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Grigori Y. Perelman

"Why did we have to struggle with the Poincare conjecture for so many years? To put it in a nutshell, the essence of it is the following. If a three-dimensional surface is reminiscent of a sphere, then it can be spread into a sphere. It is known as the Formula of the Universe because it is highly important in researching complicated physical processes in the theory of creation. The Poincare conjecture also gives an answer to the question about the shape of the Universe."

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Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. Grigori Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966.

Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school math training program. His father also had a major influence in developing his son's problem solving skills. Speaking about his father, Perelman said :

"He gave me logical and other maths problems to think about. He got a lot of books for me to read. He taught me how to play chess. He was proud of me."

His mother also helped him to develop his mathematical skills and by the age of ten ,he started taking parts in district level competition  and later he also joined a mathematical club which worked towards increasing the mathematical talents of the students.

In January 1982 Perelman was chosen as a potential member of the 1982 Soviet Mathematical Olympiad team. He attended a selection session in Chernogolovka, about 80 km north of Moscow, Perelman excelled and the next step was a two-day session in Odessa in April when they were given harder problems than those expected at the Olympiad competition. Perelman achieved full marks as he did at the International Mathematical Olympiad competition in Budapest in July. He received a gold medal and a special prize for achieving a perfect score. Being a member of the Soviet team gave Perelman automatic entry to university.

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Perelman entered Leningrad State University in autumn 1982. There he was particularly influenced by Viktor Zalgaller and Aleksandr Danilovic Aleksandrov. During his undergraduate years he assisted Rukshin as a mathematics tutor, going to summer camps, but his incredibly high standards gave even outstanding students an almost impossible time. Eventually Rukshin had to stop Perelman assisting at the summer camps. His university work, however, was exceptional and he graduated in 1987. He had already published a number of papers: Realization of abstract k-skeletons as k-skeletons of intersections of convex polyhedra in R2k-1 (Russian) (1985); (with I V Polikanova) A remark on Helly's theorem (Russian) (1986); A supplement to A D Aleksandrov's, On the foundations of geometry (Russian) (1987) in which Perelman discussed the equivalence of a Pasch-style axiom of Aleksandrov and some of its consequences; and On the k-radii of a convex body (Russian) (1987).

Being a jew, he had some difficulty in getting into the Leningrad branch of the Steklov Mathematics Institute. With the help of Aleksandr Danilovic Aleksandrov, Perelman  was accepted under Yuri Burago as his advisor. Perelman defended his thesis Saddle Surfaces in Euclidean Spaces in 1990. He had already published one of the main results of the thesis in "An example of a complete saddle surface in R4 with Gaussian curvature bounded away from zero (Russian) (1989)."

Burago contacted Mikhael Leonidovich Gromov who had been a professor at Leningrad State University, but was at this time a permanent member of the Institute des Hautes Études Scientifiques outside Paris. He explained to Gromov that he had an outstanding student and asked if an invitation could be issued for him to spend time at IHES. The invitation allowed Perelman to spend several months at IHES working with Gromov on Aleksandrov spaces. Perelman's first major paper, written jointly with Burago and Gromov, was A D Aleksandrov spaces with curvatures bounded below (1992). Tadeusz Januszkiewicz begins a review as follows:
"This is an important paper in many respects. It contains a careful and fairly detailed discussion of basic facts of the theory, including various equivalent forms of definitions. It recognizes that the home of various important theorems of Riemannian geometry is the theory of Aleksandrov spaces, that both statements and proofs become more satisfactory (but not necessarily easier) in this context, and other theorems emerge naturally to complete the picture. It develops useful tools for studying Aleksandrov spaces with curvature bounded below in full generality. Finally, it contains an ample discussion of further results and open problems."

After visiting the IHES near Paris, Perelman returned to the Steklov Mathematics Institute in Leningrad but, thanks to Gromov, Perelman was invited to the United States to talk at the 1991 Geometry Festival held at Duke University in Durham, North Carolina. He lectured on the work which he had done on Aleksandrov spaces with Burago and Gromov (which had not been published at that time). In 1992 Perelman was invited to spend the autumn semester at the Courant Institute, New York University, on a postdoctoral fellowship, and the spring 1993 semester at Stony Brook, a campus of the State University of New York, again funded by a fellowship.

