Grigori Perelman

Grigori Perelman
Perelman in 1993
Grigori Perelman
Born	Grigori Yakovlevich Perelman 13 6, 1966
Birthplace	Leningrad, Soviet Union (now Saint Petersburg, Russia)
Nationality	Russian
Occupation	Mathematician
Known for	Proof of the Poincaré conjecture, proof of the soul conjecture, geometrization of 3-manifolds
Education	Leningrad State University (PhD, 1990)
Awards	Fields Medal (offered 2006, declined), Clay Millennium Prize (offered 2010, declined)

Grigori Yakovlevich Perelman (Template:Lang-ru; born 13 June 1966) is a Russian mathematician who solved one of the most famous and enduring problems in the history of mathematics — the Poincaré conjecture — and then quietly withdrew from professional life, declining both the Fields Medal and a million-dollar Millennium Prize in the process. His work in geometric analysis, Riemannian geometry, and geometric topology fundamentally advanced the mathematical understanding of three-dimensional spaces. In 2002 and 2003, Perelman posted three preprints to the arXiv repository that outlined a proof of the Poincaré conjecture and the more general geometrization conjecture using novel techniques in the analysis of Ricci flow. His refusal to accept prizes and his subsequent retreat from public life have made him one of the most enigmatic figures in modern science. He resigned from his research post at the Steklov Institute of Mathematics in 2005 and stated in 2006 that he had left professional mathematics. Since then, Perelman has lived in seclusion in Saint Petersburg and has declined requests for interviews.

Early Life

Grigori Yakovlevich Perelman was born on 13 June 1966 in Leningrad (now Saint Petersburg), in the Soviet Union. He grew up in a Jewish family; his father was an electrical engineer, and his mother was a mathematician who taught at a technical college. Perelman's mother is credited with nurturing his early interest in mathematics, and she reportedly made personal sacrifices to support his intellectual development.

Perelman displayed exceptional mathematical talent from a young age. As a teenager, he attended Leningrad's Special Mathematics and Physics School Number 239, an institution known for producing outstanding mathematicians and scientists. His abilities attracted attention early in his academic career, and he became involved in mathematical competitions while still a secondary school student.

In 1982, at the age of sixteen, Perelman represented the Soviet Union at the International Mathematical Olympiad (IMO) held in Budapest, Hungary. He achieved a perfect score, earning a gold medal with full marks — a feat that underscored his exceptional problem-solving ability. This performance placed him among the top mathematical talents of his generation on an international stage and helped pave the way for his entry into advanced mathematical study.

The intellectual environment of Leningrad, with its deep tradition in mathematics and the sciences, provided a fertile setting for Perelman's development. The city's mathematical community, centered on institutions such as the Leningrad branch of the Steklov Institute and Leningrad State University, was renowned for producing researchers of the highest caliber. Perelman's formative years in this milieu shaped his approach to mathematics and instilled in him a rigorous, uncompromising commitment to mathematical truth that would characterize his entire career.

Education

Following his success at the International Mathematical Olympiad, Perelman enrolled at Leningrad State University (now Saint Petersburg State University), where he pursued advanced study in mathematics. He completed his doctoral dissertation, titled "Saddle Surfaces in Euclidean Spaces," in 1990 under the supervision of his advisors at the university.

After completing his doctorate, Perelman took up a research position at the Leningrad (later Saint Petersburg) branch of the Steklov Institute of Mathematics, one of the leading mathematical research institutions in Russia. This position provided him with the institutional support necessary to pursue deep and sustained research in geometry.

In the late 1980s and early 1990s, Perelman also spent periods working and collaborating with mathematicians in the United States. He held postdoctoral and visiting positions at several American universities, including stints at the Courant Institute of Mathematical Sciences at New York University, Stony Brook University, and the University of California, Berkeley. These experiences exposed him to the broader international mathematical community and allowed him to establish connections with leading researchers in his areas of interest. However, by the mid-1990s, Perelman had returned to Russia and resumed his position at the Steklov Institute, where he would carry out the work that would eventually lead to his proof of the Poincaré conjecture.

Career

Early Research: Alexandrov Spaces and the Soul Conjecture

During the 1990s, Perelman made significant contributions to the study of Alexandrov spaces, a class of metric spaces that generalize Riemannian manifolds with curvature bounded below. He worked partly in collaboration with prominent mathematicians including Yuri Burago, Mikhael Gromov, and Anton Petrunin on problems in this area. This research established Perelman's reputation as a mathematician of considerable depth and originality.

