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Born 13 June 1966 Leningrad, USSR (now St Petersburg, Russia) Summary**Grigori Perelman**is a Russian mathematician who proved the Poincaré Conjecture and who refused to accept a Fields Medal or the $1 000 000 Clay Prize.

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### Biography

**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

*Realization of abstract k-skeletons as k-skeletons of intersections of convex polyhedra in*R2k−1\mathbb{R}^{2k-1}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\mathbb{R}^{4}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

*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 MnM^{n}Mn. If such a manifold has Ricci curvature ≥ n−1n - 1n−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

*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

**I**(Eur. Math. Soc., Zürich, 2007), 66-76." href="../../Biographies/Perelman/#reference-8">8>.For an extract of Lott"s talk, giving some technical details, see Perelman"s work leading to the 2006 Fields Medal" class="elink" href="../../Extras/Perelman_Medal/" target="_blank">THIS LINK. (Note the careful choice of Lott"s language. He says the Perelman "proved the so-called Soul Conjecture," but only that he "presented proofs of the Poincaré conjecture and the geometrization conjecture.")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

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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.