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Some observers interpreted Cao and Zhu as taking credit for
Perelman's work. They later posted a revised version, with new wording, on arXiv. In addition, a page of their exposition was essentially identical to a page in one of Kleiner and Lott's early publicly available drafts; this was also amended
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Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until, eventually,
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In this paper, we shall present the
Hamilton-Perelman theory of Ricci flow. Based on it, we shall give the first written account of a complete proof of the Poincaré conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated efforts of many geometric analysts,
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and … the
Geometrization conjecture there was no consensus among the experts as to whether the Poincaré conjecture was true or false. After Thurston's work, notwithstanding the fact that it had no direct bearing on the Poincaré conjecture, a consensus developed that the Poincaré conjecture (and the
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attempted to prove the conjecture. In 1958, R. H. Bing proved a weak version of the
Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere. Bing also described some of the pitfalls in trying to prove the
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However, despite its usual phrasing in the form of a conjecture, proposing that all manifolds of a certain type are homeomorphic to the sphere, Poincaré only posed an open-ended question, without venturing to conjecture one way or the other. Moreover, there is no evidence as to which way he believed
630:. For this reason, it is not possible to read Poincaré's questions unambiguously. It is only through the formalization and vocabulary of topology as developed by later mathematicians that Poincaré's closing question has been understood as the "Poincaré conjecture" as stated in the preceding section.
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Throughout the work of
Riemann, Betti, and Poincaré, the topological notions in question are not defined or used in a way that would be recognized as precise from a modern perspective. Even the key notion of a "manifold" was not used in a consistent way in Poincaré's own work, and there was frequent
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posted a paper on arXiv in May 2006 which filled in the details of
Perelman's proof of the geometrization conjecture, following partial versions which had been publicly available since 2003. Their manuscript was published in the journal "Geometry and Topology" in 2008. A small number of corrections
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manifold with the homology of a sphere must be homeomorphic to a sphere.) This modified his negative generalization of Riemann's work in two ways. Firstly, he was now making use of the full homology groups and not only the Betti numbers. Secondly, he narrowed the scope of the problem from asking if
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spanning a bent loop of wire. Hamilton had shown that the area of a minimal surface decreases as the manifold undergoes Ricci flow. Perelman verified what happened to the area of the minimal surface when the manifold was sliced. He proved that, eventually, the area is so small that any cut after
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The two-dimensional analogue of the Poincaré conjecture says that any two-dimensional topological manifold which is closed and connected but non-homeomorphic to the two-dimensional sphere must possess a loop which cannot be continuously contracted to a point. (This is illustrated by the example of
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Essentially, an eigenvalue is like a note being played by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the cigar soliton solution, which
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Hamilton created a list of possible singularities that could form, but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps and then run the Ricci flow again, so he needed to understand the singularities and show
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This condition on the fundamental group turns out to be necessary and sufficient for finite time extinction. It is equivalent to saying that the prime decomposition of the manifold has no acyclic components and turns out to be equivalent to the condition that all geometric pieces of the manifold
718:
commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect." Work on the conjecture improved understanding of 3-manifolds. Experts in the field were often reluctant to announce proofs and tended to view any such announcement with skepticism. The 1980s and
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as a novel topological invariant, and was able to exhibit examples of three-dimensional manifolds which have the same Betti numbers but distinct fundamental groups. He posed the question of whether the fundamental group is sufficient to topologically characterize a manifold (of given dimension),
802:
These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons. In dimension three, the conjecture had an uncertain reputation until the
1257:. In the context that one makes no assumption about the fundamental group whatsoever, Perelman made a further technical study of the limit of the manifold for infinitely large times, and in so doing, proved Thurston's geometrization conjecture: at large times, the manifold has a
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In some cases, Hamilton was able to show that this works; for example, his original breakthrough was to show that if the Riemannian manifold has positive Ricci curvature everywhere, then the above procedure can only be followed for a bounded interval of parameter values,
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cylinders. With a quantitative understanding of this phenomenon, he cuts the manifold along the singularities, splitting the manifold into several pieces and then continues with the Ricci flow on each of these pieces. This procedure is known as Ricci flow with surgery.
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In this remark, as in the closing remark of the second supplement, Poincaré used the term "simply connected" in a way which is at odds with modern usage, as well as his own 1895 definition of the term. (According to modern usage, Poincaré's question is a
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to show that, on a simply-connected compact 3-manifold, any solution of the Ricci flow with surgery becomes extinct in finite time. An alternative argument, based on the min-max theory of minimal surfaces and geometric measure theory, was provided by
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on the symmetry of Betti numbers. Following criticism of the completeness of his arguments, he released a number of subsequent "supplements" to enhance and correct his work. The closing remark of his second supplement, published in 1900, said:
614:, asking if it is possible for a manifold to be simply connected without being simply connected.) However, as can be inferred from context, Poincaré was asking whether the triviality of the fundamental group uniquely characterizes the sphere.
1241:. Hence, in the simply-connected context, the above finite-time phenomena of Ricci flow with surgery is all that is relevant. In fact, this is even true if the fundamental group is a free product of finite groups and cyclic groups.
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that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself towards what are known as
849:
on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture. In the following years, he extended this work but was unable to prove the conjecture. The actual solution was not found until
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One immediate question posed was how one could be sure that infinitely many cuts are not necessary. This was raised due to the cutting potentially progressing forever. Perelman proved this cannot happen by using
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claimed a proof but then retracted it. In the process, he discovered some examples of simply-connected (indeed contractible, i.e. homotopically equivalent to a point) non-compact 3-manifolds not homeomorphic to
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for his work on the Ricci flow, but Perelman refused the medal. John Morgan spoke at the ICM on the Poincaré conjecture on August 24, 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture."
532:, which associate to any manifold a list of nonnegative integers. Riemann had showed that a closed connected two-dimensional manifold is fully characterized by its Betti numbers. As part of his 1895 paper
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looked like a strand sticking out of a manifold with nothing on the other side. In essence, Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.
924:
posted a paper on arXiv in July 2006 which gave a detailed proof of just the Poincaré Conjecture (which is somewhat easier than the full geometrization conjecture) and expanded this to a book.
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The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture.
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proved the Poincaré conjecture in four dimensions. Freedman's work left open the possibility that there is a smooth four-manifold homeomorphic to the four-sphere which is not
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However, after publication he found his announced theorem to be incorrect. In his fifth and final supplement, published in 1904, he proved this with the counterexample of the
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1419:(a number). Such numbers are called eigenvalues of that operation. Eigenvalues are closely related to vibration frequencies and are used in analyzing a famous problem:
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were made in 2011 and 2013; for instance, the first version of their published paper made use of an incorrect version of Hamilton's compactness theorem for Ricci flow.
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gives an affirmative answer to the analogous question in two dimensions. For dimensions greater than three, one can pose the Generalized Poincaré conjecture: is a
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increases, the manifold becomes easier to understand. Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part.
764:-sphere? A stronger assumption than simply-connectedness is necessary; in dimensions four and higher there are simply-connected, closed manifolds which are not
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In the 1950s and 1960s, other mathematicians attempted proofs of the conjecture only to discover that they contained flaws. Influential mathematicians such as
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shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental
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In order to avoid making this work too prolonged, I confine myself to stating the following theorem, the proof of which will require further developments:
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Morgan, John W., Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 1, 57–78
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to a 2-sphere if every loop can be continuously tightened to a point. The Poincaré conjecture asserts that the same is true for 3-dimensional spaces.
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1265:. Due to Perelman's and Colding and Minicozzi's results, however, these further results are unnecessary in order to prove the Poincaré conjecture.
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Cao, Huai-Dong & Zhu, Xi-Ping (December 3, 2006). "Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture".
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the area is that small can only be chopping off three-dimensional spheres and not more complicated pieces. This is described as a battle with a
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a compact simply-connected 3-manifold supports a Riemannian metric of positive Ricci curvature, then it must be diffeomorphic to the 3-sphere.
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From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:
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Hamilton and Perelman's work on the conjecture is widely recognized as a milestone of mathematical research. Hamilton was recognized with the
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1392:. The Ricci flow was defined by Richard S. Hamilton as a way to deform manifolds. The formula for the Ricci flow is an imitation of the
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Huai-Dong Cao; Xi-Ping Zhu (December 3, 2006). "Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture".
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2641:"A Complete Proof of the Poincaré and Geometrization Conjectures – application of the Hamilton-Perelman theory of the Ricci flow"
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to the four-sphere. This so-called smooth Poincaré conjecture, in dimension four, remains open and is thought to be very difficult.
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Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961,
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with an exposition of the complete proof of the Poincaré and geometrization conjectures. The opening paragraph of their paper stated
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1 million for the conjecture's resolution. He declined the award, saying that Hamilton's contribution had been equal to his own.
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538:(announced in 1892), Poincaré showed that Riemann's result does not extend to higher dimensions. To do this he introduced the
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Perelman, Grisha (July 17, 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds".
150:
393:). Over the next several years, several mathematicians studied his papers and produced detailed formulations of his work.
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in Szpiro's book cited below. This last part of the proof appeared in Perelman's third and final paper on the subject.
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an arbitrary manifold is characterized by topological invariants to asking whether the sphere can be so characterized.
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All three groups found that the gaps in Perelman's papers were minor and could be filled in using his own techniques.
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The primary purpose of Poincaré's paper was the interpretation of the Betti numbers in terms of his newly-introduced
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to spheres, and he hoped to apply it to prove the Poincaré Conjecture. He needed to understand the singularities.
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Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds".
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although he made no attempt to pursue the answer, saying only that it would "demand lengthy and difficult study."
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on the manifold. A minimal surface is one on which any local deformation increases area; a familiar example is a
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but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each
1763:. Translated by Baker, Roger; Christenson, Charles; Orde, Henry. Heber City, UT: Kendrick Press. pp. 1–41.
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repository in 2002 and 2003, Perelman presented his work proving the Poincaré conjecture (and the more powerful
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Perelman, Grisha (November 11, 2002). "The entropy formula for the Ricci flow and its geometric applications".
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The Surprising Resolution of the Poincaré Conjecture. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein
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Perelman proved the conjecture by deforming the manifold using the Ricci flow (which behaves similarly to the
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refer to "simply connected in the true sense of the word" as the condition of being homeomorphic to a sphere.
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Morgan, John; Tian, Gang (2015). "Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture".
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2497:. Series in Geometry and Topology. Vol. 37. Somerville, MA: International Press. pp. 119–162.
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to solve the problem. By developing a number of new techniques and results in the theory of Ricci flow,
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2001:
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Bing, R. H. (1964). "Some aspects of the topology of 3-manifolds related to the Poincaré conjecture".
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861:. In these papers, he sketched a proof of the Poincaré conjecture and a more general conjecture,
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outlining a solution of the Poincaré conjecture. Perelman's proof uses a modified version of a
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1990s witnessed some well-publicized fallacious proofs (which were not actually published in
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An exposition of attempts to prove this conjecture can be found in the non-technical book
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Over time, the conjecture gained the reputation of being particularly tricky to tackle.
577:(In a modern language, taking note of the fact that Poincaré is using the terminology of
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Familiar shapes, such as the surface of a ball (which is known in mathematics as the
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1483:. Algorithms and Computation in Mathematics. Vol. 9. Springer. pp. 46–58.
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show that the smooth Poincaré conjecture is false in dimension seven, for example.
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One question remains to be dealt with: is it possible for the fundamental group of
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The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry
2837:"Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman"
1645:"Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman"
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in the revised version, together with an apology by the journal's editorial board.
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can be continuously tightened to a point. A torus is not homeomorphic to a sphere.
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Perelman, Grisha (March 10, 2003). "Ricci flow with surgery on three-manifolds".
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2521:(2002). "The entropy formula for the Ricci flow and its geometric applications".
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Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles
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Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles
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Hamilton's program for proving the Poincaré conjecture involves first putting a
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in the space can be continuously tightened to a point, then it is necessarily a
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1759:(2004). "Foundations for a general theory of functions of a complex variable".
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2907:. Einstein Studies. Vol. 14. New York, NY: Birkhäuser. pp. 401–415.
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Each polyhedron which has all its Betti numbers equal to 1 and all its tables
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Sometimes, an otherwise complicated operation reduces to multiplication by a
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being simply connected? However, this question would carry us too far away.
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1365:(worth $ 15,000 CAD) for his work on the Ricci flow. On March 18, 2010, the
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Hamilton's program was started in his 1982 paper in which he introduced the
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was able to modify and complete Hamilton's program. In papers posted to the
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Perelman, Grigori (2003). "Ricci flow with surgery on three-manifolds".
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1946:
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is true in dimension 3, then the Poincaré conjecture must also be true.
