Grigori Yakovlevich Perelman

Born: 13 June 1966 in Leningrad, now St Petersburg, Russia

Grigori Yakovlevich Perelman's parents are Yakov Perelman, an electrical engineer, and Lubov Lvovna, who was a teacher of mathematics at a technical college. They were Jewish, which would present their son with some problems in a country where it was feared that those of Jewish descent had divided loyalty. Grigori Yakovlevich, their first child, is often known by the name Grisha. As a young child Grisha was taught to play the violin both by his mother and by a private tutor. His father also had a major influence in developing his son's problem solving skills. Speaking about his father, Perelman said (see [11]):-
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 develop his mathematical skills and, by the time he was ten, he had taken part in district mathematics competitions and shown a marked talent. Lubov sought advice about how best to develop Grisha's mathematical talents and was advised to send him to a mathematics club run by a nineteen year old coach named Sergei Rukshin. The club met twice a week at the Palace of Pioneers at the end of the school day and Rukshin, an undergraduate student at Leningrad University, had some novel ways of getting the best out of the boys who came to the club.

Rukshin quickly saw Perelman's potential even though at first there was little to distinguish him from other bright children in the group. There developed a bond, an understanding, between the two with Perelman becoming Rukshin's favourite pupil. In the summer of 1980 Rukshin tutored Perelman in English so that he could enter Leningrad's Special Mathematics and Physics School Number 239 in September of that year. To allow Perelman to get this intense tuition, learning the English covered in four years of schooling in a few weeks, the Perelman family had to remain in Leningrad over the summer rather than going to the country which would have been the norm. Lessons were conducted walking round the parks of Leningrad and successfully achieved their aim.

The class that Perelman entered in School 239 was unusual in that the group of highly talented mathematicians tutored by Rukshin were put into the same class. At the school Valery Ryzhik became both their class teacher and their mathematics teacher. Ryzhik was an extraordinarily talented mathematics teacher but the class containing Rukshin's collection of mathematical geniuses proved almost an impossible challenge for him. As well as mathematics, Ryzhik ran a chess club on one evening a week which Perelman attended, showing considerable talents at the game. When he was fifteen, Perelman attended the summer camp run by Rukshin. This was the first time that he had spent a night away from his mother but the bond between Rukshin and Perelman helped the potentially difficult situation. Rukshin not only trained his club boys to be the best solvers of mathematics problems but also tried to broaden their interests. Perelman was already interested in the violin and classical music but Perelman was able to broaden his musical interests. Although he would attend camps with Rukshin, Perelman never took part in the trips arranged by Ryzhik.

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, where in addition to the mathematical training they were subjected to stiff physical exercises in the gym. 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.

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).

One might imagine that his achievements would mean that he would be welcomed as a graduate student at the Leningrad branch of the Steklov Mathematics Institute with open arms. However, under Ivan Vinogradov's leadership the Steklov Mathematics Institute had accepted no Jews and, although it now had a new director, the old policies persisted. Aleksandr Danilovic Aleksandrov wrote to the director requesting that Perelman be allowed to undertake graduate work under his supervision at the Leningrad branch of the Steklov Mathematics Institute. The request, highly unusual coming from someone of Aleksandrov's high standing, was granted but, although Aleksandrov would be his official advisor, in practice it was Yuri Burago who took on the role. 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 Institut 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. Masha Gessen describes Perelman at this time [1]:-
By the time Perelman arrived in the United states, he was twenty-six, no longer pudgy but tall and apparently fit. His beard had passed out of its extended awkward-tuft stage and was thick, black and bushy. His hair was long. He did not believe in cutting hair or fingernails ... [he wore] the same clothes every day - most notably a brown corduroy jacket ... [he ate] a particular kind of black bread that could be procured only from a Russian store in Brooklyn Beach, where Perelman walked from Manhatten.
While Perelman was in the United States in 1992, his mother stayed with friends in New York, his father had earlier emigrated to Israel, and Perelman's young sister Lena was still being educated in St Petersburg (Leningrad returned to its original name of St Petersburg in 1991). He got to know Jeff Cheeger and Gang Tian, and the three of them regularly travelled to Princeton to attend seminars at the Institute for Advanced Study. Perelman attended a conference in Israel in 1993 then accepted a two-year Miller Research Fellowship at the University of California, Berkeley. He published some remarkable papers during these years. Elements of Morse theory on Aleksandrov spaces (Russian) (1993) investigates the local topological structure of Aleksandrov spaces. Manifolds of positive Ricci curvature with almost maximal volume (1994) solves a conjecture about a complete Riemannian manifold Mn. It such a manifold has Ricci curvature ≥ n - 1 and volume close to that of the sphere then Perelman proved it is homeomorphic to the sphere. The biggest breakthrough, however, was his paper Proof of the soul conjecture of Cheeger and Gromoll (1994) which answered a question asked by Cheeger and Gromoll twenty years earlier. Perelman was invited to address the International Congress of Mathematicians in Zürich in 1994 and he gave the lecture Spaces with curvature bounded below.

