Slashdot Mirror
Poincare Conjecture Proof Completed
Flamerule writes "A New York Times article has finally provided an update on the status of Grigori Perelman's 2003 rough proof of the Poincaré Conjecture. 3 years ago, Perelman published several papers online explaining his idea for proving the conjecture, but after giving lectures at MIT and several other schools (covered on Slashdot) he returned to Russia, where he's remained silent since. Now, mathematicians in the US and elsewhere have finally finished going over his work and have produced several papers, totaling 1000 pages, that give step-by-step, complete proofs of the conjecture. In addition to winning some or all of the $1,000,000 Millennium Prize, Perelman now seems to be the favorite to receive a Fields Medal at the International Mathematics Union meeting next week, but it's not clear that he'll even show up!"
The chinese press distorted the news:e nt_4644754.htm
http://news.xinhuanet.com/english/2006-06/04/cont
Google your friend. ANAM (I'm not a matematician), but I'll try.
According to string physicist Lubos Motl the proof indeed important to string theory. The proof based on the flow on the manifold (surface), analogous to heat dissipation - Ricci flow. This flow deform metrics (distance between points of the surface). But this process also describe renormalization of worldsheet - how the physics of the worldsheet (surface which string drawing, moving in space and time) change with changing of the observation scale. That is how phisics of string change then the scale of calculation changed.
Now that the conjecture is proved, do they change the name to "theory"? Or does the name stay put because that's what everyone knows and refers to it as?
Things that are proven, are called theorems. They do depend on axioms, but those are defined as true. Sciences about the real world that can't put up axioms (because that'd require ex facto knowledge about the real world), so they can never be conclusively "proven". Hence well call them theories, like theory of gravity, theory of evolution. A few we've called "laws" as well because they have been so extensively tested, but it is not proven in a strict formal sense.
Live today, because you never know what tomorrow brings
If any of you had read the article you would have noticed that the 1000 pages is actually a very rough figure for the sum page count of all 3 articles by various people each of which explains Perelmans result in context, thus duplicating the other 2. So in fact the full articles are about 315-470 pages each. Also what Perelman infact did was show that using the Ricci Flow technique on the 3D shapes to solve the Poincare conjecture, an idea of Hamilton's from the 80's, can work. Up till now it was thought that certain structures might degenerate to singularities and fail, but Perelman showed that those singularities would in fact all turn out ok. Poincare's conjecture is for 3D shapes, and higher dimensional generalisations have previously been solved (5+ dim by Smale in 60's, 4 dim by Freedman in 80's, both got Field's medals).
Nice bit of jingoistic xenophobia there, but that's about all that's nice about your post.
Gang Tian, who has co-wrote a guide to Perelman's proof, said in 2004: "He certainly has no interest in material things. If he gets the Fields Medal, there is the issue of whether or not he will accept it." He also refused a prize from the European Mathematical Society many years before that.
He is not being threatened, he is simply a person with little interest in material matters.
"I've got more toys than Teruhisa Kitahara."
The Jacobian, or unit volume if you will, of a hypersphere has a a highest term of sine, or cosine, which grows as you increase dimension. Specifically, for an n dimensional sphere, the highest power of sine or cosine will be sin^(n-2).
Anyway, to answer your question, integrals of sine or cosine to odd powers produce only functions of other sines and cosines. However, integrals of sine or cosine to even powers produce functions of sin(x), cos(x) and x. The x part gives you your pi, but only does so every second dimension, when the highest power is even.
Here's the integrals of (sin(x))^n, for various n
n=0: x
n=1: - cos(x)
n=2: x/2 - sin(2x)/4
n=3: 1/3 * (cos(x))^3 - cos(x)
n=4: (sin(4 x) - 8 sin(2 x) + 12 x)/32
May the Maths Be with you!