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* Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective by Terence Tao, unpublished arxiv. org preprint ( written for mathematicians ).
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Perelman's and proof
Several teams of mathematicians have verified that Perelman's proof is correct.
On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific " Breakthrough of the Year ", the first time this had been bestowed in the area of mathematics.
From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:
* Bruce Kleiner and John W. Lott posted a paper on the arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture.
They initially implied the proof was their own achievement based on the " Hamilton-Perelman theory ", but later retracted the original version of their paper, and posted a revised version, in which they referred to their work as the more modest " exposition of Hamilton – Perelman's proof ".
Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton.
This last part of the proof appeared in Perelman's third and final paper on the subject.
*: Detailed proof, expanding Perelman's papers.
* Lectures on Perelman's proof by T. Tao.
The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7. 4 in the preprint ' Ricci Flow with surgery on three-manifolds '.
This method, with full details leading to the proof of Geometrization, can be found in the exposition by B. Kleiner and J. Lott in ' Notes on Perelman's papers ' in the journal Geometry & Topology.
* Bruce Kleiner and John Lott, Notes on Perelman's Papers ( May 2006 ) ( fills in the details of Perelman's proof of the geometrization conjecture ).
1, 57-78 ( expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture )
" On 22 December 2006, the journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific " Breakthrough of the Year ", the first such recognition in the area of mathematics.
It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture, as well as in the proof of the Differentiable sphere theorem by Brendle and Schoen.
Perelman's and perspective
Perelman's work provides a political economy perspective that takes creativity, intellectual property and corporate power into account.
Perelman's and by
* Perelman's Song, by Tina S. Chang, listed on Kasman's Mathematical Fiction website, appeared Math Horizons.
Also containing proofs of Perelman's Theorem 7. 4, there is a paper of Morgan and Tian, another paper of Kleiner and Lott, and a paper by Cao and Ge.
Another caricature of Cerf, named Harry Hubris and portrayed by Bert Lahr, appears in Perelman's 1962 play The Beauty Part.
New World faces a major financial slump and the company began to restructure ; first, the Marvel Entertainment Group was sold to financier Ronald Perelman's Andrews Group, in 1989 then New World Entertainment was subsequently acquired by Perelman later in the year.
Perelman's and .
Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined.
All three groups found that the gaps in Perelman's papers were minor and could be filled in using his own techniques.
A special case of Perelman's theorems about Ricci flow with surgery is given as follows.
* Bruce Kleiner ( Yale ) and John W. Lott ( University of Michigan ): " Notes & commentary on Perelman's Ricci flow papers ".
In 1980, Ronald Perelman, the son of a wealthy Philadelphia businessman, and future " corporate raider " having made several small but successful buyouts, acquired MacAndrews & Forbes, a distributor of licorice extract and chocolate, that Perelman's father had tried and failed to acquire 10 years earlier.
Using the Pantry Pride subsidiary of his holding company, MacAndrews & Forbes Holdings, Perelman's overtures were rebuffed.
After receiving a higher offer from a white knight, private equity firm Forstmann Little & Company, Perelman's Pantry Pride finally was able to make a successful bid for Revlon, valuing the company at $ 2. 7 billion.
Under Perelman's control, Revlon sold 4 divisions: two of which were sold for $ 1 billion, its vision care division was sold for $ 574 million and its National Health Laboratories division was spun out to the public market in 1988.
He lost out at the last minute to Ronald Perelman's slightly higher bid.
There are now several different proofs of Perelman's Theorem 7. 4, or variants of it which are sufficient to prove geometrization.
Perelman's and for
There is the paper of Shioya and Yamaguchi that uses Perelman's stability theorem and a fibration theorem for Alexandrov spaces.
A Chinese drinking bird toy dating back to 1910s ~ 1930s named insatiable birdie is described in Yakov Perelman's Physics for Entertainment.
( The previous record is $ 70 million for Ron Perelman's Palm Beach estate in 2004 .).
Perelman's and ).
After appearing as an exiled actress in America So Beautiful in 2001, Aghdashloo played opposite Ben Kingsley and Jennifer Connelly in director Vadim Perelman's House of Sand and Fog ( 2003 ).
proof and Poincaré
In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by Richard Hamilton.
* Huai-Dong Cao and Xi-Ping Zhu published a paper in the June 2006 issue of the Asian Journal of Mathematics with an exposition of the complete proof of the Poincaré and geometrization conjectures.
* John Morgan and Gang Tian posted a paper on the 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.
In December 2006, the journal Science honored the proof of Poincaré conjecture as the Breakthrough of the Year and featured it on its cover.
* Clay Mathematics Institute has a description of the Poincaré Conjecture as a Millennium Problem by John Milnor ( SUNY Stony Brook ) as well as a press release about the proof.
He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961 ; in 1962 he generalized the ideas in a 107 page paper that established the h-cobordism theorem.
In 2002 Dunwoody put forward a proposed proof of the Poincaré conjecture.
The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.
For the proof of the conjectures, see the references in the articles on geometrization conjecture or Poincaré conjecture.
) Lefschetz's original proof proceeded by normal functions, which were introduced by Henri Poincaré.
The proof uses the standard filtration on the universal enveloping algebra and the Poincaré – Birkhoff – Witt theorem.
A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893.
In the first two complements to Analysis Situs, Poincaré gave a new proof in terms of dual triangulations.
The fact that this is an isomorphism of chain complexes is a proof of Poincaré Duality.
In 1908 he believed that he had found a proof of the Poincaré conjecture, but Tietze found an error.
In 2002 Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard Hamilton's Ricci flow, an idea belonging to the field of geometric analysis.
He is best known for having discovered the Ricci flow and suggesting the research program that ultimately led to the proof, by Grigori Perelman, of the Thurston geometrization conjecture and the solution of the Poincaré conjecture.
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