Perelman’s Proof

Perelman's proof of the Poincaré Conjecture demonstrated that any three-dimensional shape that is simply connected (without holes) and closed (no edges) must be a 3-sphere, a higher-dimensional analog of a sphere. He achieved this by developing new techniques in geometric analysis, specifically the Ricci flow with surgery, which gradually smooths and simplifies the shape without creating holes or other complexities. This process allowed him to classify 3D spaces, confirming that the only "hole-free" closed 3D shape is the 3-sphere, resolving a century-old mathematical question.