(Update 07.28: a further development)

It seems that I am a bit late on the bandwagon.

My friends Aleks and Aaron, both geometric analysts (at least in some point of their careers), confirmed that they had been well aware of this development.

But in any case, I just found out yesterday that the Asian Journal of Mathematics carries in its 2nd issue of volume 10, a 300+ page paper titled A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow, written by Huai-Dong Cao and Xi-Ping Zhu. The article is available for free download at the above website. A cursory glance suggest that it is written at a very introductory level and is fairly self-contained: very suitable for a graduate student to read.

In the community, it is well accepted that someone well come forward eventually and write the whole thing down--after all, for most people familiar with the progress of Ricci flow, the conjecture is already solved, albeit no complete proof explicitly written until now. This is due to, primarily, the large amount of literature and techniques needed to complete a proof of the geometrization conjecture in its entirety. To explain this, we go back and study a bit of history.

Early history: from Poincaré to Thurston

The problem was first posed by Henri Poincaré:

Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

The question came about through his own study of topology. Poincaré first made a claim in 1900 that the homology of 3-manifolds is sufficient to discern a 3-sphere; he however made a discover in 1904 that there exists a 3-dimensional topological space with the same homology as the sphere, yet with non-trivial fundamental group. A similar statement naturally follows for higher dimensions:

Every closed n-manifold which is homotopy equivalent to the n-sphere is actually homeomorphic to the n-sphere.

To put it in laymen's terms, the conjecture is about whether it is possible to continuously deform a given space to the sphere. To examine the statement, we look at the model case of 2 dimensions. The 2-dimensional sphere is the hollow ball. Suppose we have a hollow rubber ball. We can stretch it, shrink it, push on it anyway we want. It is easy to see that we can deform it so that it becomes a ellipsoid (a ball that is flatter on one side than others). We can also deform it so that it becomes a cubic surface (if you put a deflated ball in a box and then inflate it, it would take on the shape of the box). Similarly, you can put it into the shape of a soda bottle, or a lightbulb. But if you start with a ball, there's no way we can just stretch it to the shape of the inner tube. Nor can we flatten it to become a single sheet without cutting the ball open. Therefore we say that the disc, the sphere, and the torus are not homeomorphic to each other. The same concept of stretching can be applied to three dimensional (or higher dimensional) objects. It is hard to visualize, however, since the stretching mostion can only be visualized in a space of even higher dimensions. The above explains the concept of homeomorphisms. Roughly speaking, homotopy equivalence is a slightly weaker version of homeomorphism. One can think of it as the study of how circles behave on the manifold: roughly speaking, two manifolds are homotopic to one another if there exists a certain pair of maps that take a point from one manifold and gives the point on another. The pair of maps must be well behaved on loops: if we have a loop in one space that can be shrunk to a single point, its image of under the maps must also be able to be shrunk to a single point, and vice versa. For example, on the hollow ball, every sphere can be shrunk to a point; on the inner tube, however, such is not possible: you can draw a circle on a tube around the hole which, because of the presence of the hole, cannot be deformed to a single point.

Solutions for the generalized conjecture exists in many dimensions. The case for one dimension is trivial, since the homotopy equivalence would natureally induce the homeomorphism required. For two dimensions, the answer is positive and has been known for a long time (the proof is not too intimidating, using, for example, the theory of Riemann surfaces. There are also many other methods of obtaining the proof: using Riemannian geometry, one can show, via the exponential map and explicit calculations on the curvature tensor that, it is possible to put a metric with constant positive Gauss curvature on the manifold, which implies a diffeomorphism to the 2-sphere; via conformal geometry, one only needs to solve a particular partial differential equation [the solution is actually quite non-trivial in this case]). The case for high dimensions (first ≥ 7, then ≥ 5) were first tackled by Stephen Smale, who won a Fields medal for it in 1966. The 4-dimensional case was solved by Michael Freedman, who also received a Fields Medal in 86. (Strangely enough, this 20-year difference might begin to form a pattern as it is speculated that Perelman might win the one this year for his work on this conjecture.) All that's left is the three dimensional case.

