What Grigory Perelman really taught us. Mathematician Yakov Perelman: contribution to science

Grigory Yakovlevich Perelman. Born on June 13, 1966 in Leningrad (now St. Petersburg). Russian mathematician who proved the Poincaré conjecture.

By nationality - Jewish.

Father - Yakov Perelman, an electrical engineer, emigrated to Israel in 1993.

Mother - Lyubov Leibovna Shteingolts, worked as a mathematics teacher at a vocational school, after her husband left for Israel, she remained in St. Petersburg.

The younger sister is Elena (born 1976), mathematician, graduate of St. Petersburg University (1998), defended her PhD thesis at the Weizmann Institute in Rehovot in 2003, and has been working as a programmer in Stockholm since 2007.

Some sources mistakenly attribute Perelman to being related to Yakov Isidorovich Perelman, a famous physicist, mathematician and astronomer. But they are just namesakes.

Gregory's mother played the violin and instilled in him a love of classical music from an early age; he graduated from music school. He played table tennis well.

From the 5th grade, Grigory studied at the mathematics center at the Palace of Pioneers under the guidance of RGPU associate professor Sergei Rukshin, whose students won many awards at mathematical Olympiads. In 1982, as part of a team of Soviet schoolchildren, he won a gold medal at the International Mathematical Olympiad in Budapest, receiving full marks for flawlessly solving all problems.

Until the 9th grade, Perelman studied at a high school on the outskirts of Leningrad, then transferred to the 239th physics and mathematics school. I did not receive a gold medal due to a low grade in physical education.

After graduating from school without exams, he was enrolled in the Faculty of Mathematics and Mechanics of Leningrad State University. He won faculty, city and all-Union student mathematical Olympiads. All the years I studied only with “excellent” marks. For academic success he received a Lenin scholarship.

After graduating with honors from the university, he entered graduate school (scientific supervisor - A.D. Aleksandrov) at the Leningrad branch of the Mathematical Institute. V. A. Steklova (LOMI - until 1992; then - POMI).

Having defended his Ph.D. thesis on “Saddle surfaces in Euclidean spaces” in 1990, he remained to work at the institute as a senior researcher.

In 1991, he was awarded the “Young Mathematician” Prize of the St. Petersburg Mathematical Society for his work “Aleksandrov spaces with curvature bounded from below.”

In the early 1990s, Perelman came to the United States, where he worked as a researcher at various universities. He surprised his colleagues with his ascetic lifestyle; his favorite foods were milk, bread and cheese.

In 1994 proved the soul hypothesis(differential geometry). He proved several key statements in the Alexandrov geometry of spaces of curvature bounded below.

In 1996, he returned to St. Petersburg, continuing to work at POMI, where he worked alone on proving the Poincaré conjecture.

In 1996, the European Mathematical Society Prize for Young Mathematicians was awarded, but he refused to receive it.

Entropy formula for Ricci flow and its geometric applications;
- Ricci flow with surgery on three-dimensional manifolds;
- Finite decay time for solutions of Ricci flow on some three-dimensional manifolds.

The appearance on the Internet of Perelman's first article on the entropy formula for the Ricci flow caused an immediate international sensation in scientific circles. In 2003, Grigory Perelman accepted an invitation to visit a number of American universities, where he gave a series of reports on his work to prove the Poincaré conjecture.

In America, Perelman spent a lot of time explaining his ideas and methods, both in public lectures organized for him and during personal meetings with a number of mathematicians. After his return to Russia, he answered numerous questions from his foreign colleagues by email.

In 2004-2006, three independent groups of mathematicians were involved in checking Perelman’s results:

1. Bruce Kleiner, John Lott, University of Michigan;
2. Zhu Xiping, Sun Yat-sen University, Cao Huaidong, Lehigh University;
3. John Morgan, Columbia University, Gan Tian, ​​Massachusetts Institute of Technology.

All three groups concluded that the Poincaré conjecture was completely proven, but Chinese mathematicians Zhu Xiping and Cao Huaidong, along with their teacher Yau Shintong, attempted plagiarism by claiming that they had found a "complete proof". They later retracted this statement.

In December 2005, Grigory Perelman resigned from his post as a leading researcher at the Laboratory of Mathematical Physics, resigned from POMI and almost completely broke off contacts with colleagues.

In 2006, Grigory Perelman was awarded the international Fields Medal for his solution to the Poincaré conjecture - “For his contribution to geometry and his revolutionary ideas in the study of the geometric and analytical structure of the Ricci flow.” However, he refused it.

