Grigori Perelman
Grigori «Grisha» Yákovlevich Perelmán (Russian: Григорий Яковлевич Перельман) (Leningrad, USSR, June 13, 1966) is a mathematician Russian who has made historical contributions to Riemannian geometry and geometric topology. In particular, he has proved Thurston's geometrization conjecture, thereby solving the famous Poincaré conjecture, proposed in 1904 and considered one of the most important and difficult-to-prove mathematical hypotheses.
In August 2006, Perelmán was awarded the Fields Medal for "his contributions to geometry and his revolutionary ideas in the analytical and geometric structure of the Ricci flow". The Fields Medal is considered the highest honor a mathematician can receive. However, he declined both the award and attending the International Congress of Mathematicians.
On March 18, 2010, the Clay Institute of Mathematics announced that Perelmán met the criteria to receive the first million-dollar millennium problem prize for solving the Poincaré conjecture. award, stated:
"I don't want to be exposed as an animal in the zoo. I'm not a math hero. I'm not even that successful. That's why I don't want everyone looking at me."
He is considered one of the most intelligent men in the world.
Biography
Early years and family
Grigori Perelmán was born in Leningrad (now Saint Petersburg) on June 13, 1966 into a Jewish family.
His father, Jacob Perelmán, was an electrical engineer (unlike a common mistake, Jacob Isidorovich Perelmán, known as a popularizer of physics, mathematics and astronomy, is not the father of Grigori Yakovlevich Perelmán), in 1993 he emigrated to Israel. Her mother, Lyubov Leybovna Steingolts, remained in St. Petersburg, working as a mathematics teacher in vocational schools. It was she, who played the violin, who instilled in Grigori a love of classical music, which later led him to become a virtuoso violinist.
Grigori Perelmán has a younger sister, Elena (born 1976), also a mathematician, graduated in 1998 from the University of Saint Petersburg, who in 2003 defended her doctoral thesis at the Weizmann Institute of Sciences in Rehovot and since 2007, works as a programmer in Stockholm.
School
Perelmán attended secondary school outside Leningrad until ninth grade, then transferred to a specialized school, Physics and Mathematics School No. 239. He played table tennis well and also attended a music school. From the fifth grade, he studied at the Center for Mathematics, in the Palace of Pioneers, under the direction of Associate Professor of the Russian State Pedagogical University Sergey Rukshin, whose students won numerous prizes in mathematical competitions.
In 1982, as part of a team of Soviet schoolchildren, he won a gold medal at the International Mathematical Olympiad in Budapest, an international competition for high school students, receiving a perfect score for the complete solution of all problems.
In the early 1980s, at the age of 13, he achieved the highest score in the prestigious organization Mensa for people with high IQs.
University studies
Because of his academic excellence in school, he entered the faculty of mathematics and mechanics at Leningrad State University without exams, one of the leading universities in the former Soviet Union. He won the Mathematics Olympiads for the university, for the city of Leningrad and for the Soviet Union. He received the Lenin Scholarship for academic excellence and graduated from the university with honors.
Postgraduate
His postgraduate studies were conducted under the supervision of Aleksandr Danilovich Aleksandrov and Yuri Dmitrievich Burago at the Leningrad branch of the Steklov Institute of Mathematics of the Russian Academy of Sciences. In 1990 he defended his thesis on the topic « Saddle surfaces in Euclidean spaces » and received the degree of Candidate of Sciences (the Russian equivalent of a doctorate).
Research
After graduation, Perelmán continued to work in Leningrad, already as a senior researcher at the Steklov Institute of Mathematics. In the early 1990s, he worked at various universities in the US In 1992, he was invited to spend semesters at New York University and Stony Brook University. In 1993, he accepted a two-year scholarship to the University of California, Berkeley.
In 1996, he returned to St. Petersburg, to the Steklov Institute of Mathematics, where he worked alone on the proof of the Poincare conjecture.
Between 2002 and 2003, Perelmán published three of his famous articles on the Internet, in which he briefly described the original method for proving the Poincaré conjecture:
- Entropy formula for the flow of Ricci and its geometric applications.
- Ricci flux with surgery in three-dimensional varieties.