Our story takes a turn as William Thurston joins the chase in the 70s. He originally studied knot-theory, from which he developed a theory of hyperbolic 3-manifolds. Using similar techniques, he is led to his famous "geometrization conjecture". The exact statement is difficult (and I don't claim to even understand what it says fully), but the basic idea is the following: every 3-manifold can be decomposed as a "sum" of smaller pieces; each of the smaller pieces must be one of the 8 fundamental types of manifolds known as Thurston model geometries. As a special case, Thurston's conjecture implies the Poincaré conjecture.

At the same time that the beautiful interplay between geometry and topology led Thurston to his conjecture, significant progress has been made in the field of geometric analysis by the likes of S.T. Yau, R. Schoen, C. Taubes, K. Uhlenbeck, S. Donaldson, and many more. The most important piece relevant to the geometrization conjecture, however, is provided by Richard Hamilton. In the 80s, Hamilton began to study the Ricci flow. The flow evolves the Riemannian metric of a manifold by its Ricci curvature. It is, in a sense, a geometric analogue of the Heat flow.

The Ricci flow of Hamilton

The heat flow is described by the heat equation, a consequence of Fourier's law of conduction. In physical terms, the heat flow described the tendency for heat to average itself on an object. A heuristic description is like this: if we think of heat as money, and points as people, we can say that the dynamics described by the heat flow is as follows,

Every second, each person polls his neighbors, and asks: "hey guys, how much money do you have?" After getting the answer, he averages them out. If he finds that he has more money than the average, he gives some of his money away to his neighbors; and if he finds that he has less money than the average, he steals some from those people around him.

In a group of people with no external source of income nor any sort of expenditure, one can easily see that this dynamic would stablize after a while and end up with everyone having the same amount of money (this corresponds to the sourceless/sinkless heat equation which stablizes to a constant on a compact manifold). In the case where there is a source and a sink (think of it as some people having net income, some people balances their income with expenditures, and some people have a net spending; of course, we do not count the amount one gives away or steals as a result of the above dynamic as part of the income or expenditure), one would observe some sort of trickle down effect.

The Ricci flow behaves similarly: if a point has excessive curvature, it tries to pass it to points near it; if a point has not enough curvature (compared to its neighbors), it tries to obtain it. The net effect should be that the Ricci flow smoothes out bumps in the manifold and makes the entire manifold uniformly curved (of course, the precise description is a lot more difficult due to the structure of the equation). Indeed, Hamilton showed early on that if one starts with a manifold that has positive Ricci curvature everywhere, it would eventually smooth out to some object derivable from the sphere.

The problem is that if one starts with some portions that had negative Ricci curvature to begin with. The dynamics of the Ricci flow is more complex than that of the heat flow, and in this case, there could be accumulation of curvature in a single point. When Hamilton laid down his program for using the Ricci flow to prove the geometrization conjecture, he realized that those accumulation points are precisely the big hurdles he needs to pass. By assumine those hurdles can be cleared, Hamilton could already write down the remaining steps needed to prove the Thurston conjecture. And this is where Perelman came in. The result of G. Perelman's seclusive studies is precisely the steps required to clear those hurdles.

Back to today

Arguably, the largest contributors to the resolution of the geometrization and Poincaré conjectures are Hamilton and Perelman. But a problem arises: Hamilton's program would be incomplete without the work of Perelman, yet Perelman's work is very difficult to understand, and several small parts of it are still not published. While the community has already accepted the individual works of both parties to be essentially correct, no one has actually sat down and written the entire proof. So the work of Cao and Zhu is raising a few eyebrows in the community, particular since many people have been expecting the first such paper to come from Perelman himself. And, to get into a little bit of politics, there's the additional problem of recognition. If it were purely a case of academic prestige, everyone know that Hamilton and Perelman are the guys to put the credit to. But to this proble, there are two strings attached: the Fields medal and the Millenium Prize from the Clay Foundation. The former is perhaps less of a concern since it is an aware for significant contribution, but the latter, being a specific prize for specific problems, might be difficult. In particular, since this is most certainly the first Clay prize to be given out, it would be interesting to see how the committee evaluates the relative merits of the many people who made contribution to the solution of the problem.