To understand the problems that Perelman was beginning to think about around this time, we give the description of the Poincaré Conjecture and the Thurston Geometrization Conjecture from [16]:-

A 2-manifold with positive curvature can be deformed into a 2-sphere; one with zero curvature can be deformed into a torus; and one with negative curvature can be deformed into a torus with more than one hole. The Poincaré Conjecture, which originated with the French mathematician Henri Poincaré in 1904, concerns 3-dimensional manifolds, or 3-manifolds. ... Can every simply connected 3-manifold be deformed into the 3-sphere? The Poincaré Conjecture asserts that the answer to this question is yes. Just as with 2- manifolds, one could also hope for a classification of 3-manifolds. In the 1970s, Fields Medalist William Thurston made a new conjecture, which came to be called the Thurston Geometrization Conjecture and which gives a way to classify all 3-manifolds. The Thurston Geometrization Conjecture provides a sweeping vision of 3-manifolds and actually includes the Poincaré Conjecture as a special case. Thurston proposed that, in a way analogous to the case of 2-manifolds, 3-manifolds can be classified using geometry. But the analogy does not extend very far: 3-manifolds are much more diverse and complex than 2-manifolds.
A possible approach to attacking the Poincaré Conjecture had been developed by Richard Hamilton who had introduced a significant idea in 1982 when he began to study a particular equation he called the Ricci flow. 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 [11]:-
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. Quickly he received invitations to visit the Stony Brook campus of the State University of New York and the Massachusetts Institute of Technology. He began making plans for the visits and, before setting off, he posted a second paper Ricci flow with surgery on three-manifolds on the web continuing his proof. He arrived in the United States in April 2003 and went first to the Massachusetts Institute of Technology where he gave talks on his work for most days in the two weeks he was there. He spent two similar weeks at Stony Brook followed by visits to Columbia University and Princeton University where he gave lectures. He turned down all offers of professorships that were made to him, becoming annoyed at the pressure some put on him to accept.

He returned to St Petersburg at the end of April 2002 and, in July, put Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, the third instalment 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. However 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.
John Lott described Perelman's work leading to the award of a Fields Medal in a lecture he gave to the International Congress of Mathematicians in Zürich in August 2006 [8].

Perelman refused the invitation to be a plenary speaker at the 2006 International Congress of Mathematicians. He also refused the award of the Fields Medal, the first person to have done so. If his hope had been to avoid publicity he was highly unsuccessful since huge public interest was generated and he was hounded by the press. In March 2010 the Clay Mathematics Institute announced that Perelman had met the conditions for the award of one million US dollars which they had offered 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.
Let us end this biography by quoting Mikhael Gromov (see [1]):-
[Perelman] has moral principles to which he holds. And this surprises people. They often say he acts strangely because he acts honestly, in a nonconformist manner, which is unpopular in this community - even though it should be the norm.

Article by: J J O'Connor and E F Robertson

July 2011
MacTutor History of Mathematics