- Finite decadence time for Ricci flow solutions in some three-dimensional varieties.
The appearance of Perelman's first paper on the entropy formula for Ricci flow on the Internet caused an immediate international sensation in scientific circles. In 2003, Grigory Perelman accepted an invitation to visit several American universities, where he presented a series of reports on his work on the proof of the Poincaré hypothesis. In the United States, Perelmán spent much time explaining his ideas and methods, both in public lectures organized for him and during personal meetings with various mathematicians. After his return to Russia, he answered numerous questions from his foreign colleagues by email.
Between 2004 and 2006, three independent groups of mathematicians undertook to verify Perelman's results:
- Bruce Kleiner, John Lott, University of Michigan;
- Zhu Xiping, University of Sun Yat-sen, Cao Huaydong, University of Leahai;
- John Morgan, Columbia University, Gan Tian, Massachusetts Institute of Technology.
The three groups concluded that Poincaré's hypothesis was totally proven, but Chinese mathematicians, Zhu Xiping and Cao Huaydong, along with their teacher Yau Shintun, tried the plagiarism, stating that they had found "complete evidence" Some time later, they retracted their statements.
In December 2005, Grigory Perelmán resigned his position as a leading researcher in the mathematical physics laboratory, he resigned from the Steklov Mathematics Institute and almost completely broke contacts with his colleagues.
In September 2011, it was learned that the mathematician refused to accept the offer to become a member of the Russian Academy of Sciences.
Geometrization and Poincaré conjectures
Until 2002, Perelmán was better known for his work in comparison theorems in Riemannian geometry. Among his notable achievements was the demonstration of Soul's conjecture.
The problem
Poincaré's conjecture, proposed by the French mathematician Henri Poincaré in 1904, was the most famous open problem in the topology. In relatively simple terms, the conjecture indicates that if a closed three -dimensional variety is sufficiently similar to a sphere in the sense that each loop in the variety can be transformed into a point, then it will be considered that it is really just a three -dimensional sphere. For some time, it has been known that the analogous result is true in major dimensions; However, the case of three -dimensional varieties has turned out to be the most difficult of all because, speaking colloquially, when a three -dimensional variety is topologically manipulated, there are very few dimensions to move "problematic regions" out of the way without interfering with something else.
In 1999, the Clay Institute announced Millennium's winning problems: a prize of one million dollars for the demonstration of one of the conjectures, including Poincaré's. It was accepted by all that a successful demonstration of the conjecture of Poincaré would constitute a milestone in the history of mathematics, comparable to the demonstration of Andrew Wiles of the last Fermat theorem or even greater reach.
The demonstration of Perelmán
In November 2002, Perelmán wrote in the Arxiv the first of a series of free access articles in which he claimed to have described a demonstration of the geometrization conjecture, a result that includes the conjecture of Poincaré as a particular case.
Perelmán modified the Richard Hamilton program for the demonstration of the conjecture, in which the central idea was the notion of Ricci's flow. The basic idea of Hamilton is to formulate a "dynamic process" in which a given three -dimensional variety is transformed geometrically, so that this distortion process is governed by a differential equation analogous to the heat equation. The heat equation describes the behavior of scalar quantities such as temperature; She states that high temperature concentrations will be dispersed until a uniform temperature is reached along the object. Similarly, Ricci's flow describes the behavior of a tensioning amount, Ricci's curvature tensioner. Hamilton's hope was that, under the flow of Ricci, the concentrations of great curvature were dispersed until a uniform curvature over the entire three -dimensional variety. If this is so, starting with any three -dimensional variety and if the magic of the ricci flow is used, some "normal form" would finally be obtained. According to William Thurson, this normal form must be a small number of possibilities, each with a different flavor of geometry called Thurston models geometries.
This is similar to formulating a dynamic process that gradually "disturbs" a given square matrix and that, with certainty, will result after a finite time in its rational canonical form.
Hamilton's idea had attracted much attention but no one had managed to demonstrate that the process would not "hang" develop "singularities"... until Perelmán's articles sketched a program to overcome these obstacles. According to Perelmán, a modification of the standard RICCI flow, called Ricci flow with surgery , can systematically remove singular regions as they develop, in a controlled way.