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1303: in this section. Unsourced material may be challenged and removed.
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2233:(1958). "Necessary and sufficient conditions that a 3-manifold be S".
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orientable is simply connected, i.e., homeomorphic to a hypersphere.
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marked Perelman's proof of the Poincaré conjecture as the scientific
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in 1904, the theorem concerns spaces that locally look like ordinary
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A Report on the Poincaré Conjecture. Special lecture by John Morgan.
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Taubes, Gary (July 1987). "What happens when hubris meets nemesis".
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in recognition of his proof. Perelman rejected that prize as well.
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480:
450:. In terms of the vocabulary of that field, it says the following:
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The Poincaré conjecture was a mathematical problem in the field of
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2903:(2018). "The Surprising Resolution of the Poincaré Conjecture".
1803:(1870). "Sopra gli spazi di un numero qualunque di dimensioni".
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Matveev, Sergei (2007). "1.3.4 Zeeman's Collapsing Conjecture".
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the major contributors are unquestionably Hamilton and Perelman.
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The Poincaré Conjecture: In Search of the Shape of the Universe
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418:, having included the Poincaré conjecture in their well-known
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3021:. Clay Mathematics Monographs. Vol. 3. Providence, RI:
2839:. Clay Mathematics Institute. March 18, 2010. Archived from
2337:"The Poincaré Conjecture 99 Years Later: A Progress Report"
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A History of Algebraic and Differential Topology, 1900–1960
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1361:. In August 2006, Perelman was awarded, but declined, the
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Leroy P. Steele Prize for Seminal Contribution to Research
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M., Halverson, Denise; Dušan, Repovš (23 December 2008).
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The American Heritage Dictionary of the English Language
865:, completing the Ricci flow program outlined earlier by
1637:
857:
In late 2002 and 2003, Perelman posted three papers on
1929:. History of Mathematics. Vol. 37. Translated by
1480:
Algorithmic Topology and Classification of 3-Manifolds
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2587:; John W. Lott (2008). "Notes on Perelman's Papers".
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Comptes Rendus des Séances de l'Académie des Sciences
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Grundlagen für eine allgemeine Theorie der Functionen
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have geometries based on the two Thurston geometries
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its Ricci curvature, and one hopes that, as the time
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581:
in an unusual way, this says that a closed connected
294:
288:
282:
256:
250:
807:put it into a framework governing all 3-manifolds.
303:
285:
247:
2583:
1388:The first step is to deform the manifold using the
300:
244:
2700:(2006). "Ricci Flow and the Poincaré Conjecture".
1208:
1169:
1143:
1116:
1090:
1033:
828:
707:Włodzimierz Jakobsche showed in 1978 that, if the
668:
2877:"Russian mathematician rejects $ 1 million prize"
2773:
2274:. Vol. II. New York: Wiley. pp. 93–128.
1124:, and more significantly, that there are numbers
3594:
3113:
3098:
3083:
949:honored the proof of Poincaré conjecture as the
894:published a paper in the June 2006 issue of the
3157:"Poincaré and the early history of 3-manifolds"
2720:
2692:
2446:"Three-manifolds with positive Ricci curvature"
1872:
1870:
1868:
1228:Perelman provided a separate argument based on
2874:
2631:
2287:"The Bing–Borsuk and the Busemann conjectures"
2059:Proceedings of the London Mathematical Society
3288:
3162:Bulletin of the American Mathematical Society
2485:
1724:"Russian mathematician rejects million prize"
201:
2317:: CS1 maint: multiple names: authors list (
2128:Rendiconti del Circolo Matematico di Palermo
2101:
2099:
2097:
1913:
1911:
1909:
1907:
1865:
1833:
1341:On November 13, 2002, Russian mathematician
1034:{\displaystyle \partial _{t}g_{ij}=-2R_{ij}}
2938:
2380:"$ 1 million mathematical mystery "solved""
1532:
676:, the prototype of which is now called the
3295:
3281:
3057:
3009:
2899:
2752:
1678:
1676:
956:
208:
194:
36:
3174:
3151:
3119:
3104:
3089:
3030:
3000:
2961:
2808:"Highest Honor in Mathematics Is Refused"
2758:
2705:
2677:
2600:
2568:
2547:
2526:
2461:
2302:
2174:
2168:
2123:"Cinquième complément à l'analysis situs"
2094:
1904:
1612:
1591:
1587:
1585:
1319:Learn how and when to remove this message
656:
438:Neither of the two colored loops on this
3191:
2868:
2562:
2541:
2517:
2440:
2377:
2162:
2117:
2106:
2048:
1917:
1876:
1839:
1330:
832:
433:
2284:
1988:
1986:
1984:
1982:
1755:
1740:
1673:
1476:
14:
3595:
3128:
3019:Ricci Flow and the Poincaré Conjecture
2946:(2008). "Notes on Perelman's papers".
2730:Ricci Flow and the Poincaré Conjecture
2402:
2358:
2331:
2054:"Second complément à l'analysis situs"
1998:Henri Poincaré: A Scientific Biography
1806:Annali di Matematica Pura ed Applicata
1721:
1715:
1690:[The last "no" Dr. Perelman].
1582:
1369:awarded Perelman the $ 1 million
1345:posted the first of a series of three
507:
422:list, offered Perelman their prize of
161:Navier–Stokes existence and smoothness
3276:
2805:
1799:
976:equations" for improving the metric;
824:Geometrization conjecture) were true.
318:
2269:
2229:
1992:
1979:
1301:adding citations to reliable sources
1272:
863:Thurston's geometrization conjecture
151:Birch and Swinnerton-Dyer conjecture
1709:Google Translated archived link at
1684:"Последнее 'нет' доктора Перельмана
24:
3261:, Professor of Mathematics at the
3253:, Professor of Mathematics at the
2932:
2806:Chang, Kenneth (August 22, 2006).
2777:; David Gruber (August 28, 2006).
1761:Collected Papers: Bernhard Riemann
1597:"The Poincaré Conjecture – Proved"
1111:
987:
618:confusion between the notion of a
601:to reduce to the identity without
25:
3644:
3628:Conjectures that have been proved
3226:
2378:Matthews, Robert (9 April 2002).
1421:can you hear the shape of a drum?
750:classification of closed surfaces
182:Yang–Mills existence and mass gap
2450:Journal of Differential Geometry
1886:Journal de l'École Polytechnique
1651:. March 18, 2010. Archived from
1277:
669:{\displaystyle \mathbb {R} ^{3}}
634:his question would be answered.