During the time, he was in the United States, he started thinking about the Poincare Conjecture and attended a lecture by Richard Hamilton. When Perelman was going to lectures at the Institute for Advanced Study he attended a lecture there by Hamilton and talked with him after the lecture. Perelman recalled :
"I really wanted to ask him something. He was smiling, and he was quite patient. He actually told me a couple of things that he published a few years later. He did not hesitate to tell me. Hamilton's openness and generosity -- it really attracted me. I can't say that most mathematicians act like that. I was working on different things, though occasionally I would think about the Ricci flow. You didn't have to be a great mathematician to see that this would be useful for geometrization. I felt I didn't know very much. I kept asking questions."

When he was a Miller fellow at Berkeley, Perelman attended some further lectures by Hamilton and he began to understand why Hamilton could not make any further progress towards proving the Poincaré Conjecture using the Ricci flow.

While he was in the United States, Perelman received several requests asking him to apply for professorships. These came from top institutions such as Stanford and Princeton. He was offered a full professorship, without making any application, by Tel Aviv University in Israel, but he turned down all the offers and returned to the St Petersburg branch of the Steklov Mathematics Institute after his Miller fellowship came to an end in the summer of 1995. Basically he was able to live on the savings he had made from the money paid to him in the United States which was quite considerable since he had lived exceptionally frugally. He refused to accept a European Mathematical Society prize in 1996. Perelman had realised that Hamilton was making no progress with the Poincaré Conjecture when he read a paper Hamilton published in 1995 and, in the following year, he wrote to Hamilton explaining that he might have a way round the problem and offering to collaborate with him. When he received no reply, Perelman seems to have decided to work on solving the Poincaré Conjecture alone.

On 11 November 2002, Perelman put his paper The Entropy Formula for the Ricci Flow and Its Geometric Applications on the web. Although he did not claim in the paper to be able to solve the Poincaré Conjecture, when experts in the subject read it they realised that he had made the breakthrough necessary to solve the Conjecture.

At the end of April 2002 and, in July, he put Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, the third installment of his work, on the web. It took some time for experts in the field to convince themselves that Perelman had solved the Poincaré Conjecture and a little longer to work through the details to see that he had also solved the Thurston Geometrization Conjecture. He continued working at the Steklov Mathematics Institute in St Petersburg where he was promoted to Senior Researcher. In December 2005 he resigned, saying that he was disappointed in mathematics and wanted to try something else. In August 2006 he was awarded a Fields medal:

   "For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow."
In March 2010 the Clay Mathematics Institute offered Perelman the millenium prize for the solution of the Poincaré Conjecture. In July 2010 Perelman refused to accept the million dollars, saying:

"I do not like their decision, I consider it unfair. I consider that the American mathematician Hamilton's contribution to the solution of the problem is no less than mine. "

When he was awarded the Fields Medal and invited to accept the medal, he declined and is quoted to have said:

"Everybody understood that if the proof is correct, then no other recognition is needed. I'm not interested in money or fame. I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me".

Nevertheless, on 22 August 2006, Perelman was publicly offered the medal at the International Congress of Mathematicians in Madrid "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." He did not attend the ceremony, and declined to accept the medal, making him the first and only person to decline this prestigious prize.

He had previously turned down a prestigious prize from the European Mathematical Society, allegedly saying that he felt the prize committee was unqualified to assess his work, even positively.

On 18 March 2010, Perelman was awarded a Millennium Prize for solving the problem. On June 8, 2010, he did not attend a ceremony in his honor at the Institut Océanographique, Paris to accept his $1 million prize. According to Interfax, Perelman refused to accept the Millennium prize in July 2010. He considered the decision of Clay Institute unfair for not sharing the prize with Richard Hamilton, and stated that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust."

Perelman's proof was rated one of the top cited articles in Math-Physics in 2008.

Perelman is quoted in an article in The New Yorker saying that he is disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to the efforts of Fields medalist Shing-Tung Yau to downplay Perelman's role in the proof and play up the work of Cao and Zhu. Perelman added, "I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them 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 who are regarded as aliens. It is people like me who are isolated."

According to Perelman, every theoretical development of mathematicians has applied relevance.

"Why did we have to struggle with the Poincare conjecture for so many years? To put it in a nutshell, the essence of it is the following. If a three-dimensional surface is reminiscent of a sphere, then it can be spread into a sphere. It is known as the Formula of the Universe because it is highly important in researching complicated physical processes in the theory of creation. The Poincare conjecture also gives an answer to the question about the shape of the Universe."

"I've learned to compute hollowness. Me and my colleagues are studying the mechanisms that fill social and economic hollowness. Hollowness is everywhere, it can be computed, and this opens large opportunities. I know how to control the Universe. Why would I run after a million, tell me?"

 

 


Collected from various online and offline sources.

 


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