It is known that singularities (including those that occur, vaguely speaking, after the flow has occurred during an infinite amount of time) they must occur in many cases. However, mathematicians hope that, assuming that the geometrization conjecture is true, any singularity that develops in a finite time essentially "tightening" throughout certain spheres that correspond to the decomposition in cousins of 3-variety. If this is so, any singularities of "infinite time" must result from certain collapse pieces of JSJ decomposition. Perelmán's work apparently demonstrates this statement and thus demonstrates the geometrization conjecture.
Verification
Since 2003, Perelmán's program has attracted more and more attention from the mathematical community. In April 2003, he accepted an invitation to visit the Massachusetts Institute, Princeton University, Stony Brook University, Columbia University and Harvard University, where he gave a series of talks about his work. However, then, then From his return to Russia, it has been said that he has gradually stopped responding to the emails of his colleagues.
On May 25, 2006, Bruce Kleiner and John Lott, both from the University of Michigan, placed an article in the Arxiv that claims to add the details of Perelmán's demonstration of the geometrization conjecture.
In June 2006, the Asian Mathematics Magazine ( Asian Journal of Mathematics ) published an article by Professor Xi-Ping Zhu, from the University of Sun Yat- Sen, in China, and Huai-Dong Cao, of the University of Lehight in Pennsylvania, USA This article aimed "to give the last touches to the complete demonstration of Poincaré's conjecture."
The true magnitude of Zhu and Cao's contribution, as well as the ethics of Yau's intervention, have been controversial. Yau is both editor -in -chief of the Asian Mathematics Magazine as a doctoral advisor of Cao. Sylvia Nasar and David Gruber, in a letter for the The New Yorker , have suggested that Yau tried to be associated, directly or indirectly, with the demonstration of the conjecture and pressed the editors of the magazine to accept Zhu and Cao's article unusually rapidly. Others have wondered if «the little time between the date of presentation... and the date of acceptance for publication »for the magazine was sufficient to allow the article to be" serious reviewed. " However, in relation to the conjecture of Poincaré, the authors also revealed an apparently not reported accusation in the press before the appearance (online) of their article [3]. They wrote:
On April 13, this year, the thirty-one mathematicians of the editorial board of the Asian Journal of Mathematics they received a brief email from Yau and the editor of the magazine informing them that they had three days to comment on an article by Xi-Ping Zhu and Huai-Dong Cao entitled The Hamilton-Perelmán Theory of Ricci Flow: The Poincaré and Geometrization Conjectures (The Hamilton-Perelman theory of the flow of Ricci. Poincaré and geometry conjecturesThat Yau planned to publish in the magazine. The mail did not include a copy of the article, reports of arbitrators or a summary. At least one member of the board asked to see the article, but was told it was not available.
To date, no member of the editorial board of the RAM has objected to this fact nor has there been any explanation for the change of title to A Complete Proof of the Poincaré and Geometrization Conjectures: Application of the Hamilton-Perelmán Theory of the Ricci Flow (A complete proof of the Poincaré and geometrization conjectures. Application of the Hamilton-Perelmán theory of the Ricci flow). Yau responded by saying that the article had been peer-reviewed in the usual way, and that the journal "has very high standards". Cao has said: "Hamilton and Perelmán have done the most fundamental work. They are the giants and our heroes. In my mind there is no doubt that Perelmán deserves the Fields Medal. We only follow the tracks of Hamilton and Perelmán and explain the details. I hope that everyone who reads our article agrees that we have given a fair account. Cao also defended Yau, saying that Yau had noted that Perelmán deserved the Fields Medal, The New Yorker reporters added.
In July 2006, John Morgan of Columbia University and Gang Tian of the Massachusetts Institute of Technology posted a paper on the arXiv titled Ricci Flow and the Poincaré Conjecture (The Ricci flow and the Poincaré conjecture). In it, they claim to provide a "detailed proof of the Poincaré conjecture". On August 24, 2006, Morgan gave a talk at the ICM in Madrid on the Poincaré conjecture.
The above work seems to show that Perelmán's sketch can indeed be expanded into a full proof of the geometrization conjecture.