278:
240:
3176:10.1090/S0273-0979-2012-01385-X
2893:
2847:
2829:
2820:
2799:
2767:
2746:
2714:
2686:
2665:
2625:
2577:
2556:
2535:
2511:
2434:
2425:
2396:
2371:
2352:
2325:
2278:
2263:
2223:
2155:
2111:
2042:
1793:
1722:Ritter, Malcolm (1 July 2010).
1288:needs additional citations for
829:Hamilton's program and solution
744:Generalized Poincaré conjecture
131:Generalized Poincaré conjecture
2732:. Clay Mathematics Institute.
2495:Collected Papers on Ricci Flow
2272:Lectures on Modern Mathematics
2105:cf. Stillwell's commentary in
1734:
1567:Merriam-Webster.com Dictionary
1554:
1526:
1497:
1470:
1203:
1197:
1161:
1085:
1073:
953:and featured it on its cover.
943:In December 2006, the journal
369:The eventual proof built upon
46:2-dimensional surface without
13:
1:
3023:American Mathematical Society
2875:Malcolm Ritter (2010-07-01).
1935:American Mathematical Society
1614:10.1126/science.314.5807.1848
1542:(5th ed.). HarperCollins
1464:
841:on a two-dimensional manifold
737:
491:-dimensional sphere) or of a
473:has trivial fundamental group
27:Theorem in geometric topology
3498:CRISPR genome-editing method
2913:10.1007/978-1-4939-7708-6_13
2857:. Clay Mathematics Institute
2648:Asian Journal of Mathematics
1117:{\displaystyle T<\infty }
897:Asian Journal of Mathematics
637:
7:
2291:Mathematical Communications
1939:London Mathematical Society
1452:
1268:
1170:{\displaystyle t\nearrow T}
429:
10:
3649:
3572:James Webb Space Telescope
2161:The opening paragraphs of
2002:Princeton University Press
1696:(in Russian). July 1, 2010
1649:Clay Mathematics Institute
1367:Clay Mathematics Institute
1091:{\displaystyle t\in [0,T)}
960:
815:It is my view that before
741:
701:Christos Papakyriakopoulos
502:
416:Clay Mathematics Institute
350:Originally conjectured by
3618:Millennium Prize Problems
3552:developed at record speed
3313:
3307:Breakthroughs of the Year
3233:"The Poincaré Conjecture"
2191:10.1007/978-0-8176-4907-4
1209:{\displaystyle c_{t}g(t)}
1177:, the Riemannian metrics
805:geometrization conjecture
387:geometrization conjecture
142:Millennium Prize Problems
126:
113:Geometrization conjecture
105:
97:
87:
79:
69:
59:
35:
2489:; Chow, B.; Chu, S. C.;
2072:10.1112/plms/s1-32.1.277
1927:and Its Five Supplements
1259:thick-thin decomposition
951:Breakthrough of the Year
931:On August 22, 2006, the
591:Poincaré homology sphere
552:Poincaré duality theorem
481:three-dimensional sphere
459:Every three-dimensional
420:Millennium Prize Problem
412:Breakthrough of the Year
373:'s program of using the
364:three-dimensional sphere
3427:Human genetic variation
3362:Whole genome sequencing
3195:; Nadis, Steve (2019).
3129:Szpiro, George (2008).
2972:10.2140/gt.2008.12.2587
2949:Geometry & Topology
2790:On-line version at the
2611:10.2140/gt.2008.12.2587
2403:Szpiro, George (2008).
2184:Birkhäuser Boston, Inc.
1750:University of Göttingen
1516:Oxford University Press
1505:"Poincaré, Jules-Henri"
957:Ricci flow with surgery
520:initiated the study of
356:three-dimensional space
3531:Single-cell sequencing
3435:Cellular reprogramming
2463:10.4310/jdg/1214436922
1845:"Sur l'Analysis situs"
1338:
1210:
1171:
1145:
1118:
1092:
1035:
908:
854:published his papers.
842:
837:Several stages of the
826:
821:hyperbolic 3-manifolds
709:Bing–Borsuk conjecture
670:
607:
575:
528:. They introduced the
522:topological invariants
485:
443:
3346:Accelerating universe
3255:University of Warwick
3201:Yale University Press
2855:"Poincaré Conjecture"
2589:Geometry and Topology
2236:Annals of Mathematics
1754:English translation:
1712:(archived 2014-04-20)
1512:UK English Dictionary
1357:program developed by
1334:
1230:curve shortening flow
1211:
1172:
1146:
1144:{\displaystyle c_{t}}
1119:
1093:
1036:
935:awarded Perelman the
903:
836:
813:
704:Poincaré conjecture.
671:
595:
557:
452:
437:
3613:Theorems in topology
3479:Cancer immunotherapy
3444:Ardipithecus ramidus
3263:University of Oxford
3067:Walker & Company
1923:Papers on Topology:
1297:improve this article
1181:
1155:
1128:
1102:
1064:
983:
760:homeomorphic to the
651:
620:topological manifold
579:simple-connectedness
461:topological manifold
3522:neutron star merger
3510:gravitational waves
3418:Poincaré conjecture
2654:(2). Archived from
1607:(5807): 1848–1849.
1359:Richard S. Hamilton
867:Richard S. Hamilton
781:h-cobordism theorem
766:homotopy equivalent
508:Poincaré's question
455:Poincaré conjecture
371:Richard S. Hamilton
230:Poincaré conjecture
171:Poincaré conjecture
166:P versus NP problem
32:
31:Poincaré conjecture
18:Poincaré Conjecture
3633:1904 introductions
3603:Geometric topology
3563:protein structures
2813:The New York Times
2779:"Manifold destiny"
2141:10.1007/bf03014091
1819:10.1007/BF02420029
1339:
1206:
1167:
1141:
1114:
1088:
1048:is the metric and
1031:
843:
678:Whitehead manifold
666:
644:J. H. C. Whitehead
448:geometric topology
444:
320:[pwɛ̃kaʁe]
226:geometric topology
177:Riemann hypothesis
64:Geometric topology
30:
3590:
3589:
3550:COVID-19 vaccines
3506:First observation
3374:Molecular circuit
3210:978-0-300-23590-6
3199:. New Haven, CT:
3144:978-0-452-28964-2
3076:978-0-8027-1654-5
3042:978-0-8218-4328-4
2922:978-1-4939-7708-6
2787:. pp. 44–57.