In 2007, the British newspaper The Daily Telegraph published a list of “One Hundred Living Geniuses”, in which Grigory Perelman ranks 9th. In addition to Perelman, only 2 Russians were included in this list - Garry Kasparov (25th place) and Mikhail Kalashnikov (83rd place).

In March 2010, the Clay Mathematics Institute awarded Grigory Perelman a US$1 million prize for his proof of the Poincaré conjecture, marking the first time in history that the prize had been awarded for solving one of the Millennium Problems.

In June 2010, Perelman ignored a mathematical conference in Paris, at which the Millennium Prize was supposed to be awarded for proving the Poincaré conjecture, and on July 1, 2010, he publicly announced his refusal of the prize. He motivated as follows: “I refused. You know, I had a lot of reasons in both directions. That's why it took me so long to decide. In short, the main reason is disagreement with the organized mathematical community. I don't like their decisions, I think they are unfair. I believe that the contribution of the American mathematician Hamilton to solving this problem is no less than mine.”

“Simply, the essence of Poincaré’s theory can be stated as follows: if a three-dimensional surface is somewhat similar to a sphere, then it can be straightened into a sphere. Poincaré's statement is called the “Formula of the Universe” because of its importance in the study of complex physical processes in the theory of the universe and because it provides an answer to the question of the shape of the Universe. That’s why they struggled with its proof for so many years. I know how to control the Universe. And tell me, why should I run for a million?”, he said in an interview.

Such a public assessment of the merits of Richard Hamilton by the mathematician who proved the Poincaré conjecture can be an example of nobility in science, since, according to Perelman himself, Hamilton, who collaborated with Yau Shintun, noticeably slowed down in his research, encountering insurmountable technical difficulties.

In September 2011, the Clay Institute, together with the Henri Poincaré Institute (Paris), created a position for young mathematicians, the money for which will come from the Millennium Prize awarded but not accepted by Grigory Perelman.

In 2011, Richard Hamilton and Demetrios Christodoulou were awarded the so-called. The $1,000,000 Shao Prize in Mathematics, also sometimes called the Nobel Prize of the East. Richard Hamilton was awarded for creating a mathematical theory, which was then developed by Grigory Perelman in his work to prove the Poincaré conjecture. Hamilton accepted the award.

In 2011, Masha Gessen’s book about the fate of Perelman, “Perfect Severity. Grigory Perelman: genius and the task of the millennium,” based on numerous interviews with his teachers, classmates, co-workers and colleagues.

In September 2011, it became known that the mathematician refused to accept the offer to become a member of the Russian Academy of Sciences.

Personal life of Grigory Perelman:

Not married. Have no children.

Leads a secluded life, ignores the press. Lives in St. Petersburg in Kupchin with his mother.

There were reports in the press that since 2014 Gregory has been living in Sweden, but later it turned out that he only visits there sporadically.

On November 11, 2002, an article from St. Petersburg appeared on one of the major scientific publication portals on the Internet. mathematician Grigory Perelman, in which he gave proof of the Poincaré conjecture. Thus, the hypothesis became the first solved problem of the millennium - the so-called mathematical questions, the answers to which have not been found for many years. Eight years later, the Clay Mathematics Institute awarded the scientist a prize of one million US dollars for this achievement, but Perelman refused it, saying that he did not need the money and, moreover, did not agree with the official mathematical community. The poor mathematician's refusal of a large sum caused surprise in all layers of society. For this and for his reclusive lifestyle, Perelman is called the strangest Russian scientist. the site found out how Grigory Perelman lives and what he does today.

Mathematician No. 1

Now Grigory Perelman is 51 years old. The scientist leads a secluded life: he practically never leaves home, does not give interviews, and is not officially employed anywhere. The mathematician never had close friends, but people who know Perelman claim that he was not always like this.

“I remember Grisha as a teenager,” says Perelman’s housemate, Sergey Krasnov. - Although we live on different floors, we see each other sometimes. Previously, we could talk to his mother, Lyubov Leibovna, but now I rarely see her. She and Grigory periodically go out for a walk, but are always at home. When we see each other, they quickly nod and move on. They don't communicate with anyone. And during his school years, Grisha was no different from other boys. Of course, even then he was actively interested in science and spent a lot of time reading books, but he also found time for other things. I studied music, hung out with friends, and played sports. And then he sacrificed all his interests to mathematics. Was it worth it? Don't know".

Grigory always took first place in mathematics Olympiads, but one day victory eluded him: in the eighth grade at the All-Union Olympiad, Perelman became only second. Since then, he abandoned all his hobbies and recreation, immersing himself in books, reference books and encyclopedias. He soon caught up and became the #1 young mathematician in the country.

