​Grigori Yakovlevich Perelman, born on June 13, 1966, in Leningrad (now Saint Petersburg), Russia, is a mathematician renowned for his groundbreaking contributions to geometry and topology. His most celebrated achievement is the proof of the Poincaré conjecture, a problem that stood unsolved for nearly a century and was one of the seven "Millennium Prize Problems" designated by the Clay Mathematics Institute. Perelman's work has profoundly influenced the field of mathematics, yet his reclusive nature and rejection of prestigious awards have added an aura of mystery to his persona.​

Early Life and Education

Perelman's prodigious talent in mathematics was evident from a young age. His mother, Lyubov, a dedicated mathematician, recognized his potential and enrolled him in Sergei Rukshin's renowned after-school mathematics training program. This program was instrumental in nurturing young mathematical talents in Leningrad. Perelman attended the Leningrad Secondary School 239, a specialized institution focusing on advanced mathematics and physics. In 1982, at just 16 years old, he represented the Soviet Union in the International Mathematical Olympiad in Budapest, achieving a perfect score and earning a gold medal. ​

Following his secondary education, Perelman pursued studies at the School of Mathematics and Mechanics at Leningrad State University. Under the guidance of esteemed mathematicians Aleksandr Aleksandrov and Yuri Burago, he completed his Ph.D. in 1990. His doctoral thesis, titled "Saddle Surfaces in Euclidean Spaces," delved into the intricacies of Riemannian geometry, laying the groundwork for his future endeavors. ​

Early Research and the Soul Conjecture

In the early stages of his career, Perelman focused on Alexandrov spaces, which are generalizations of Riemannian manifolds with curvature bounded from below. Collaborating with notable geometers such as Yuri Burago and Mikhail Gromov, he made significant strides in understanding these spaces. ​

A landmark achievement during this period was his proof of the soul conjecture in 1994. This conjecture posited that any complete, non-compact Riemannian manifold of non-negative sectional curvature is diffeomorphic to the normal bundle of a compact, totally geodesic submanifold, termed the "soul." Perelman's proof provided a comprehensive understanding of the topological structure of such manifolds, resolving a question that had remained open for two decades. ​

The Poincaré Conjecture and Geometrization Conjecture

The Poincaré conjecture, formulated by Henri Poincaré in 1904, was a central question in topology. It hypothesized that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Despite numerous attempts, the conjecture resisted proof for nearly a century. Parallelly, William Thurston's geometrization conjecture proposed a comprehensive framework for understanding the structure of 3-manifolds, suggesting that every closed 3-manifold can be decomposed into pieces with uniform geometric structures. ​

Perelman's approach to these conjectures involved the Ricci flow, a process introduced by Richard S. Hamilton. The Ricci flow describes the deformation of the metric of a Riemannian manifold in a way analogous to heat diffusion, aiming to "smooth out" irregularities in curvature. Perelman introduced novel techniques to analyse the Ricci flow, particularly in handling singularities that develop over time. His work culminated in a series of preprints posted on the arXiv between 2002 and 2003, wherein he outlined proofs for both the Poincaré and geometrization conjectures. ​

Verification and Recognition

Perelman's proofs underwent rigorous scrutiny by the mathematical community. Teams of mathematicians worldwide dedicated years to verifying the correctness of his arguments. By 2006, consensus was reached affirming the validity of his proofs, marking a monumental milestone in mathematics. ​

In recognition of his ground breaking work, the International Mathematical Union awarded Perelman the Fields Medal in 2006, often regarded as the "Nobel Prize of Mathematics." However, Perelman declined the honor, stating, "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." He also expressed discontent with the ethical standards in the mathematical community. ​

Similarly, in 2010, the Clay Mathematics Institute announced that Perelman had met the criteria for the Millennium Prize, which included a $1 million reward for solving the Poincaré conjecture. Perelman declined this prize as well, citing that his contribution was no greater than that of Richard S. Hamilton, whose foundational work on the Ricci flow was instrumental to his own. ​

Personal Life and Withdrawal from Academia

Following his monumental contributions, Perelman resigned from his position at the Steklov Institute of Mathematics in 2005 and withdrew from professional mathematics. He has since led a reclusive life in Saint Petersburg, avoiding public appearances and declining interviews. His decision to step away from the mathematical community has been the subject of much speculation, with some attributing it to his dissatisfaction with the academic world's ethical standards. ​

Legacy and Impact

Perelman's work has left an indelible mark on mathematics. His proof of the Poincaré conjecture not only resolved a century-old problem but also validated the broader geometrization conjecture, providing a comprehensive understanding of the structure of 3-manifolds. His innovative techniques in analysing the Ricci flow have opened new avenues in geometric analysis