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

In August 2006, Perelman was offered the Fields Medal[1] 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.

[12] In 1982, not long after his sixteenth birthday, he won a gold medal as a member of the Soviet team at the International Mathematical Olympiad hosted in Budapest, achieving a perfect score.

[citation needed] 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 the late 1980s and early 1990s, with a strong recommendation from the geometer Mikhail Gromov,[14] Perelman obtained research positions at several universities in the United States.

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

[15] 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.

After proving 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 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.

[P93] Despite the lack of smoothness in Alexandrov spaces, Perelman and Anton Petrunin were able to consider the gradient flow of certain functions, in unpublished work.

In 1994, Perelman gave a short proof of Cheeger and Gromoll's conjecture by establishing that, under the condition of nonnegative sectional curvature, Sharafutdinov's retraction is a submersion.

Poincaré suggested that a converse might be true: if a closed three-dimensional manifold has the property that any loop can be contracted into a point, then it must be topologically equivalent to a 3-sphere.

In 1982, William Thurston developed a novel viewpoint, making the Poincaré conjecture into a small special case of a hypothetical systematic structure theory of topology in three dimensions.

In three seminal articles published in the 1980s, Hamilton proved that his equation achieved analogous phenomena, spreading extreme curvatures and uniformizing a Riemannian metric, in certain geometric settings.

In completely general settings, it is inevitable that "singularities" occur, meaning that curvature accumulates to infinite levels after a finite amount of "time" has elapsed.

Following Shing-Tung Yau's suggestion that a detailed understanding of these singularities could be topologically meaningful, and in particular that their locations might identify the spheres and tori in Thurston's conjecture, Hamilton began a systematic analysis.

[24] Throughout the 1990s, he found a number of new technical results and methods,[25] culminating in a 1997 publication constructing a "Ricci flow with surgery" for four-dimensional spaces.

Yau has identified this article as one of the most important in the field of geometric analysis, saying that with its publication it became clear that Ricci flow could be powerful enough to settle the Thurston conjecture.

The first, valid in any dimension, was based on a novel adaptation of Peter Li and Shing-Tung Yau's differential Harnack inequalities to the setting of Ricci flow.

Other results in Perelman's first preprint include the introduction of certain monotonic quantities and a "pseudolocality theorem" which relates curvature control and isoperimetry.

The first half of Perelman's second preprint, in addition to fixing some incorrect statements and arguments from the first paper, used his canonical neighborhoods theorem to construct a Ricci flow with surgery in three dimensions, systematically excising singular regions as they develop.

As an immediate corollary of his construction, Perelman resolved a major conjecture on the topological classification in three dimensions of closed manifolds which admit metrics of positive scalar curvature.

In order to settle the Thurston conjecture, the second half of Perelman's second preprint is devoted to an analysis of Ricci flows with surgery, which may exist for infinite time.

Sir John Ball, president of the International Mathematical Union, approached Perelman in Saint Petersburg in June 2006 to persuade him to accept the prize.

I'm not even that successful; that is why I don't want to have everybody looking at me.Nevertheless, on 22 August 2006, at the International Congress of Mathematicians in Madrid, 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".

He considered the decision of the Clay Institute unfair for not sharing the prize with Richard S. Hamilton,[5] and stated that "the main reason is my disagreement with the organized mathematical community.

[52] Russian media speculated that he periodically visits his sister in Sweden, while living in Saint Petersburg and taking care of his elderly mother.

Masha Gessen, author of a biography about Perelman, Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century, was unable to meet him.

[54] A Russian documentary about Perelman in which his work is discussed by several leading mathematicians, including Mikhail Gromov, Ludwig Faddeev, Anatoly Vershik, Gang Tian, John Morgan and others, was released in 2011 under the title "Иноходец.

[citation needed] In April 2011, Aleksandr Zabrovsky, producer of "President-Film" studio, claimed to have held an interview with Perelman and agreed to shoot a film about him, under the tentative title The Formula of the Universe.