Reclusion

Krasnov states: none of the residents of their house doubted that Perelman would become a great scientist. “When we found out that Grisha proved the Poincaré conjecture, which no other person in the world could do, we were not even surprised,” the pensioner admits. - Of course, we were very happy for him, we decided: finally Grigory will make his way into the people, make a dizzying career! Well done, he deserves it! But he chose a different path for himself.”

Perelman refused a cash prize in the amount of a million dollars, justifying his decision by disagreement with the official mathematical community, while adding that he did not need the money.

After Perelman’s name thundered throughout the world, the mathematician was invited to the USA. In America, the scientist gave presentations, exchanged experience with foreign colleagues and explained his methods for solving mathematical problems. He quickly became bored with publicity. Returning to Russia, Perelman voluntarily left his post as a leading researcher at the laboratory of mathematical physics, resigned from the St. Petersburg branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, and reduced his communication with colleagues to zero. A few years later, they wanted to make Perelman a member of the Russian Academy of Sciences, but he refused. Having stopped almost all contacts with the outside world, the scientist locked himself in his apartment in Kupchino, on the outskirts of St. Petersburg, where he lives with his mother.

“Grisha was tortured by attention”

Nowadays, mathematicians very rarely leave home and spend whole days solving new problems. “Grisha and his mother live on Lyubov Leibovna’s pension,” says Krasnov. - We, the residents of the house, in no way condemn Grisha - they say, the man is in the prime of his life, but does not bring money to the family, does not help his old mother. There is no such. He is a genius, and geniuses cannot be condemned. Once they even wanted to chip in with the whole house to help them financially. But they refused - they said that they had enough. Lyubov Leibovna always said that Grisha is unpretentious: he wears jackets or boots for decades, and for lunch, macaroni and cheese is enough for him. Well, it’s not necessary, it’s not necessary.”

According to neighbors, any person in Perelman’s place would become unsociable and closed: although the mathematician has not given rise to discussion for a long time, his person still cannot be ignored. “Some journalists are on duty at Perelman’s door 24 hours a day,” Sergei Petrovich is indignant. - Once they waited until Grisha and his mother were leaving the apartment for a walk. A huge cameraman literally dared Lyubov Leibovna to film the situation in their apartment - their apartment was not very tidy, and they decided to focus on this. And the young correspondent attacked Perelman himself with questions. They didn't go out for a long time after that. Everyone will be stressed here! Grisha was tormented with attention!”

Residents of the house are confident that Perelman will make a name for himself by making a new discovery in the field of mathematics. “His work will not be in vain,” says Krasnov. “You just need to leave him alone and let him live in peace.”

Russian mathematician who proved the Poincaré conjecture

Russian scientist who proved the Poincaré conjecture, one of the fundamental problems of mathematics. Candidate of Physical and Mathematical Sciences. He worked at the Leningrad (St. Petersburg) department of the Steklov Mathematical Institute and taught at a number of US universities. Since 2003, he has not worked and hardly communicates with outsiders.

Grigory Yakovlevich Perelman was born on June 13, 1966 in Leningrad. His father was an electrical engineer who immigrated to Israel in 1993. Mother remained in St. Petersburg, worked as a mathematics teacher at a vocational school.

Perelman graduated from high school No. 239 with in-depth study of mathematics. In 1982, as part of a team of schoolchildren, he participated in the International Mathematical Olympiad in Budapest. In the same year, he was enrolled in the Faculty of Mathematics and Mechanics of Leningrad State University without exams. He won faculty, city and all-Union student mathematical Olympiads. He received a Lenin scholarship and graduated from the university with honors.

In November 2002 - July 2003, Perelman posted three scientific articles on the website arXiv.org, which in an extremely condensed form contained a solution to one of the special cases of William Thurston's geometrization hypothesis, leading to a proof of the Poincaré conjecture. The proof of this conjecture (which states that every simply connected closed three-dimensional manifold is homeomorphic to a three-dimensional sphere) was considered one of the fundamental problems of mathematics. The method of studying Ricci flow described by the scientist was called the Hamilton-Perelman theory. These works by Perelman did not receive official scientific publication status, since arXiv.org is a preprint library and not a peer-reviewed journal. Perelman made no attempts to officially publish these works.

In 2003, Perelman gave a series of lectures in the United States on his work, after which he returned to St. Petersburg and settled in his mother’s apartment in Kupchino. In December 2005, he resigned from his post as a leading researcher at the Laboratory of Mathematical Physics, resigned from the Mathematical Institute and almost completely broke off contacts with colleagues.