Dennis Overbye of the New York Times has said that "there is a growing sense, a cautious optimism that mathematicians have finally reached a milestone not just for mathematics, but for human thought." Nigel Hitchin, Professor Professor of Mathematics at Oxford University, has said that "I think for many months or even years people have been saying that they were convinced by the argument. I think it's a done deal."
The Fields Medal and the Millennium Prize
In May 2006, a committee of nine mathematicians voted to award Perelmán a Fields Medal for his work on the Poincaré conjecture. The Fields Medal is the highest award in mathematics; two to four medals are awarded every four years.
Sir John Ball, president of the International Mathematical Union, approached Perelman in Saint Petersburg in June 2006 to persuade him to accept the prize. After 10 hours of persuasion over two days, he gave up. Two weeks later, Perelmán summarized the conversation like this: «He proposed three alternatives: accept and come; accept and do not come, and we will send you the medal later; third, do not accept or come. From the beginning I told him that he had chosen the third ». He went on to say that the award "was completely irrelevant to me. Everyone understands that if the proof is correct, then no further acknowledgment is needed."
On August 22, 2006, Perelmán was publicly offered the medal at the International Congress of Mathematicians in Madrid, "for his contributions to geometry and his revolutionary ideas on the analytical and geometric structure of the Ricci flow." He did not attend the ceremony and declined the medal.
He had previously turned down a prestigious prize from the European Mathematical Society, reportedly saying that he felt the prize committee was not qualified to evaluate his work, even positively.
Perelmán should also receive a share of the millennium prize (probably shared with Richard Hamilton). Although you have not sought formal publication of your proof in a peer-reviewed mathematics journal, as the prize rules require, many mathematicians believe that the scrutiny your sketch has been subjected to exceeds the peer review implicit in peer review. normal.[citation needed] The Clay Mathematics Institute has explicitly said that the awarding council may change the formal requirements, in which case Perelmán would be eligible to receive part of the prize.[citation required]
On March 18, 2010, Perelman won the millennium award for solving the problem. He also declined this award. According to the Interfax news agency, Perelmán believed the award was unfair, since "his contribution to solving the Poincaré conjecture was no greater than that of the mathematician Richard Hamilton". I accept the prize until it is offered."
Retirement from mathematics
Since spring 2003, Perelmán has not worked at the Steklov Institute. His friends are said to have claimed that he currently finds mathematics a painful topic of discussion; some even say that he has given up mathematics altogether. According to a recent interview, Perelman is currently unemployed, living with his mother, Lubov, in a poor apartment in Saint Petersburg. It is also said that he is not really disappointed in the mathematics, but rather immersed in the Galilean idea that «The humble reasoning of one is worth more than the authority of thousands»; thus, he has preferred to isolate himself, continue studying and not submit to arbitrary or mathematical authorities. [citation needed ]
Although Perelmán says in an article in The New Yorker that he is disappointed in the ethical standards of the field of mathematics, the article implies that Perelmán is referring particularly to Yau's efforts to lessen his role in the demonstration and praise the work of Cao and Zhu. Perelmán has said that «I cannot say that I am outraged. Other people do worse things. Of course, there are many mathematicians who are more or less honest. But of them, almost all are conformists. They are more or less honest, but they tolerate those who are not honest ».He has also said that «it is not people who break ethical standards that are considered strange. It's people like me who are isolated."
This, combined with the possibility of being awarded a Fields Medal, caused him to give up professional mathematics. He has said that « when he was not conspicuous, he had a choice. Even to do something ugly” (a scandal about the lack of integrity of the mathematical community) “or, if he didn't do this kind of thing, to be treated like a pet. Now that I've become a very conspicuous person, I can't be a pet and say nothing. That's why I had to resign".
Professor Marcus du Sautoy of the University of Oxford has said that “it has become somewhat isolated from the mathematical community. He has become disillusioned with mathematics, which is very unfortunate. He is not interested in money. The big prize for him is to prove his theorem. »
He is currently retired from mathematics. The last news that was had of him was a photo of him taken on June 20, 2007 in the Saint Petersburg metro. However, in April 2011 he gave an interview.
Literary character
The figure of Perelmán has inspired various literary works, such as the novel Perelmán's conjecture (2011) by Juan Soto Ivars.