This entry describes the solution of the Poincaré conjecture at a level intended for the general public. The proof described is that of Grigori Perelman using the Ricci flow developed by Richard Hamilton. Links to other expositions for general readers are included below, together with links to the original research papers.
The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected (so lassos cannot tie around it) is a 3-dimensional sphere. To understand the statement one needs to understand "manifold", "compact", "no boundary", "simply connected" and "3-dimensional sphere" and so all these concepts are described below. Perelman and Hamilton proved the conjecture by deforming the manifold using something called the Ricci flow (which behaves similarly to the heat equation that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself (like hot mozzarella) towards what are known as singularities. Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery") causing the separate pieces to form into ball-like shapes. Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and wondering whether the surgery might need to be repeated infinitely many times.
A one-dimensional sphere is a circle, which can be thought of as the set of points, (x, y), in two dimensions that satisfy the equation x2 + y2 = r 2, where r is the radius. A two-dimensional sphere is the surface of a globe, or the set of points, (x, y, z) in three dimensions that satisfy the equation x2 + y2 + z2 = r 2. And a three-dimensional sphere is the set of points in four dimensions, (x, y, z, w), that satisfy the equation x2 + y2 + z2 + w2 = r 2.
A manifold is a space created by gluing together pieces of Euclidean space, called charts. For example you could take two 2-dimensional disks and curve them around to hemispheres and then glue them together to make a 2-dimensional sphere.
You could also build a torus (the surface of a bagel) using a rectangular chart as seen in this image.
You can build a 3-dimensional sphere using a pair of solid 3-dimensional balls: Identify each point of the boundary of the first ball with the corresponding point of the second ball.
Other manifolds can be created in similar ways. See manifold for an easy and advanced description. Manifolds can be warped or distorted using diffeomorphisms.
We say a manifold has an edge or a boundary, if one of the charts is not glued to another on all sides. One of the conditions in the Poincaré conjecture is that there be no edges, just like in the sphere and the torus.
A compact manifold is bounded and does not extend to infinity. Both an infinitely long cylinder and an infinite plane are examples of manifolds which are not compact. In Poincaré's Conjecture it is required that the manifolds be compact. See compact for an advanced definition.
A manifold is simply connected if any loop drawn on the space can be deformed to a point without leaving the manifold. An example of a simply connected manifold is a sphere. If you try to wrap a lasso around a sphere it will slide off. An example of a manifold which is not simply connected is a torus. One can tie a lasso around a bagel and catch hold of it. Nothing short of untying the lasso or cutting the bagel will let it loose. See simply connected for an easy and advanced description.
Putting all these terms together, we can now understand the statement of the Poincaré conjecture:
The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected must be a 3-dimensional sphere.
The first step is to deform the manifold using the Ricci flow. The Ricci flow was used by Richard Hamilton as a way to deform manifolds and he used it to prove that many compact manifolds were diffeomorphic to spheres. He did not prove they were all diffeomorphic to spheres. The Ricci flow is an imitation of the Heat flow equation which describes the way heat permeates a room. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. These singularities have been likened to the strands in mozzarella cheese.
Hamilton was able to list a number of possible singularities that could form but he was concerned as to whether he had found all possible singularities. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again. But he needed to understand the singularities. Grigori Perelman examined the singularities and discovered they were very simple manifolds: essentially three dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line.
This was proven using something Perelman called the "Reduced Volume" which is closely related to an eigenvalue of a certain "elliptic equation". Eigenvalues are difficult to describe without calculus but they are part of a famous problem: Can you hear the shape of a drum?. Essentially an eigenvalue is like a note being played by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the cigar soliton, which looked like a strand sticking out of a manifold with nothing on the other side. In essence Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.
Completing the proof, Perelman takes any compact, simply connected, three dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until eventually he is left with a collection of round three dimensional spheres. Then he rebuilds the original manifold by connecting the spheres together with three dimensional cylinders, morphs them into a round shape and sees that, despite all the initial confusion, the manifold was in fact diffeomorphic to a sphere.
Two immediate questions were then: how can one be sure there aren't infinitely many cuts necessary? That the cutting does not progress forever? Perelman proved this using soap films on the manifold and showing that the areas of the soap films decreases as the manifold undergoes Ricci flow. Eventually the area is so small that any cut after the area is that small can only be chopping off three dimensional spheres and not more complicated pieces. This is described as a battle with a Hydra in Szpiro's book cited below.