After Perelman's work appeared, several groups of mathematicians began to verify the correctness of his proofs. Over four years of checking and detailing Perelman's calculations, leading experts in this field did not find any errors. On August 22, 2006, Perelman was awarded the Fields Medal "for his contributions to geometry and revolutionary advances in understanding the analytical and geometric structure of the Ricci flow." Perelman refused to accept the award and communicate with journalists. At the same time, he said that he said goodbye to the scientific community and no longer considered himself a professional mathematician.

In December 2006, Perelman's proof of Poincare's theory was named the main scientific breakthrough of the year by Science magazine.

For the proof of the Poincaré conjecture, the Clay Mathematical Institute (USA) awarded a prize of one million dollars, the “Millennium Prize”. According to the prize's rules, Perelman can be awarded the award upon publication of his work in a peer-reviewed journal. Despite this, in March 2010 he was announced as the winner of the prize, but as The Daily Mail reported at the end of the same month, Perelman refused this prize as well. Nevertheless, in June 2010, the award ceremony was held: a symbolic certificate of the Millennium Prize was given to the Russian mathematician Mikhail Gromov, who worked in France, and Francois Poincaré, the grandson of Henri Poincaré, who put forward the hypothesis confirmed by Perelman. At the end of the same month, Perelman officially notified the Clay Institute of his final refusal of the award. The mathematician cited disagreement with the unfair decisions of the mathematical community as the reason for the refusal. At the same time, he emphasized that his contribution to the proof of the Poincaré conjecture was no greater than that of Hamilton.

In April 2011, after a long silence, Perelman gave his first interview to the Israeli journalist and executive producer of the Moscow film company "President Film" Alexander Zabrovsky. In this interview, the mathematician, when asked why he did not take a million dollars, replied that this money was nothing for “the man who rules the Universe.” In addition, Perelman agreed to star in Zbarovsky’s film, which was supposed to be about “the cooperation and confrontation of the three main world mathematical schools: Russian, Chinese and American, which are the most advanced along the path of studying and managing the Universe.”

In September 2011, it became known that the Clay Institute had decided to allocate a million dollars intended for Perelman to scholarships for young gifted mathematicians. In the same month, the academic council of the St. Petersburg branch of the Steklov Institute nominated Perelman for the post of academician of the Russian Academy of Sciences, but the scientist did not react to this initiative and was not included in the list of candidates for academician.

In The Sunday Telegraph's list of 100 living geniuses published in October 2007, Perelman shared ninth place with Brazilian architect Oscar Niemeyer and American minimalist composer Philip Glass.

Used materials

Perelman refused to become an academician of the Russian Academy of Sciences. - Interfax, 03.10.2011

Modest math man"s million goes to young scholars. - Russia Today, 23.09.2011

Math genius’ $1m to go to charity. - Voice of Russia, 22.09.2011

RAS: to elect Perelman as an academician, his consent is required. - Vesti.Ru, 14.09.2011

Irina Tumakova. Grigory Perelman has been nominated as an academician. - News, 13.09.2011

Anna Veligzhanina. Interview with mathematician Grigory Perelman: Why do I need a million dollars? I can control the Universe. - TVNZ, 28.04.2011

The hero of the new issue of the “Icon of the Era” column is Russian mathematician Grigory Perelman. What is known about him is that he gave up a million dollars by proving the Poincaré Conjecture, which, in turn, is known to be extremely difficult to understand. Moreover, the sequence here is exactly this - the fact of refusing money excited the respectable public much more than “some kind of abstract mathematical calculation.” Now that the hype around this decision has subsided, let’s figure out who Grigory Perelman is for mathematics and what mathematics is for him.

Grigory Perelman

Born in 1966 in Leningrad

mathematician

Life path

The Soviet Union had an outstanding mathematical tradition, so it is impossible to talk about Perelman's childhood without mentioning the phenomenon of Soviet mathematical schools. In them, talented children were trained under the guidance of the best mentors; such an environment served as fertile ground for future outstanding achievements. However, despite the competent organization of the learning process, there was also discrimination inherent in the Soviet system, when even having an unusual surname could cost a place in the city’s national team or admission to a university.

Henri Poincaré

Perelman grew up in an intelligent family and showed interest in mathematics from childhood. However, once he got into the mathematical circle, he did not immediately become a leader. The first failures spurred him to work harder and influenced his character - unyielding and stubborn. These qualities helped the scientist solve the main problem of his life.

