Russian mathematician, Grigory Perelman, may have solved the Poincare Conjecture, one of the toughest problems in the world. He could win $1 million but has done nothing to get it.
The conjecture tries to explain the behaviour of multi-dimensional shapes in space. Clay Mathematics Institute based in Massachusetts offered the prize to anyone who could solve the problem.
Perelman from St. Petersburg has posted his results on the Internet and did not talk to anybody about it, showing no interest in the money, Reuters quoted Keith Devlin, Professor of Mathematics at Stanford University in California, as saying.
“There is good reason to believe that Perelman’s approach is correct. But there won’t be a golden moment when he is suddenly accepted as being right. There will just be a drift in that direction,” Devin said at the annual meeting of the British Association for the Advancement of Science.
Mathematicians are still struggling to understand whether Perelman has solved the conjecture.
By Jascha Hoffman, Globe Correspondent, 12/30/2003
BERKELEY, Calif. -- A reclusive Russian mathematician appears to have answered a question that has stumped mathematicians for more than a century
After a decade of isolation in St. Petersburg, over the last year Grigory Perelman posted a few papers to an online archive. Although he has no known plans to publish them, his work has sent shock waves through what is usually a quiet field.
At two conferences held during the last two weeks in California, a range of specialists scrutinized Perelman's work, trying to grasp all the details and look for potential flaws.
If Perelman really has proved the so-called Poincare Conjecture, as many believe he has, he will become known as one of the great mathematicians of the 21st century and will be first in line for a $1 million prize offered by the Clay Mathematics Institute in Cambridge.
Colleagues say Perelman, who did not attend the California conferences and did not respond to a request for comment, couldn't care less about the money, and doesn't want the attention. Known for his single-minded devotion to research, he seldom appears in public; he answers e-mails from mathematicians, but no one else.
"What mathematicians enjoy is the chase of really difficult problems," said Hyam Rubinstein, a mathematician who came from Australia to attend meetings at the Mathematical Sciences Research Institute in Berkeley and the American Institute of Mathematics in Palo Alto, Calif., hoping to better understand Perelman's solution. "This problem is like the Mount Everest of math conjectures, so everyone wants to be the first to climb it."
The Poincare Conjecture, named after the Frenchman who proposed it in 1904, is the question that essentially founded the field of topology, the "rubber-sheet geometry" that looks at the properties of surfaces that don't change no matter how much you stretch or bend them.
To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe.
Dozens of the best mathematicians of the last century tried with all kinds of approaches to solve the conjecture. Some thought they had it for months, even years, but counter-examples and flaws just kept springing up. Simply-stated but elusive to prove -- like Fermat's Last Theorem -- this conjecture has spurred the development of whole branches of mathematics.
A decade ago, after some work in the United States that colleagues described as "brilliant," Perelman gave up a promising career to work in seclusion in St. Petersburg. Although he appears occasionally, most recently for lectures at the Massachusetts Institute of Technology and several other US schools last spring, he keeps a very low profile.
Even in mathematical circles, surprisingly little is known about him, and those who know him often don't want to speak publicly about his work.
At any rate, he seems to have used his time alone wisely. While working out the Poincare Conjecture, Perelman also seems to have established a much stronger result, one that could change many branches of mathematics. Called the "Geometrization Conjecture," it is a far-reaching claim that joins topology and geometry, by stating that all space-like structures can be divided into parts, each of which can be described by one of three kinds of simple geometric models. Like a similar result for surfaces proved a century ago, this would have profound consequences in almost all areas of mathematics.
As the foundation for his proof, Perelman used a method called Ricci flow, invented in the mid-1980s by Columbia University mathematician Richard Hamilton, which breaks a surface into parts and smooths these parts out, making them easier to understand and classify.
Although some mathematicians find it disturbing that Poincare's simple question could have such a complicated answer, Hamilton is not worried. After so many failed proofs, he said, "no one expected it to be easy."
Hamilton calls Perelman's work original and powerful -- and is now running a seminar at Columbia devoted to checking Perelman's proof in all its detail.
If the proof is vetted, the Clay Mathematics Institute may face a difficult choice. Its rules state that any solution must be published two years before being considered for the $1 million prize. Perelman's work remains unpublished and he appears indifferent to the money. Hamilton, on the other hand, did the foundational work on which the proof is based -- but that was over a decade ago. And, as with any major finding, many people have contributed in some degree.
Huge financial prizes raise the stakes for assigning credit for major proofs like this one. For the time being, however, researchers are sharing their approaches with a sense of openness. And the mood is one of cautious optimism that Perelman's approach, even if flawed, will eventually be the one that works.
It takes years for a solution to make the leap from being just another claim to actually being considered "true." Perelman's work will be digested by a wide range of mathematicians in the next few years, said University of California at Davis mathematician Joel Hass. Steps that Perelman pushed through by brute force will be replaced with simpler methods, and his work will be integrated into other fields, Hass said.
And while the equivalent of the Poincare conjecture has already been proven for dimensions four and up, no one yet has any idea how to classify all the spaces that appear in higher dimensions. This state of ignorance is what prods mathematicians to keep working.