In 1994, Perelman proved the soul conjecture in Riemannian geometry, a problem that had remained open for approximately twenty years since it was posed by Jeff Cheeger and Detlef Gromoll. The soul conjecture concerns the structure of complete, non-compact Riemannian manifolds with non-negative sectional curvature. Perelman's proof was recognized as a major achievement in differential geometry and brought him to the attention of the broader mathematical community.

In 1996, Perelman was awarded the prize of the European Mathematical Society for young mathematicians, but he declined the award. This early refusal foreshadowed his later, more widely publicized rejections of major prizes. The reasons for this initial refusal were not entirely clear, though it suggested that Perelman had already formed views about mathematical recognition that differed from those prevailing in the professional community.

The Poincaré Conjecture and Geometrization

The Poincaré conjecture, formulated by the French mathematician Henri Poincaré in 1904, is a statement about the characterization of the 3-sphere among closed three-dimensional manifolds. In its simplest formulation, the conjecture asserts that every simply connected, closed three-dimensional manifold is homeomorphic to the three-dimensional sphere. By the early 2000s, the Poincaré conjecture had resisted proof for nearly a century and had become one of the most famous unsolved problems in mathematics. In 2000, the Clay Mathematics Institute designated it as one of seven Millennium Prize Problems, offering a prize of one million dollars for a correct solution.

The key tool in Perelman's approach was Ricci flow, a technique for deforming the metric of a Riemannian manifold in a manner analogous to heat diffusion. Ricci flow had been introduced and developed by the American mathematician Richard S. Hamilton beginning in the 1980s, partly with the aim of attacking the Poincaré conjecture and the more general geometrization conjecture formulated by William Thurston. Hamilton had made significant progress, establishing the Ricci flow program and proving important special cases, but critical obstacles remained — particularly in understanding the singularities that could form during the flow.

In November 2002, Perelman posted the first of three preprints to the arXiv online repository. These papers, which appeared in November 2002, March 2003, and July 2003, outlined a proof of the geometrization conjecture — which subsumes the Poincaré conjecture as a special case — using new techniques in the analysis of Ricci flow. Perelman introduced several innovative ideas, including a monotonicity formula for what he called the "entropy" of the Ricci flow and a technique he described as "Ricci flow with surgery," which allowed the flow to be continued past singularities by performing topological surgeries on the manifold.

Perelman did not submit his papers to a peer-reviewed journal. Instead, he made them freely available online and gave a series of lectures at American institutions in 2003 to explain his work, including talks at the Massachusetts Institute of Technology, Stony Brook University, and Princeton University. During these lectures, mathematicians were able to question Perelman directly about the details of his arguments.

The mathematical community subsequently undertook an extensive effort to verify Perelman's work. Several teams of mathematicians published detailed expositions that filled in the arguments sketched in Perelman's preprints. Over the following years, these verification efforts confirmed that Perelman's proof was correct. By 2006, a consensus had emerged among experts that the Poincaré conjecture and the geometrization conjecture had been proven.

Withdrawal from Mathematics

In 2005, Perelman resigned from his research position at the Steklov Institute of Mathematics in Saint Petersburg. In 2006, he stated that he had quit professional mathematics entirely, expressing disappointment with what he perceived as a decline in ethical standards within the field.

Perelman's withdrawal from the mathematical community was gradual but ultimately complete. After giving lectures in the United States in 2003, he returned to Russia and became increasingly reclusive. He stopped responding to correspondence from other mathematicians and declined invitations to speak at conferences and universities. His departure from the Steklov Institute and his public statements about leaving mathematics surprised many in the community, given that he was at the height of his career and that his work had resolved one of the most important open questions in the discipline.

Reports from Russian media in the years following his withdrawal described Perelman as living modestly with his mother in a small apartment in Saint Petersburg, largely cut off from the mathematical world. He has not published any new mathematical work since 2003, and he has consistently declined requests for interviews from journalists and media outlets around the world.