2739:978-0-8218-4328-4
2519:Perelman, Grigori
2442:Hamilton, Richard
2418:978-0-452-28964-2
2239:. Second Series.
2011:978-0-691-15271-4
2000:. Princeton, NJ:
1956:978-0-8218-5234-7
1947:10.1090/hmath/037
1757:Riemann, Bernhard
1742:Riemann, Bernhard
1658:on March 22, 2010
1647:(Press release).
1570:. Merriam-Webster
1410:elliptic equation
1329:
1328:
1321:
1239:William Minicozzi
969:Riemannian metric
550:, along with the
540:fundamental group
218:
217:
136:
135:
118:Zeeman conjecture
50:is topologically
16:(Redirected from
3640:
3583:
3575:
3566:
3553:
3544:
3533:
3525:
3512:
3500:
3492:
3481:
3473:
3464:
3455:
3447:
3437:
3429:
3421:
3412:
3403:
3392:
3384:
3382:RNA interference
3376:
3364:
3356:
3348:
3340:
3332:
3297:
3290:
3283:
3274:
3273:
3265:, and presenter
3259:Marcus du Sautoy
3222:
3188:
3178:
3148:
3125:
3123:
3110:
3108:
3095:
3093:
3080:
3054:
3034:
3006:
3004:
2991:
2965:
2956:(5): 2587–2855.
2927:
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2709:
2690:
2684:
2683:
2681:
2669:
2663:
2662:
2660:
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2629:
2623:
2622:
2604:
2595:(5): 2587–2855.
2581:
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2554:
2553:
2551:
2539:
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2532:
2530:
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2465:
2438:
2432:
2429:
2423:
2422:
2400:
2394:
2393:
2391:
2390:
2384:NewScientist.com
2375:
2369:
2368:
2356:
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2349:
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2341:
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2316:
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2227:
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2220:
2172:
2166:
2159:
2153:
2152:
2115:
2109:
2103:
2092:
2091:
2046:
2040:
2039:
1990:
1977:
1976:
1915:
1902:
1901:
1882:"Analysis situs"
1874:
1863:
1862:
1837:
1831:
1830:
1797:
1791:
1790:
1753:
1738:
1732:
1731:
1728:The Boston Globe
1719:
1713:
1706:
1703:
1701:
1680:
1671:
1670:
1665:
1663:
1657:
1641:
1635:
1634:
1616:
1589:
1580:
1579:
1577:
1575:
1558:
1552:
1551:
1549:
1547:
1530:
1524:
1523:
1518:. Archived from
1501:
1495:
1494:
1474:
1459:Manifold Destiny
1434:minimal surfaces
1371:Millennium Prize
1343:Grigori Perelman
1336:Grigori Perelman
1324:
1317:
1313:
1310:
1304:
1281:
1273:
1215:
1213:
1212:
1207:
1193:
1192:
1176:
1174:
1173:
1168:
1150:
1148:
1147:
1142:
1140:
1139:
1123:
1121:
1120:
1115:
1097:
1095:
1094:
1089:
1040:
1038:
1037:
1032:
1030:
1029:
1008:
1007:
995:
994:
852:Grigori Perelman
785:Michael Freedman
728:Poincaré's Prize
675:
673:
672:
667:
665:
664:
659:
604:
600:
572:
514:Bernhard Riemann
391:William Thurston
379:Grigori Perelman
345:four-dimensional
339:that bounds the
329:characterization
322:
317:
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309:
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239:
210:
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196:
156:Hodge conjecture
138:
137:
92:Grigori Perelman
40:
33:
29:
21:
3648:
3647:
3643:
3642:
3641:
3639:
3638:
3637:
3593:
3592:
3591:
3586:
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3556:
3547:
3536:
3528:
3515:
3503:
3495:
3484:
3476:
3467:
3458:
3453:quantum machine
3450:
3440:
3432:
3424:
3415:
3406:
3395:
3387:
3379:
3367:
3359:
3351:
3343:
3338:Dolly the sheep
3335:
3326:
3319:
3309:
3301:
3247:Open University
3229:
3211:
3193:Yau, Shing-Tung
3153:Stillwell, John
3145:
3121:math.DG/0307245
3106:math.DG/0303109
3091:math.DG/0211159
3077:
3043:
3011:Morgan, John W.
3002:math.DG/0612069
2935:
2933:Further reading
2930:
2923:
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2883:
2873:
2869:
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2800:
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2751:
2747:
2740:
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2715:
2707:math.DG/0607607
2691:
2687:
2679:math.DG/0612069
2670:
2666:
2658:
2643:
2630:
2626:
2602:math.DG/0605667
2582:
2578:
2570:math.DG/0307245
2561:
2557:
2549:math.DG/0303109
2540:
2536:
2528:math.DG/0211159
2516:
2512:
2505:
2493:, eds. (2003).
2439:
2435:
2430:
2426:
2419:
2401:
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2386:
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2357:
2353:
2344:
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2330:
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2310:
2309:
2283:
2279:
2268:
2264:
2249:10.2307/1970041
2228:
2224:
2201:
2176:Dieudonné, Jean
2173:
2169:
2163:Poincaré (1904)
2160:
2156:
2116:
2112:
2107:Poincaré (2010)
2104:
2095:
2047:
2043:
2012:
1991:
1980:
1957:
1931:Stillwell, John
1919:Poincaré, Henri
1916:
1905:
1875:
1866:
1838:
1834:
1798:
1794:
1771:
1739:
1735:
1720:
1716:
1699:
1697:
1682:
1681:
1674:
1661:
1659:
1655:
1643:
1642:
1638:
1593:Mackenzie, Dana
1590:
1583:
1573:
1571:
1560:
1559:
1555:
1545:
1543:
1531:
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1503:
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1471:
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1455:
1325:
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1305:
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1000:
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990:
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965:
959:
831:
746:
740:
685:Georges de Rham
660:
655:
654:
652:
649:
648:
640:
628:smooth manifold
602:
598:
571:
563:
548:homology groups
510:
505:
458:
432:
335:, which is the
315:
281:
272:
271:
243:
234:
233:
214:
127:Generalizations
122:
55:
28:
23:
22:
15:
12:
11:
5:
3646:
3636:
3635:
3630:
3625:
3623:Henri Poincaré
3620:
3615:
3610:
3605:
3588:
3587:
3585:
3584:
3576:
3567:
3554:
3545:
3534:
3526:
3513:
3501:
3493:
3482:
3474:
3465:
3463:clinical trial
3456:
3448:
3438:
3430:
3422:
3413:
3404:
3393:
3385:
3377:
3365:
3357:
3349:
3341:
3333:
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3321:
3311:
3310:
3300:
3299:
3292:
3285:
3277:
3271:
3270:
3228:
3227:External links
3225:
3224:
3223:
3209:
3189:
3169:(4): 555–576.
