Grigori Perelman

Perelman's first works to have a major Impact on the mathematical literature were in the field of Alexandrov spaces, the concept of which dates back to the 1950s.

In a very well-known paper coauthored with Yuri Burago and Mikhael Gromov, Perelman established the modern foundations of this field, with the notion of Gromov–Hausdorff convergence as an organizing principle.

In a followup unpublished paper, Perelman proved his "stability theorem," asserting that in the collection of all Alexandrov spaces with a fixed curvature bound, all elements of any sufficiently small metric ball around a compact space are mutually homeomorphic.

Vitali Kapovitch, who described Perelman's article as being "very hard to read," later wrote a detailed version of Perelman's proof, making use of some further simplifications.

Perelman developed a version of Morse theory on Alexandrov spaces.

Grigori Yakovlevich Perelman (Григорий Яковлевич Перельман; born 13 June 1966) is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology.

Grigori Yakovlevich Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents, Yakov (who now lives in Israel) and Lyubov (who still lives in Saint Petersburg with Grigori).

Grigori's mother Lyubov gave up graduate work in mathematics to raise him.

Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school mathematics training program.

His mathematical education continued at the Leningrad Secondary School 239, a specialized school with advanced mathematics and physics programs.

Grigori excelled in all subjects except physical education.

In the late 1980s and early 1990s, with a strong recommendation from the geometer Mikhail Gromov, Perelman obtained research positions at several universities in the United States.

In 1982, as a member of the Soviet Union team competing in the International Mathematical Olympiad, an international competition for high school students, he won a gold medal, achieving a perfect score.

He continued as a student of The School of Mathematics and Mechanics at the Leningrad State University, without admission examinations, and enrolled at the university.

In 1987, the year he began graduate studies, he published an article controlling the size of circumscribed cylinders by that of inscribed spheres.

Surfaces of negative curvature were the subject of Perelman's graduate studies.

His first result was on the possibility of prescribing the structure of negatively-curved polyhedral surfaces in three-dimensional Euclidean space.

He proved that any such metric on the plane which is complete can be continuously immersed as a polyhedral surface.

Later, he constructed an example of a smooth hypersurface of four-dimensional Euclidean space which is complete and has Gaussian curvature negative and bounded away from zero.

Previous examples of such surfaces were known, but Perelman's was the first to exhibit the saddle property on nonexistence of locally strictly supporting hyperplanes.

As such, his construction provided further obstruction to the extension of a well-known theorem of Nikolai Efimov to higher dimensions.

In the 1990s, partly in collaboration with Yuri Burago, Mikhael Gromov, and Anton Petrunin, he made contributions to the study of Alexandrov spaces.

After completing his PhD in 1990, Perelman began work at the Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences, where his advisors were Aleksandr Aleksandrov and Yuri Burago.

In 1991, Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society for his work on Aleksandrov's spaces of curvature bounded from below.

In 1992, he was invited to spend a semester each at the Courant Institute in New York University, where he began work on manifolds with lower bounds on Ricci curvature.

From there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley, in 1993.

In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years.

After having proved the soul conjecture in 1994, he was offered jobs at several top universities in the US, including Princeton and Stanford, but he rejected them all and returned to the Steklov Institute in Saint Petersburg in the summer of 1995 for a research-only position.

In his undergraduate studies, Perelman dealt with issues in the field of convex geometry.

His first published article studied the combinatorial structures arising from intersections of convex polyhedra.

With I. V. Polikanova, he established a measure-theoretic formulation of Helly's theorem.

He had previously rejected the prestigious prize of the European Mathematical Society in 1996.

In 2002 and 2003, he developed new techniques in the analysis of Ricci flow, and proved the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous open problem in mathematics for the past century.

The full details of Perelman's work were filled in and explained by various authors over the following several years.

In 2005, Perelman abruptly quit his research job at the Steklov Institute of Mathematics, and in 2006 stated that he had quit professional mathematics, due to feeling disappointed over the ethical standards in the field.

He lives in seclusion in Saint Petersburg, and has not accepted offers for interviews since 2006.

In August 2006, Perelman was offered the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."

On 22 December 2006, the scientific journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year", the first such recognition in the area of mathematics.

On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture.

On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton, the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture.