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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!"
What kind of strange rabbits have these topologists seen? The rabbits I've seen have a hole from end to end through them called the digestive tract.
AccountKiller
$1,000,000, 1,000 pages, those numbers are apprpriately round for the occasion.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
wait! Don't dung beetles roll their dung into balls? And what does that make it? A sphere! There's some connection here, I swear. Whoa...
The incredulity that this mathematician might have been more interested in the challenge of the work than fame and fortune in the Western world practically oozes from each sentence.
I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake? Perhaps he's gone on to other challenges, or he's wrapped up in some research that has his complete attention. Heck, perhaps he just enjoys math for its own sake and doesn't want to deal with all the side-effects of notoriety.
Interested in a Flash-based MAME front end? Visit mame.danzbb.com
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?
First of all, I highly doubt that all of those 1000 pages are devoted to solving the Poincare Conjecture. Perelman, if I remember correctly, studies Ricci curvature flows which is a large area of mathematics in its own right. In the course of his research, he discovered some things that led to this proof of the Poincare Conjecture. I would expect that the 1000 pages referred to by this article deal with many different consequences of Perelman's work. Mathematicians like to do things in full generality, so they would have studied broader consequences instead of focussing for so long on only one result.
Secondly, I would invite you to write down a complete proof of some well-known mathematical fact, the Stone-Weierstrass theorem say. You must prove this from first principles, starting with axiomatic set theory. I would be very surprised if you even managed to finish and even more surprised if the proof came in at under 1000 pages. This highlights what was mentioned by a sibling of mine: mathematics is divided into small steps and you would never dream of trying to prove something all at once.
Thirdly, this is the first ever proof of the Poincare conjecture. It is quite common in mathematics that a nicer proof of a known fact will be found.
Don't you hate meta-sigs?
Haha.. oh that's rich. "Please Mr. Perelman--flee from the military-industrial complex. Come to a sanctum of human rights and democracy. Come to ... [wait for it] ... America!"
The reason they can't find him in Russia is because he's already living in Sweden.
I think there is a world market for maybe five personal web logs.
In Eastern Europe we don't pick up mushrooms to get narcotic high. It is merely a popular ingredient in our cuisine. The guy got his priorities right. No matter how rich and famous you are, in the West you cant get exactly the same ingredients for East European food. As mushrooms based meals are so delicious, I wouldn't be bothered to travel somewhere to get some stupid price when there is high season for mushrooms.
I'm not a geometer, but here is my understanding of the proof:
The Ricci Flow was defined by Richard Hamilton in 1981 as a step towards classifying topological compact 3-manifolds. Classifying 3-manifolds would certainly decide The Poincare Conjecture, as it states that all simply connected compact 3-manifolds are homeomorphic to the sphere. This is an important special case: most proofs of the classification of compact 2-manifolds start out by proving the an analogous statement for the 2-sphere. The Ricci Flow is a differential equation which defines how the shape of a manifold changes in time: given an arbitrary manifold M(0), you can apply the differential equation to it to get manifolds M(t) for (some) positive t, which gradually change shape. However, the Ricci Flow is not volume preserving, so you "renormalize" so that M(t) has constant volume.
The Ricci Flow has the useful property that it tends to make manifolds smoother and smoother. For example, if you started out with a lumpy ball, you would eventually get a smooth ball. It was hoped that it could be proved that if the initial manifold was a compact simply connected 3-manifold, then as t increased, the manifold would tend towards a 3-sphere. Unfortunately, while locally solutions to differential equations always exist, they don't necessarily exist for all time, and for some starting manifolds, eventually you would get to a road-block: a t for which M(t) could not be defined. What Perlman (hopefully) showed was that all road-blocks were of certain types, and that a surgery could be formed that would modify the manifold but not it's topological nature, and then you could again apply the Ricci Flow, until the manifold became a sphere.
Note that this method is useful beyond proving the Poincare Conjecture, as it (again, hopefully) describes all road blocks to extending the Ricci Flow, so that the same tools can be applied to any 3-manifold, and not just simply connected ones. In this manner, assuming Perlman made no mistakes (or that any mistakes can be corrected), it is possible to apply the same arguments to prove the Geometrization Conjecture of Thurston, which classifies 3-manifolds.
David
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).
According to The Guardian
A Scottish physicist two centuries ago sees a strange bump-like waveform in a canal. It persists for over three miles, moving at nearly constant speed along the canal trench. He writes a paper, calling it a soliton wave and two Dutch mathematicians find a nonlinear partial differential equation that describes its motion. The equation, the Korteweg-De Vries Equation, proves fiendishly hard to solve. Finally, the crew working on the hydrogen bomb, finish the job early, so Ulam decides to use ENIAC to help him solve the Korteweg-De Vries Equation. He attains the first analytic solutions, and the study of soliton waves begins in earnest.
How does this earn a quid? Well, solitons model the way that blips of light move down a fiber-optic cable. The military decides that DARPA-net could run on fiber-optic cables, and uses them in building the early internet. Cellular telephone companies begin using fiber-optic cables to pack 100,000 phone conversations into a single pipe in such a way that they all get separated on the other end of the pipe-- one of the great engineering marvels of our time. We owe the modern internet, cell phones, anything that uses fiber-optics, to the solution of the Korteweg-De Vries equation. There was a similar burst of technology earlier in the last century when some closed-form solutions of the Schrödinger Equation were found.
Truth is, when we solve a major math problem like the Poincaré conjecture, billions of dollars of revenue are generated by new technologies that spring into being because of the new scientific understanding that the solution affords us. A thousand Adam Sandlers will not generate the amount of capital that the solution of the Poincaré conjecture will generate, especially considering that Perelman has shown the world that the Millenium Prize Problems are actually solvable.
"Indeed, it is wise never to consider any form of electronic data as final." --Arnold Robbins
Nothing extraordinary really. In USSR, mathematics (as well physics) was just one of the top prioritized subjects. As one of my german friends compared me and his son, we soviet pupils have had about twice more mathematics during our school times.
Mathematics is not about numbers and problems - it teaches brain to think. Nothing more.
All hope abandon ye who enter here.
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!
1) I met him at the Mathematical Sciences Research Institute in Berkeley at a workshop sometime around 1994 and he at that point had ridiculously long fingernails and was quite unkempt, even by the quite weak standards applied to research mathematicians. That was a while ago, of course and that was probably one of his first visits to the US. He gave an incomprehensible energetic talk so what most people commented on was his nails.
2) In 2003 or so, during a limited lecture tour about his proof of the Poincare Conjecture, he responded deftly and hilariously to a comment of Misha Gromov in the audience. Gromov is one of the most difficult people to have in a talk- he is a great mathematician with not much patience and has derailed or rerouted talks by many great researchers, who sometimes get quite flustered. I can't remember the exact wording of the exchange, which is too bad since it was precious, but Gromov asked something like "I don't see how that goes, I'd like to see some more details" and Grisha responded with something like "well, yes, you would" and carried on as he had intended.
(forgive me)
In Soviet Russia, mathematics teaches you.
- None can love freedom heartily, but good men; the rest love not freedom, but license. -- John Milton