3149:
3143:
3126:
3111:
3096:
3081:
3075:
3055:
3041:
3007:
2992:
2940:Kleiner, Bruce
2934:
2931:
2929:
2928:
2921:
2892:
2867:
2846:
2843:on 2010-03-22.
2828:
2819:
2798:
2784:The New Yorker
2766:
2745:
2738:
2713:
2685:
2664:
2661:on 2012-05-14.
2633:Cao, Huai-Dong
2624:
2585:Kleiner, Bruce
2576:
2555:
2534:
2510:
2503:
2484:Reprinted in:
2456:(2): 255–306.
2433:
2424:
2417:
2395:
2370:
2351:
2324:
2277:
2262:
2222:
2199:
2182:. Boston, MA:
2167:
2154:
2110:
2093:
2066:(1): 277–308.
2041:
2010:
1978:
1955:
1925:Analysis Situs
1903:
1864:
1832:
1792:
1769:
1733:
1714:
1672:
1636:
1595:(2006-12-22).
1581:
1553:
1525:
1522:on 2022-09-02.
1496:
1490:978-3540458999
1489:
1468:
1466:
1463:
1462:
1461:
1454:
1451:
1327:
1326:
1285:
1283:
1276:
1270:
1267:
1263:graph manifold
1235:Tobias Colding
1205:
1202:
1199:
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1006:
1003:
999:
993:
989:
961:Main article:
958:
955:
926:
925:
914:
913:
902:
901:
885:
830:
827:
797:exotic spheres
742:Main article:
739:
736:
697:Edwin E. Moise
693:Wolfgang Haken
663:
658:
642:In the 1930s,
639:
636:
567:
535:Analysis Situs
512:In the 1800s,
509:
506:
504:
501:
431:
428:
404:. The journal
352:Henri Poincaré
216:
215:
213:
212:
205:
198:
190:
187:
186:
185:
184:
179:
174:
168:
163:
158:
153:
145:
144:
134:
133:
128:
124:
123:
121:
120:
115:
109:
107:
103:
102:
99:
98:First proof in
95:
94:
89:
88:First proof by
85:
84:
81:
80:Conjectured in
77:
76:
74:Henri Poincaré
71:
70:Conjectured by
67:
66:
61:
57:
56:
41:
26:
9:
6:
4:
3:
2:
3645:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3600:
3598:
3582:
3577:
3573:
3568:
3564:
3560:
3555:
3551:
3546:
3543:
3540:
3535:
3532:
3527:
3523:
3519:
3514:
3511:
3507:
3502:
3499:
3494:
3491:
3490:comet mission
3489:
3483:
3480:
3475:
3471:
3466:
3462:
3457:
3454:
3449:
3446:
3445:
3439:
3436:
3431:
3428:
3423:
3419:
3414:
3410:
3405:
3402:
3400:
3394:
3391:
3386:
3383:
3378:
3375:
3371:
3366:
3363:
3358:
3355:
3350:
3347:
3342:
3339:
3334:
3331:understanding
3330:
3325:
3324:
3322:
3318:
3317:
3312:
3308:
3306:
3298:
3293:
3291:
3286:
3284:
3279:
3278:
3275:
3268:
3264:
3260:
3256:
3252:
3248:
3244:
3243:
3238:
3234:
3231:
3230:
3220:
3216:
3212:
3206:
3202:
3198:
3194:
3190:
3186:
3182:
3177:
3172:
3168:
3164:
3163:
3158:
3154:
3150:
3146:
3140:
3136:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3092:
3087:
3082:
3078:
3072:
3068:
3064:
3060:
3059:O'Shea, Donal
3056:
3052:
3048:
3044:
3038:
3033:
3028:
3024:
3020:
3016:
3012:
3008:
3003:
2998:
2993:
2989:
2985:
2981:
2977:
2973:
2969:
2964:
2959:
2955:
2951:
2950:
2945:
2941:
2937:
2936:
2924:
2918:
2914:
2910:
2906:
2902:
2901:O'Shea, Donal
2896:
2882:
2878:
2871:
2856:
2850:
2842:
2838:
2832:
2823:
2815:
2814:
2809:
2802:
2795:
2793:
2786:
2785:
2780:
2776:
2775:Nasar, Sylvia
2770:
2761:
2756:
2749:
2741:
2735:
2731:
2727:
2723:
2717:
2708:
2703:
2699:
2695:
2689:
2680:
2675:
2668:
2657:
2653:
2649:
2642:
2639:(June 2006).