"It's interesting how a really good problem can sometimes be much better than a really good answer," Rubinstein said with a grin.
J'avais lu ce truc sur Yahoo-Actualités il y a 2 ou 3 jours, et j'avais hesité à le poster ici, croyant que ça n'interesserait personne.
Je te remercie de me demontrer que j'avais tort.
Citation:
La conjecture mathématique de Poincaré résolue pour la gloire ?
EXETER, Angleterre (Reuters) - Un Russe pourrait avoir résolu un des problèmes mathématiques les plus ardus au monde, la conjecture de Poincaré, décrochant ainsi un prix d'un million de dollars dont il semble n'avoir cure.
Si Grigori Perelman, originaire de Saint-Pétersbourg, a effectivement résolu ce problème de topologie relatif aux objets tridimensionnels, il pourrait remporter la récompense offerte par l'institut de mathématiques Clay, dans le Massachusetts.
Mais l'homme a simplement publié ses résultats sur internet, laissant le soin à la communauté scientifique d'infirmer ou de confirmer ses travaux.
"Il est raisonnable de penser que l'approche de Perelman est correcte. Mais le problème est qu'il ne veut en parler avec personne et n'a pas manifesté le moindre intérêt pour l'argent", a déclaré Keith Devlin, docteur en mathématiques à l'université de Stanford (Californie).
Qu'est-ce que la conjecture de Poincaré?
Explication:
Citation:
Prenez une pomme, et imaginez un ruban autour de cette pomme. En faisant glisser le ruban tout doucement, il est possible de le comprimer en un point de la pomme, sans couper le ruban ni le faire quitter la surface de la pomme. Prenez maintenant un anneau, et imaginez un ruban enfilé autour de l'anneau. Cette fois, il est impossible, sans couper le ruban ou l'anneau, de réduire juste par glissement et compression le ruban en un point. En langage mathématique, on dit que la pomme est une surface simplement connexe, alors que l'anneau ne l'est pas.
Poincaré savait il y a un peu moins d'un siècle que cette propriété caractérisait topologiquement la sphère parmi les surfaces de l'espace. Autrement dit, si une surface (fermée) de l'espace est simplement connexe, elle peut être déformée continûement en la sphère (une déformation continue peut être assimilée à ce que l'on est capable de réaliser avec de la pâte à modeler, sans couper une boule de pâte en deux). Poincaré posa alors en 1904 la question suivante : est-ce que cette propriété caractérise encore la sphère 3-dimensionnelle dans l'espace à 4 dimensions, ou plus généralement la sphère n-dimensionnelle dans l'espace à (n+1) dimensions. En langage plus mathématique :
Est-ce qu'une variété compacte de l'espace à n+1 dimensions est homéomorphe à la sphère n-dimensionnelle?
Bien sûr, il faut être un petit peu mathématicien pour comprendre ce que peut être la sphère dans l'espace à 4, 5 ou plus, dimensions. Et bizarrement, ce problème a été plus simple à résoudre pour les valeurs de n supérieure à 4. Il fut en effet résolu par Zeeman, Stallings et Smale pour n>4 vers 1961-1962, puis par Freedman en 1982 pour n=4. Ce fut plus simple à résoudre certes que le cas n=3, mais cela valut tout de même à Freedman la médaille Fields!
La conjecture initiale de Poincaré (le cas n=3) restait donc irrésolu, et le Clay Mathematics Institute la choisit en l'an 2000 parmi les 7 problèmes du millénaire dont la résolution est primée 1 million de dollars. Il semblerait que le mathématicien russe Grigori Perelman, du prestigieux Steklov Institute of Mathematics de Saint-Petersbourg, ait produit récemment une preuve de la conjecture de Poincaré.
Plus précisément, Perelman aurait résolu une conjecture encore plus générale que la conjecture de Poincaré, à savoir la conjecture de Thurston, dans deux articles qui circulent actuellement comme preprint. Ces articles s'annoncent toutefois, selon des experts du sujet, très techniques à lire, et il faudra sans doute de nombreux mois avant que la preuve soit avérée (ou non). Selon les règles du Clay Mathematics Institute, la preuve de Perelman doit pouvoir passer 2 ans sans être contestée avant que le prix puisse lui être remis.
Evidemment, le CONDITIONNEL est utilisé à chaque fois...
On dit que Perelman aurait démontré ..mais ça date de 2003 ça...
C'est dû au fait qu'il faut en moyenne 2 ans pour que la communauté des mathématiciens decide de si la demo est correcte ou pas.
Plusieurs demos ont été proposées par le passé (dont une par Poincaré lui-même), et toutes se sont avérées fausses.
Faut être très prudent avec ces trucs-là, car
Citation:
Ces articles s'annoncent toutefois, selon des experts du sujet, très techniques à lire, et il faudra sans doute de nombreux mois avant que la preuve soit avérée (ou non). Selon les règles du Clay Mathematics Institute, la preuve de Perelman doit pouvoir passer 2 ans sans être contestée avant que le prix puisse lui être remis.
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