Following a gold medal at the International Mathematical Olympiad in Budapest in 1982 and a brilliant graduation (there were not enough GTO standards passed for the gold medal) followed by Mathematical and Mechanics of St. Petersburg State University, and later graduate school, where Perelman also studied exclusively with “excellent” marks. When the Soviet Union ceased to exist, the scientist was faced with reality: science was experiencing a severe crisis. An internship in the USA unexpectedly took place, where the young scientist first met Richard Hamilton. The American mathematician made serious progress in solving the famous Poincaré problem. Moreover, he even outlined a plan, following which this decision could be reached. Perelman managed to communicate with him, and Hamilton made an indelible impression on him: he was open and spared no effort in explaining.

Institute building named after. Steklova in St. Petersburg

Despite offers to stay, at the end of the internship, Perelman returned to Russia, to his home apartment in a nine-story building in St. Petersburg in Kupchino (the notorious "ghetto" in the south of the city), and began working at the Mathematical Institute. Steklova. In his free time, he reflected on the Poincaré Hypothesis and the ideas that Hamilton had told him about. At this time, the American, judging by the publications, was unable to advance further in his reasoning. Soviet education gave Perelman the opportunity to look at the problem from the other side, using his own approach. Hamilton no longer responded to letters, and this became the “green light” for Perelman: he began working on solving the Hypothesis.

Every simply connected compact three-dimensional manifold without boundary is homeomorphic to a three-dimensional sphere.

The Poincaré conjecture belongs to topology - that branch of mathematics that studies the most general properties of space. Like any other branch of mathematics, topology is extremely specific and precise in its formulations. Any simplifications and retellings in a “more accessible form” distort the essence and have little in common with the original. That is why, in the framework of this article, we will not talk about the well-known thought experiment with a mug, which, through continuous deformation, turns into a donut. Out of respect for the main character, we simply admit that it is difficult to explain the Poincaré Hypothesis to people far from mathematics. And for those who are ready to devote time and effort to this, we will provide several materials for independent study.

• Uspensky V.V. "From Poincaré to Perelman", part one
• Uspensky V.V. "From Poincaré to Perelman", part two
• Uspensky V.V. “From Poincaré to Perelman”, part three

The three-dimensional sphere is the object referred to in the formulation of the Poincaré Hypothesis

It took Perelman seven years to solve this problem. He did not recognize conventions and did not submit his works to scientific journals for review (a common practice among scientists). In November 2002, Perelman published the first part of his calculations on arXiv.org, followed by two more. In them, in an extremely concise form, a problem even more general than the Poincaré Conjecture was solved - this is the Thurston Geometrization Conjecture, from which the first was a simple consequence. However, the scientific community received these works with caution. I was confused by the brevity of the solution and the complexity of the calculations that Perelman presented.

After the publication of the decision, Perelman again went to the United States. For several months he held seminars at various universities, talking about his work and patiently answering all questions. However, the main purpose of his trip was to meet with Hamilton. It was not possible to communicate with the American scientist a second time, but Perelman again received an invitation to stay. He received a letter from Harvard asking him to send them his resume, to which he irritably replied: “If they know my work, they don’t need my CV. If they need my CV, they don't know my work."

Fields Medal

The next few years were marred by an attempt by Chinese mathematicians to claim credit for the discovery.(their interests were supervised by Professor Yau, a brilliant mathematician, one of the creators of the mathematical apparatus of String Theory), the unbearably long wait for verification of the work, which was carried out by three groups of scientists, and the hype in the press.

All this went against Perelman’s principles. Mathematics attracted him with its categorical honesty and unambiguity, which is the basis of this science. However, the intrigues of his colleagues, concerned about recognition and money, shook the scientist’s faith in the mathematical community, and he decided not to study mathematics anymore.

And although Perelman’s contribution was ultimately appreciated, and Yau’s claims were ignored, the mathematician did not return to science. No Fields Medal (analogous to the Nobel Prize for mathematicians), nor the Millennium Prize (million dollars) he didn't accept. Perelman was extremely skeptical about the hype in the press and minimized contacts with former colleagues. To this day he lives in the same apartment in Kupchino.

Timeline

Born in Leningrad.

As part of a team of schoolchildren, he participated in the International Mathematical Olympiad in Budapest.

Perelman was invited to spend a semester each at New York University and Stony Brook University.

Returned to the institute. Steklova.

november
2002 -
July 2003

Perelman posted three scientific articles on the website arXiv.org, which in an extremely condensed form contained a solution to one of the special cases of William Thurston’s Geometrization Hypothesis, leading to a proof of the Poincaré Hypothesis.

Perelman gave a series of lectures in the United States on his works.

Perelman's results were verified by three independent groups of mathematicians. All three groups concluded that Poincaré's Problem had been successfully solved, but Chinese mathematicians Zhu Xiping and Cao Huaidong, along with their teacher Yau Shintang, attempted plagiarism, claiming that they had found a "complete proof".