2638:
2634:
2628:
2620:
2616:
2612:
2608:
2603:
2598:
2594:
2590:
2586:
2580:
2571:
2566:
2559:
2550:
2545:
2538:
2529:
2524:
2520:
2514:
2506:
2504:1-57146-110-8
2500:
2496:
2492:
2488:
2481:
2477:
2473:
2469:
2464:
2459:
2455:
2451:
2447:
2443:
2437:
2428:
2420:
2414:
2410:
2406:
2399:
2385:
2381:
2374:
2366:
2362:
2355:
2338:
2334:
2328:
2320:
2314:
2305:
2300:
2296:
2292:
2288:
2281:
2273:
2266:
2258:
2254:
2250:
2246:
2242:
2238:
2237:
2232:
2226:
2218:
2214:
2210:
2206:
2202:
2200:0-8176-3388-X
2196:
2192:
2188:
2185:
2181:
2177:
2171:
2164:
2158:
2150:
2146:
2142:
2138:
2134:
2130:
2129:
2124:
2120:
2114:
2108:
2102:
2100:
2098:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2060:
2055:
2051:
2045:
2037:
2033:
2029:
2025:
2021:
2017:
2013:
2007:
2003:
1999:
1995:
1989:
1987:
1985:
1983:
1974:
1970:
1966:
1962:
1958:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1914:
1912:
1910:
1908:
1899:
1895:
1891:
1887:
1883:
1879:
1873:
1871:
1869:
1860:
1856:
1852:
1851:
1846:
1842:
1836:
1828:
1824:
1820:
1816:
1812:
1808:
1807:
1802:
1801:Betti, Enrico
1796:
1788:
1784:
1780:
1776:
1772:
1770:0-9740427-2-2
1766:
1762:
1758:
1751:
1747:
1743:
1737:
1729:
1725:
1718:
1711:
1708:
1705:
1694:
1691:
1688:
1687:
1679:
1677:
1669:
1654:
1650:
1646:
1640:
1632:
1628:
1624:
1620:
1615:
1610:
1606:
1602:
1598:
1594:
1588:
1586:
1569:
1568:
1563:
1557:
1541:
1540:
1535:
1529:
1521:
1517:
1513:
1511:
1506:
1500:
1492:
1486:
1482:
1481:
1473:
1469:
1460:
1457:
1456:
1450:
1448:
1444:
1439:
1435:
1429:
1425:
1422:
1418:
1413:
1411:
1408:of a certain
1407:
1401:
1399:
1398:diffeomorphic
1395:
1394:heat equation
1391:
1386:
1384:
1383:singularities
1379:
1378:heat equation
1374:
1372:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1337:
1333:
1323:
1320:
1312:
1302:
1298:
1292:
1291:
1286:This section
1284:
1280:
1275:
1274:
1266:
1264:
1260:
1256:
1252:
1248:
1242:
1240:
1236:
1231:
1226:
1222:
1220:
1200:
1194:
1189:
1185:
1164:
1158:
1151:such that as
1136:
1132:
1108:
1105:
1082:
1079:
1076:
1070:
1067:
1057:
1055:
1051:
1047:
1026:
1023:
1019:
1015:
1012:
1009:
1004:
1001:
997:
991:
979:
978:
977:
975:
970:
964:
954:
952:
948:
947:
941:
938:
934:
929:
923:
919:
916:
915:
910:
909:
907:
899:
898:
893:
889:
888:Huai-Dong Cao
886:
882:
878:
877:Bruce Kleiner
875:
874:
873:
870:
868:
864:
860:
855:
853:
848:
840:
835:
825:
822:
818:
812:
810:
806:
800:
798:
794:
790:
789:diffeomorphic
786:
782:
778:
777:Stephen Smale
773:
771:
767:
763:
759:
757:
751:
745:
735:
733:
732:George Szpiro
729:
724:
722:
721:peer-reviewed
717:
712:
710:
705:
702:
698:
694:
690:
686:
681:
679:
661:
645:
635:
631:
629:
625:
621:
615:
613:
606:
594:
592:
587:
584:
580:
574:
570:
566:
560:
556:
553:
549:
544:
541:
537:
536:
531:
530:Betti numbers
527:
523:
519:
515:
500:
496:
494:
490:
484:
482:
478:
474:
470:
466:
462:
456:
451:
449:
441:
436:
427:
425:
421:
417:
414:in 2006. The
413:
409:
408:
403:
399:
394:
392:
388:
384:
380:
376:
372:
367:
365:
361:
357:
353:
348:
346:
342:
338:
334:
330:
326:
321:
311:
275:
267:
237:
231:
227:
223:
211:
206:
204:
199:
197:
192:
191:
189:
188:
183:
180:
178:
175:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
143:
140:
139:
132:
129:
125:
119:
116:
114:
111:
110:
108:
104:
100:
96:
93:
90:
86:
82:
78:
75:
72:
68:
65:
62:
58:
53:
49:
45:
39:
34:
19:
3542:made visible
3487:
3451:2010: First
3442:
3417:
3398:
3370:Nanocircuits
3314:
3304:
3267:Melvyn Bragg
3240:
3196:
3166:
3160:
3130:
3062:
3032:math/0607607
3018:
2963:math/0605667
2953:
2947:
2904:
2895:
2884:. Retrieved
2870:
2859:. Retrieved
2849:
2841:the original
2831:
2822:
2811:
2801:
2791:
2782:
2769:
2748:
2729:
2722:Morgan, John
2716:
2694:Morgan, John
2688:
2667:
2656:the original
2651:
2647:
2627:
2592:
2588:
2579:
2558:
2537:
2513:
2494:
2453:
2449:
2436:
2427:
2404:
2398:
2387:. Retrieved
2383:
2373:
2364:
2360:
2354:
2343:. Retrieved
2333:Milnor, John
2327:
2313:cite journal
2294:
2290:
2280:
2271:
2265:
2243:(1): 17–37.
2240:
2234:
2225:
2179:
2170:
2157:
2132:
2126:
2119:Poincaré, H.
2113:
2063:
2057:
2050:Poincaré, H.
2044:
1997:
1994:Gray, Jeremy
1926:
1922:
1889:
1888:. 2e Série.
1885:
1878:Poincaré, H.
1848:
1841:Poincaré, H.
1835:
1810:
1804:
1795:
1760:
1745:
1736:
1727:
1717:
1707:
1698:. Retrieved
1695:
1689:
1685:
1667:
1662:November 13,
1660:. Retrieved
1653:the original
1639:
1604:
1600:
1572:. Retrieved
1565:
1556:
1544:. Retrieved
1537:
1528:
1520:the original
1508:
1499:
1479:
1472:
1430:
1426:
1414:
1402:
1387:
1375:
1363:Fields Medal
1340:
1315:
1309:October 2013
1306:
1295:Please help
1290:verification
1287:
1254:
1250:
1246:
1243:
1227:
1223:
1218:
1058:
1053:
1049:
1045:
1043:
966:
944:
942:
937:Fields Medal
930:
927:
904:
895:
881:John W. Lott
871:
856:
844:
814:
801:
783:. In 1982,
774:
769:
761:
755:
747:
727:
725:
713:
706:
682:
641:
632:
616:
608:
596:
588:
576:
568:
564:
561:
558:
545:
533:
518:Enrico Betti
511:
497:
488:
486:
477:homeomorphic
454:
453:
445:
405:
395:
368:
349:
229:
222:mathematical
219:
170:
52:homeomorphic
3608:3-manifolds
3581:GLP-1 Drugs
3470:Higgs boson
3390:Dark energy
3251:Ian Stewart
3242:In Our Time
3237:BBC Radio 4
2637:Xi-Ping Zhu
2231:Bing, R. H.
2020:j.ctt1r2fwt
1813:: 140–158.
918:John Morgan
892:Xi-Ping Zhu
819:'s work on
809:John Morgan
716:John Milnor
624:PL manifold
337:hypersphere
3597:Categories
3539:black hole
3239:programme
3015:Tian, Gang
2944:Lott, John
2886:2011-05-15
2861:2018-10-04
2792:New Yorker
2760:1512.00699
2491:Yau, S.-T.
