- all simply connected three-manifolds (i.e. those containing no non-contractible one-dimensional submanifolds) can be continuously deformed to an "S^3".
Note that the topology of three-dimensional manifolds is the maximally difficult part of the research of topology of manifolds of different dimensions; this is also where knot theory occurs. It is not trivial to give a convincing conceptual explanation why it is so but it is much easier to show historical evidence for the difficult character of "n=3".
If you replace "S^3" by "S^n" in the conjecture above, all statements with "n" different from "3" have been proved years ago. For "n=1", the conjecture trivially follows from the fact that the circle is the only compact one-manifold. For "n=2" it follows from some classical facts about the Riemann surfaces. The "n=4" case was proved by Freedman in 1982, earning him a Fields medal in 1986. The "n=5" case was demonstrated much earlier, in 1961, by Zeeman. The "n=6" was proved by Stallings in 1962, and a 1961 proof by Smale covered all cases where "n > 6"; later, this proof was improved to include all cases "n > 4".
As you can see, only the "n=3" case was waiting. This is where Russian mathematician Grigori Perelman comes to rescue: see John Lott's website for details about Perelman's work (thanks, Mike Ros). In fact, Perelman, who has worked on the problems for 7 years in isolation, seems to have established an even stronger result, namely Thurston's geometrization procedure.
The difficult "n=3" case is the analogue of the whole string/M-theory at a generic values of the couplings, while the cases in which "n" is very small or very large, far enough from "n=3", correspond to certain weakly-coupled and low-energy limits of string/M-theory that could have been understood much earlier.
Richard Hamilton of Columbia University, the original father of the Ricci flow approach, now says that the work of two Chinese scholars, Cao Huaidong and Zhu Xiping - recall that the first names written in the sentence are their last names - the work that was published in the American journal called Asian Journal of Mathematics is "very important" for completing the proof. Shing-Tung Yau of Harvard, who has also worked on these problems, is "very positive" about the work of Cao and Zhu. He knows it well enough so that he described it at a seminar during the second day of Strings 2006 - a day freshly summarized by Victor Rivelles. The third day - with Brian Greene & cosmology, among other things - is also available.
Why Strings 2006? You should have learned, by now, that all good ideas in theoretical physics and continuous mathematics are parts of string theory, and most of the famous mathematicians in geometry and related fields are literally or essentially string theorists: Witten, Yau, Perelman, Atiyah, Singer, and many others. In this particular case, the inclusion in string theory occurs because the proof involves the Ricci flows - a method to deform one manifold to another that is essentially equivalent to the renormalization group flows on the string worldsheet.
The theories on the worldsheet must be conformal which is string theory's way to demand the spacetime equations of motion - the beta-functions are the equations of motion. If the worldsheet theory is not conformal, you will not get a consistent string-theoretical background but such a theory can still be useful for solving mathematical problems (or for understanding time-dependent configurations). Also recall, from initial chapters of string theory textbooks, that the beta function for the spacetime metric in a nonlinear sigma models is equal to the Ricci tensor (plus corrections) - which is the reason behind the terminology of "Ricci flows".
The proof is then rather simple and here's a sketch: take a non-linear sigma model (string theory) on your simply-connected three-manifold, and flow it to the infrared. The manifold can be seen to become increasingly smooth and the only possible endpoint is the three-sphere.
When we say that Cao and Zhu's contribution is very important, it might be a good idea, because of certain reasons, to quantify how important it is. ;-) Shing-Tung Yau and Yang Le, two mathematicians who know what they're talking about, propose the following quantification - note that 100% means $1 million:
- 50% Hamilton
- 25% Perelman
- 30% Yau, Zhu, Cao
Congratulations to Prof. Yau who has appeared in the list, too. Yau himself had convinced Hamilton to think about the conjecture at the very beginning, and his own contributions to mathematics and its fruitful co-existence with physics are great. There's not enough space to describe them here in detail. He is arguably among the mathematicians who have attended the largest number of physics seminars, too.
The numbers don't yet add up to 100% - the wrong precise values are probably a contribution of a journalistic imbecile - but I am sure that with a few extra years of work, they could fine-tune the details. :-) Perelman does not care about pathetic things like "money" so I am afraid that the mathematicians' consensus will ultimately eat the 5% to get the correct sum from his portion.
Regardless whether there are any risks that people will find serious holes in Zhu and Cao's work, this success has already ignited the jealousy of numerous morons. For example, "pubkeybreaker" believes that the Chinese mathematicians must be incompetent because he also believes that competent mathematicians would never write the following sentence (which is true according to everything we know):
- "This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow."
Well, "pubkeybreaker's" beliefs that this dumb comparative literature means anything in math are as silly as similar statements at a certain anti-physics blog: namely breathtakingly idiotic. The journalists should be very careful before they are influenced by random posters on the Internet. "Pubkeybreaker" also conjectures:
- Competent mathematicians do not pat themselves on the back in their own paper. They let OTHERS judge the work.
Quite on the contrary, "Pubkeybreaker". Competent mathematicians are defined as those who do not have to wait for others to decide about the validity - and indeed, also the value - of a proof. Also, I find their appraisal modest. They could also try to paint the initial ideas by Hamilton and the contributions of Perelman as incomplete non-rigirous heuristic speculations and they could try to present the proof purely as their work. Of course, I will consider the proof to be primarily Hamilton's and Perelman's victory. But I also understand that in mathematics, rigor plays an important role and there is a significant difference between a slightly incomplete proof and a complete proof.
The fact that a full proof of the Poincaré conjecture is the crowning achievement of virtually anything that contributes to the proof should be completely obvious to anyone who knows what questions in pure mathematics are important. It is important to say that there doesn't seem to be any disagreement about the credit and most of it will go to Perelman - but some people just like to generate vitriolic comments.