The precise reasons behind Perelman's decision have been the subject of considerable speculation. In a 2006 interview — one of the last he gave — he reportedly expressed frustration with what he viewed as a lack of integrity among some members of the mathematical establishment, particularly in the context of the attribution of credit for the proof of the Poincaré conjecture. A controversy had arisen involving Chinese mathematicians Huai-Dong Cao and Xi-Ping Zhu, who in 2006 published a paper that some initially presented as providing the first complete proof of the geometrization conjecture, before it was clarified that their work was a detailed exposition of Perelman's proof. This episode, along with other disputes over credit, appears to have contributed to Perelman's disillusionment with the profession.

Personal Life

Grigori Perelman is known for living an extremely private and reclusive life. Since his withdrawal from professional mathematics in 2006, he has resided in Saint Petersburg, Russia, reportedly living with his mother in modest circumstances. He has consistently avoided public appearances and has not granted interviews to journalists or media organizations.

Perelman's lifestyle stands in marked contrast to the level of attention his mathematical achievements have attracted. Despite being offered prizes worth more than one million dollars, he has shown no interest in financial reward or public recognition. His reported daily life is characterized by simplicity and seclusion, and he has shown no inclination to re-enter the mathematical community or engage with the broader public.

Masha Gessen, a Russian-American journalist, wrote a biography of Perelman titled Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century (published in 2009), which attempted to shed light on his life and personality. The book was written without Perelman's cooperation, as he declined to participate in the project. Gessen drew on interviews with Perelman's former classmates, teachers, and colleagues to construct a portrait of the mathematician.

Recognition

Fields Medal (2006)

In August 2006, the International Mathematical Union announced that Perelman had been awarded the Fields Medal, the most prestigious prize in mathematics, at the International Congress of Mathematicians held in Madrid, Spain. The citation recognized "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." Perelman declined the medal, becoming the first person in the history of the award to do so. In explaining his decision, he stated: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."

Science Breakthrough of the Year (2006)

In December 2006, the journal Science named Perelman's proof of the Poincaré conjecture the "Breakthrough of the Year." This was the first time the journal had bestowed this recognition on an achievement in mathematics, underscoring the significance of the result and the broad interest it had generated within the scientific community.

Clay Millennium Prize (2010)

On 18 March 2010, the Clay Mathematics Institute announced that Perelman had met the criteria to receive the first-ever Clay Millennium Prize, carrying a value of one million US dollars, for the resolution of the Poincaré conjecture. On 1 July 2010, Perelman publicly rejected the prize. He stated that he considered the decision of the Clay Institute's board to be unfair, arguing that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton, whose pioneering work on Ricci flow had laid the foundation for Perelman's proof. This refusal attracted worldwide media attention and further cemented Perelman's reputation as a figure who placed mathematical principle above material reward.

European Mathematical Society Prize (1996)

In 1996, the European Mathematical Society offered Perelman its prize for young mathematicians, in recognition of his proof of the soul conjecture and his contributions to the study of Alexandrov spaces. Perelman declined this award as well, predating his later, more publicized refusals by a decade.

Legacy

Grigori Perelman's proof of the Poincaré conjecture represents one of the landmark achievements in the history of mathematics. The Poincaré conjecture had been open for nearly a century and was one of seven problems designated by the Clay Mathematics Institute as Millennium Prize Problems — problems considered among the most important unsolved questions in the discipline. Perelman's resolution of the conjecture — and, more broadly, of Thurston's geometrization conjecture — provided a complete classification of closed three-dimensional manifolds, resolving a central question in geometric topology.

The techniques Perelman introduced, particularly his methods for handling singularities in Ricci flow through the surgery procedure and his entropy monotonicity formula, have had a lasting impact on the field of geometric analysis. These ideas extended and completed the program initiated by Richard Hamilton, and they have continued to influence research in differential geometry and topology in the years since Perelman's preprints were posted.

Perelman's decision to decline both the Fields Medal and the Millennium Prize has been the subject of extensive discussion, both within the mathematical community and in the broader public sphere. His refusals have been interpreted in various ways — as a statement about the ethics of credit attribution in mathematics, as an expression of a pure commitment to mathematics for its own sake, and as a reflection of his personal values and temperament. The episode has prompted reflection on the culture of prizes and recognition in academia.

The Poincaré conjecture remains the only one of the seven Millennium Prize Problems to have been solved as of 2025. Perelman's proof, and his subsequent withdrawal from public life, have made him a figure of unusual interest both within and beyond mathematics. His story has been the subject of books, documentaries, and numerous articles, and it continues to generate discussion about the nature of mathematical genius, the value of recognition, and the relationship between the individual and the institutions of science.