Geometrization

The geometrization conjecture, proposed by William Thurston in 1982, states that every closed 3-manifold can be decomposed into pieces, each of which admits one of eight possible geometric structures. This means that the wild diversity of 3-manifolds is not arbitrary but is constrained by the same geometries that describe homogeneous spaces: spherical, Euclidean, hyperbolic, and five others. The conjecture was proved by Grigori Perelman in 2003 using Ricci flow, and it implies the Poincaré conjecture as a special case.

The geometrization theorem is a classification result of extraordinary scope: it says that the topology of three-dimensional space is not infinitely complicated but is generated by a finite alphabet of geometric templates. In this sense, it is the 3-manifold equivalent of the periodic table: the elements are few, but the compounds are infinite.