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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!"
Talking about 1 million prizes from the Clay Institute, these two people claim they deserve one with 13 pages (>$63k/page)
http://arxiv.org/abs/math.AG/0608265
but of course many of us are a bit suspicious.
I am in the field, and I am pretty sure that there is no application of this conjecture to any branch of physics at the moment (in particular, for string theory). See also this:3 4
http://www.math.columbia.edu/~woit/wordpress/?p=4
(Peter's answer to Cynthia question)
P.S. what is this crap?
"Slashdot requires you to wait between each successful posting of a comment to allow everyone a fair chance at posting a comment.
It's been 10 minutes since you last successfully posted a comment"[...]
I will wait for the reader-friendly version. Reader's Digest, Simon Singh, Mario Livio where are you all?
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
According to The Guardian
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.
I'm trying to glean what some of the practical implications could be of this discovery.
It seems to me at this Ricci Flow differential equation could be quite useful practically. For example, in pattern recognition, if a computer could build a 3d model of an object using multiple vantage points, then simplify the object to one of the handfull of object types described by Perlman using the Ricci Flow, then this simple catagorization might help in the identification of complex objects (e.g. a donut really is a donut, even if it's been heavily frosted).
Do you know if Perlman's technique for handling the singularities will help with the numerical implementation of this process? Or are these issues numerically simple to solve - but only challenging to solve in proof?
TFA mentions he has distanced himself from others in the Math community because he has become disillusioned. I read into that my own experience, which involved professors trying to hit on me, others trying to get me to write/edit their papers and then taking the credit, others who weave tall tales with just enough truth to fool grant money providers.
One of my colleagues now believes that Science is actually performing a random walk on the landscape of Truth. Occasionally, the walk stumbles over something meaningful, and it's called progress.
For pattern (image) recognition the geometry is quite important, since usual applications are essentially trying to mimick the human behaviour, and humans in practical life are more geometers than topologists.
I do not believe in karma. "Funny"=-6. Do good and forbid evil. Yours, Oft-Offtopic Flamebaiting Troll.
I really don't know anything specific about Perlman's technique, so I have no idea how effective or abstract it is (i.e. whether a computer could implement it).
As far as object recognition goes, remember that what we mean when we say a 3-dimensional manifold is a space that has three dimensions everywhere, not an object which can be embedded in 3-dimensional space. In fact, a 3-dimensional manifold may require as many as 7 spacial dimensions to be embedded in ordinary euclidean space, and even more may be required if the embedding actually preserves distance, and not just topological properties.
What you seem to be referring to is to have a computer tell what an object is by looking at it's surface, which is a 2-manifold, not a three manifold. There are mathematical programs that can identify the type of a surface, and these use triangulations rather than Ricci Flow, but I'm not sure if such methods have ever been used to identify real-world objects.
If you're looking for real world applications of Ricci Flow or Perlman's surgery methods, I think the closest you'll get for the moment is theoretical physics. Of course, I could be wrong - sometimes seemingly very abstract mathematics has turned out to be very useful.
David