Poincaré Conjecture

Though I’m a little bit late on this, Science Magazine recently published a great article on the scientific breakthroughs of 2006. Topping the list was the proof of the Poincare Conjecture, which I’ve posted about several times on this blog. You can read their synopsis of the breakthrough proof here. It turns out that from the media’s perspective the drama behind the proof is almost greater than the mathematical result. Basically there was a lot of name calling among some members of the mathematical community concerning who made certain contributions toward the eventual proof. Sad. Apart from the soap opera, the author explains the Poincaré Conjecture in a very accessible way, which should be understandable by anyone who’s interested in reading it. This proof will be a huge deal for mathematics over the coming decades, and should help mathematicians better understand topics such as the “Navier-Stokes equation [of fluid dynamics] and the Einstein equation [of general relativity].”

Perhaps the most interesting thing to note is that the article focuses not only on the result of the problem (the proof itself), but also the methods used to solve the problem. This is an hugely understated part of the mathematical process. I’m of the opinion that when the general populace thinks about math that they are fixated on two things: the problem and the answer. What people tend to overlook is the process of problem solving. In math, there are not always clear-cut methods that explain how to get from point A to point B. A lot of thought is sometimes necessary to figure out how to traverse the path. The Poincaré Conjecture is a monumental achievement not only because of the end result, but also because of the original steps the solvers of the problem (especially Grigori Perelman) took to get there. These steps will be used in other problems; they are not exclusively tied to this one specific problem. Once again, congratulations to Perelman and the other mathematicians who had a hand in making this historic achievement!

I’m sure most of you have heard by now, but on Tuesday Grigori Perelman refused to accept the Fields Medal for his work on the Poincaré Conjecture. I’m sure there are people with all sorts of opinions about this, but the First Post has an interesting take on the situation. Also, if you’re looking for another explanation of why any of this matters anyway, I suggest reading Jordan Ellenberg’s Slate article here.

If you have no idea at all what I’m talking about, then you should read my first and second posts on the subject, which should supply a bit of mathematical background information and redirect you to more extensive resources.

It looks like the New York Times has a story today about the proof of the Poincaré Conjecture. You can read the article here. I just love how the media always plays up the “insatiably crazed introvert” image of mathematicians, in this case overly dramatizing Grigory Perelman, the Russian mathematician responsible for much of the proof methodology. I blogged about this a while back, but this article definitely gives a much more in-depth explanation of the history of the conjecture. Happy reading everyone!

The Xinhua News Agency website is reporting that two Chinese mathematicians, Zhu Xiping and Cao Huaidong, have proved the Poincaré Conjecture. Or rather, I should say, that the proof is currently being scrutinized by a panel from the Asian Journal of Mathematics, where the proposed proof will be published upon consensus.

According to its MathWorld entry, the Poincaré Conjecture states “that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes”, where “a three-sphere is simply a generalization of the usual sphere to one dimension higher”.

The Clay Mathematics Institute also had listed the conjecture as one of its Millenium Problems, each of which carried a one million dollar reward for a proof. Their explanation of the problem puts it this way:

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is “simply connected,” but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

As the article makes clear, the movement toward this proof spanned both decades and nationality. American, Russian, and Chinese mathematicians were instrumental in progression of ideas which finally cracked the conjecture. I hope that this example illustrates the global and community effort of current mathematicians in their work.

Congratulations to Zhu Xiping and Cao Huaidong!