2487:Cao, H. D.
2480:0504.53034
2389:2007-05-05
2345:2007-05-05
2217:0673.55002
2149:35.0504.13
2135:: 45–110.
2080:31.0477.10
2036:1263.01002
1973:1204.55002
1898:26.0541.07
1859:24.0506.02
1827:03.0301.01
1787:1101.01013
1748:(Thesis).
1562:"Poincaré"
1534:"Poincaré"
1465:References
1406:eigenvalue
1390:Ricci flow
1355:Ricci flow
974:Ricci flow
963:Ricci flow
847:Ricci flow
839:Ricci flow
738:Dimensions
689:R. H. Bing
398:Shaw Prize
375:Ricci flow
327:about the
106:Implied by
3472:discovery
3411:in action
3409:Evolution
3354:Stem cell
2988:119133773
2726:Gang Tian
2698:Gang Tian
2619:119133773
2304:0811.0886
1892:: 1–121.
1631:121869167
1438:soap film
1162:↗
1112:∞
1071:∈
1013:−
988:∂
922:Gang Tian
772:-sphere.
754:homotopy
638:Solutions
612:tautology
526:manifolds
469:connected
463:which is
341:unit ball
224:field of
3537:2019: A
3518:GW170817
3461:HPTN 052
3235: –
3155:(2012).
3061:(2007).
3017:(2007).
2881:Phys.Org
2728:(2007).
2444:(1982).
2367:: 66–77.
2361:Discover
2335:(2004).
2178:(1989).
2121:(1904).
2052:(1900).
1996:(2013).
1921:(2010).
1880:(1895).
1843:(1892).
1744:(1851).
1693:Interfax
1623:17185565
1574:9 August
1546:9 August
1453:See also
1269:Solution
817:Thurston
626:, and a
583:oriented
430:Overview
400:and the
333:3-sphere
173:(solved)
48:boundary
3561:brings
3488:Rosetta
3320:journal
3316:Science
3305:Science
3219:3930611
3185:2958930
3051:2334563
2980:2460872
2794:website
2472:0664497
2257:1970041
2209:0995842
2088:1576227
2028:2986502
1965:2723194
1779:2121437
1700:5 April
1601:Science
1447:Sormani
1347:eprints
946:Science
811:wrote:
758:-sphere
723:form).
503:History
479:to the
407:Science
347:space.
331:of the
325:theorem
323:) is a
316:French:
220:In the
44:compact
3579:2023:
3570:2022:
3565:to all
3557:2021:
3548:2020:
3529:2018:
3516:2017:
3504:2016:
3496:2015:
3485:2014:
3477:2013:
3468:2012:
3459:2011:
3441:2009:
3433:2008:
3425:2007:
3416:2006:
3407:2005:
3399:Spirit
3396:2004:
3388:2003:
3380:2002:
3368:2001:
3360:2000:
3352:1999:
3344:1998:
3336:1997:
3327:1996:
3217:
3207:
3183:
3141:
3073:
3049:
3039:
2986:
2978:
2919:
2736:
2617:
2501:
2478:
2470:
2415:
2255:
2215:
2207:
2197:
2147:
2086:
2078:
2034:
2026:
2018:
2008:
1971:
1963:
1953:
1896:
1857:
1825:
1785:
1777:
1767:
1629:
1621:
1510:Lexico
1487:
1417:scalar
1044:where
793:Milnor
768:to an
699:, and
471:, and
465:closed
228:, the
3574:debut
3420:proof
3401:rover
3135:Plume
3116:arXiv
3101:arXiv
3086:arXiv
3027:arXiv
2997:arXiv
2984:S2CID
2958:arXiv
2755:arXiv
2702:arXiv
2674:arXiv
2659:(PDF)
2644:(PDF)
2615:S2CID
2597:arXiv
2565:arXiv
2544:arXiv
2523:arXiv
2409:Plume
2340:(PDF)
2299:arXiv
2297:(2).
2253:JSTOR
2016:JSTOR
1656:(PDF)
1627:S2CID
1443:Hydra
1351:arXiv
1098:with
972:the "
859:arXiv
493:torus
440:torus
383:arXiv
60:Field
3205:ISBN
3139:ISBN
3071:ISBN
3037:ISBN
2917:ISBN
2734:ISBN
2499:ISBN
2413:ISBN
2319:link
2195:ISBN
2006:ISBN
1951:ISBN
1937:and
1765:ISBN
1702:2016
1664:2015
1619:PMID
1576:2019
1548:2019
1485:ISBN
1253:and
1237:and
1109:<
920:and
890:and
879:and
748:The
622:, a
516:and
424:US$
360:loop
101:2002
83:1904
3508:of
3372:or
3329:HIV
3171:doi
2968:doi
2909:doi
2607:doi
2476:Zbl
2458:doi
2245:doi
2213:Zbl
2187:doi
2145:JFM
2137:doi
2076:JFM
2068:doi
2032:Zbl
1969:Zbl
1943:doi
1894:JFM
1855:JFM
1823:JFM
1815:doi
1783:Zbl
1609:doi
1605:314
1445:by
1349:on
1299:by
933:ICM
795:'s
730:by
524:of
489:two
475:is
389:of
343:in
3599::
3559:AI
3257:,
3249:,
3215:MR
3213:.
3203:.
3181:MR
3179:.
3167:49
3165:.
3159:.
3137:.
3133:.
3069:.
3065:.
3047:MR
3045:.
3035:.
3025:.
3013:;
2982:.
2976:MR
2974:.
2966:.
2954:12
2952:.
2942:;
2915:.
2879:.
2810:.
2781:.
2724:;
2696:;
2652:10
2650:.
2646:.
2635:;
2613:.
2605:.
2593:12
2591:.
2474:.
2468:MR
2466:.
2454:17
2452:.
2448:.
2411:.
2407:.
2382:.
2363:.
2315:}}
2311:{{
2295:13
2293:.
2289:.
2251:.
2241:68
2211:.
2205:MR
2203:.
2193:.
2143:.
2133:18
2131:.
2125:.
2096:^
2084:MR
2082:.
2074:.
2064:32
2062:.
2056:.
2030:.
2024:MR
2022:.
2014:.
2004:.
1981:^
1967:.
1961:MR
1959:.
1949:.
1941:.
1933:.
1906:^
1884:.
1867:^
1853:.
1847:.
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