On Nov. 11, 2002, Grigori "Grisha" Perelman posted the first of three papers on arXiv that completed the missing pieces of Richard Hamilton's Ricci-flow program and thereby proved the century-old Poincaré conjecture for three-dimensional manifolds. His work analyzed and controlled singularities under Ricci flow, allowing a finite surgery process that reduces the manifold to a sphere. The proof was vetted by the community—famously summarized in a 2006 work by John Morgan and Gang Tian—and Perelman declined major awards and largely withdrew from public life.
Quiet Genius: How Grigori Perelman Solved the Poincaré Conjecture
On Nov. 11, 2002, Grigori "Grisha" Perelman posted the first of three papers on arXiv that completed the missing pieces of Richard Hamilton's Ricci-flow program and thereby proved the century-old Poincaré conjecture for three-dimensional manifolds. His work analyzed and controlled singularities under Ricci flow, allowing a finite surgery process that reduces the manifold to a sphere. The proof was vetted by the community—famously summarized in a 2006 work by John Morgan and Gang Tian—and Perelman declined major awards and largely withdrew from public life.
04:54 PM, 11/11/2025Science
Quiet Genius: How Grigori Perelman Solved the Poincaré Conjecture
On Nov. 11, 2002, a reclusive mathematician in St. Petersburg posted a paper to the public mathematics server arXiv that would become the cornerstone of one of the most celebrated breakthroughs in modern mathematics.
Published under the name "Grisha Perelman" and titled "The entropy formula for the Ricci flow and its geometric applications," the paper was the first of three closely related works (2002–2003) that together completed a long-standing program to resolve the Poincaré conjecture.
What the conjecture says (in plain terms)
Henri Poincaré's conjecture concerns three-dimensional spaces (3-manifolds). Informally: if every closed loop on a 3D space can be continuously shrunk to a point without cutting the loop or tearing the space, then that space is, topologically, a 3-sphere. Proving this statement for 3D shapes was a central challenge in topology for much of the 20th century.
Ricci flow and the obstruction
In the 1980s, Richard Hamilton proposed using a geometric evolution called Ricci flow to smooth the curvature of a manifold over time. A helpful analogy—used by reporters and some mathematicians—is to think of Ricci flow like applying heat with a hair dryer to smooth wrinkles in shrink-wrap: the flow evens out curvature so shapes become more regular.
Ricci flow worked well in many cases, but difficult manifolds developed singularities—regions where curvature blows up and the evolution cannot continue. Topologists devised a controlled "surgery" procedure to cut out singular regions and keep flowing, but it remained unclear whether singularities could appear endlessly and prevent the process from terminating.
Perelman’s contribution
Perelman's papers introduced several new ideas and technical tools that closed the gap in Hamilton's program. He proved monotonicity formulas (notably an entropy-type functional) and a no-collapsing result for Ricci flow, and analyzed the structure of singularities in enough detail to show they reduce to simple, well-understood pieces (like spherical or cylindrical regions). This made it possible to perform a finite sequence of surgeries and continue the flow to a canonical end state, thereby establishing that the original manifold is a 3-sphere when the hypotheses are satisfied.
Perelman's arguments were original and highly technical; it took years for the geometry and topology community to check and formalize them. In 2006, John Morgan and Gang Tian published a 473-page work that clarified and completed parts of the verification, and the broader mathematical community accepted that Perelman's work, building on Hamilton's foundation, proved the Poincaré conjecture.
The man and the response
Grigori (Grisha) Perelman had spent the prior decade doing postdoctoral research at several institutions in the United States before returning to St. Petersburg in the mid-1990s and joining the Steklov Institute of Mathematics. Colleagues described him as intensely private, uninterested in money or acclaim, and fond of hiking and mushroom hunting. One biographical anecdote often repeated is his terse response to a reporter in 2010:
"You are disturbing me. I am picking mushrooms."
In recognition of his proof, Perelman was offered the Fields Medal (2006) and the Clay Millennium Prize (the $1 million prize associated with the Poincaré problem). He declined these honors, reportedly citing objections about how credit for the resolution was being assigned. Perelman resigned from his Steklov position in 2005 and largely withdrew from public mathematical life.
Significance
The resolution of the Poincaré conjecture closed a century-old question and showcased a powerful synthesis of geometric analysis and topology. Perelman's work transformed the study of 3-manifolds and remains a landmark example of a deep, elegant solution emerging from modest, unassuming circumstances.
Timeline (brief): 1961 — Smale solves the higher-dimensional analogue; 1980s — Hamilton formulates Ricci flow program; Nov. 11, 2002 — Perelman posts first paper on arXiv; 2002–2003 — two more Perelman papers and talks; 2006 — Morgan & Tian publish extensive verification; mid-2000s — Perelman declines major prizes and retreats from public life.