# Perelman's work leading to the 2006 Fields Medal

John Lott described Perelman's work leading to the award of a Fields Medal in a lecture he gave to the International Congress of Mathematicians in Zurich in August 2006. Here is an extract from his lecture:

Grigory Perelman has been awarded the Fields Medal for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.

Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Aleksandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman's results in Aleksandrov geometry are summarized in his 1994 ICM talk. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture.

Soul Conjecture (conjectured by J Cheeger and D Gromoll in 1972, proved by Perelman in 1994):
Let $M$ be a complete connected noncompact Riemannian manifold with nonnegative sectional curvatures. If there is a point where all of the sectional curvatures are positive, then $M$ is diffeomorphic to Euclidean space.

In the 1990s, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints posted on the arXiv in 2002-2003 [The entropy formula for the Ricci flow and its geometric applications; Ricci flow with surgery on three-manifolds; Finite extinction time for the solutions to the Ricci flow on certain three-manifolds], Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture.

The Poincaré conjecture dates back to 1904. The version stated by Poincaré is equivalent to the following.

Poincaré conjecture:
A simply connected closed (= compact boundaryless) smooth 3-dimensional manifold is diffeomorphic to the 3-sphere.

Thurston's geometrization conjecture is a far-reaching generalization of the Poincaré conjecture. It says that any closed orientable 3-dimensional manifold can be canonically cut along 2-spheres and 2-tori into 'geometric pieces'. There are various equivalent ways to state the conjecture. We give the version that is used in Perelman's work.

Geometrization conjecture:
If $M$ is a connected closed orientable 3-dimensional manifold, then there is a connected sum decomposition

$M = M_{1} \# M_{2} \# ... \# M_{N}$

such that each $M_{i}$ contains a 3-dimensional compact submanifold-with-boundary $G_{i}$ a subset of $M_{i}$ with the following properties:
1. $G_{i}$ is a graph manifold.
2. The boundary of $G_{i}$, if nonempty, consists of 2-tori that are incompressible in $M_{i}$.
3. $M_{i} - G_{i}$ admits a complete finite-volume Riemannian metric of constant negative curvature.

In the statement of the geometrization conjecture, $G_{i}$ is allowed to be the empty set or $M_{i}$. (For example, if $M = S^{3}$ then we can take $M_{1} = G_{1} = S^{3}$.) The geometrization conjecture implies the Poincaré conjecture. Thurston proved that the geometrization conjecture holds for Haken 3-manifolds.

Last Updated July 2011