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Russian May Have Solved Poincare Conjecture
nev4 writes "Reuters (via Yahoo News) reports that Grigori Perelman from St. Petersburg, Russia appears to have solved the Poincare Conjecture. The Poincare Conjecture is one of the 7 Millenium Problems (another is P vs NP, also covered on /. recently). Solving a Millenium Problem carries a reward of $1M, but apparently Perelman isn't interested..." nerdb0t provides some background in the form of this MathWorld page from 2003.
True math genius and the desire for money (and fame and babes, etc.) seem to be mutually exclusive traits and I think that's rather inspiring (and damned practical).
/. come form "anonymous cowards" sitting in their offices at MIT. What a god.
Take the case of Paul Erdos who was essentially homeless, but published over 1500 papers and is considered one of the all time greats in the field.
Perelman just casually posted his solution out to the web in much the same way that some of the most brilliant posts on
"...all the labours of the ages, all the devotion, all the inspiration, all the noonday brightness..." yada yada
But there's a snag. He has simply posted his results on the Internet and left his peers to work out for themselves whether he is right -- something they are still struggling to do.
Okay, so tell me how this is any different from every l33t user that tells me how to get my dual flat panel setup working under Xandros without editing the X files manually? Sounds like these kids just tried their hands at mathematics, too.
What's your damage, Heather?
I'm tired of seeing these 'please make me famous even though I didn't really prove it' threads. The little boy has cried wolf too many times. We don't care unless it's really solved.
Editors, I'm talking to you.
I can't believe slashdot would run a story with that title. "Perelman May Have Solved Poincare Conjecture" would have been much more dignified. You would never see "Muppet May Have Solved Poincare Conjecture" would you? Please, Perelman is a mathematician first, Russian second.
I want a new world. I think this one is broken.
"Grigori Perelman May Have Solved Poincare Conjecture"
I've noticed that these kinds of announcements often make a point of appending a nationality to the name of the person involved in the discovery. Surely this proof builds on mathematical knowledge from around the world. Or was Grigori Perelman standing solely on the shoulders of "fellow Russian" mathematicians? I highly doubt it...
As Balzac said, "there goes another novel."
"...all the labours of the ages, all the devotion, all the inspiration, all the noonday brightness..." yada yada
It's very easy. A rubber band around a sphere can slide along the surface so that the circle it forms becomes smaller and smaller, until it converges into a point. But if a rubber band is wrapped around a torus (doughnut) like a link in a chain (so that it goes through the hole in the doughnut), you can't slide it along the surface to make it any smaller than the cross-section of the torus nor can you detach it without cutting the band or the pastry.
The Poincare Conjecture involves hypothetical 4-dimensional shapes with the same properties, and isn't very easy.
Well, it could just be that the drive to do math, or whatever, is a subtle emergent thing, so when a stronger pull exists, like the time requirements due to a family, the drive towards academics becomes diluted. Plus, theres the peace and quiet of no kids/spouse running around, which is much more conducive to spending time thinking about a hard problem than constant ruckus.
(Ok, in reality, that's kinda short-sighted, as you could buy $1 million of computer time, but maybe he doesn't like computers.)
Computer time will only help with P problems, or P elements of NP problems. Great mathematicians seem to be NP-solving machines. A hundred years of computing time on the best computer might releive some of their tedium but would actually have an insignificant impact on their ability to solve problems.
The rest of us lesser beings might consider spending out time building a super-high resolution MRI machine. We'd want to be able to image every atom in a person's brain and record a year's worth of data at something like 100k samples per second. The MRI should be light and comfortable so our test subject could wear it comfortably for that year.
Once the suerp-MRI machine is ready, we manufacture it into a comfortable yet stylish (to the eyes of mathematicians) hat, and invite a prize-winning mathematician to wear it for a year.
At the end of the year, we need to locate some prize-winning neuroscientists to help us decode our brain scans and prize-winning computer scientists to help us build it.
The n-sphere (which mathematicians generally denote by S^n) can be thought of as `all points in (n+1)-dimensional space which are at unit distance from the origin'. So S^2 is the surface of a solid 3-dimensional ball. This sometimes surprises people, who expect this to be S^3 but the key observation here is that the 2 refers to the intrinsic dimension of the object, rather than the extrinsic dimension of any space you might happen to put (`embed') the object in. The fact that we often think of the 2-sphere as being embedded in 3-dimensional space doesn't change the fact that it's inherently a 2-dimensional object. An ant wandering around on it still only has two degrees of freedom.
The 3-sphere (S^3) locally looks like ordinary, flat, Euclidean 3-space, but on a larger scale it kind of doubles back on itself - if you keep walking (or floating) in a `straight line' (well, actually the 3-dimensional analogue of a `great circle', but never mind) in any direction, then you'll eventually get back to where you started.
The Poincaré Conjecture says
This, by itself, isn't particularly enlightening to the non-topologist, but what it actually boils down to is:
What does this mean?
Well, an `n-manifold' is a space which locally looks like ordinary, flat, Euclidean n-dimensional space. So a 3-manifold is a space (like S^3) which locally looks like ordinary 3-space (but which might twist back on itself in a peculiar way on a larger scale).
`Closed' means that the 3-manifold doesn't have a boundary - no matter how far you walk, you're not going to run into a brick wall, or fall off the end. `Compact' is a bit more technical, but in this context essentially means you don't get odd shooting-off-to-infinity stuff you have to deal with.
And `simply-connected' means that the first homotopy group (the `fundamental group' of the space) is trivial. What that means is that any closed loop (of string, if you like), in the manifold, can be continuously shrunk down to a point. Here `continuous' means that you're not allowed to cut or glue the string while you're doing it.
To use a 2-dimensional analogy, the 2-sphere (the surface of the 3-dimensional ball, remember, or alternatively a British doughnut) is simply-connected, because given any closed loop in the surface, you can shrink it down to a point without it getting snagged on anything. Whereas the 2-torus (the surface of an American doughnut) isn't, because you can't shrink all closed loops down to a point - one which goes all the way round the central hole, for example, can't be shrunk.
Finally, `homeomorphic' is basically a technical word for `topologically equivalent' - we allow continuous deformations (stretching, twisting, etc, but not cutting or pasting), rotations, reflections, or any combination of these.
So, the (classical) Poincaré Conjecture is essentially a technical way of saying ``If it looks like a 3-sphere then, basically, it is''. (For certain definitions of `is', and `looks like'.)
The analogous conjecture in n-dimensional space is known to be true for n=1 (trivial), 2 (pretty simple), and 5 and above (the 5-dimensional case was proved by Zeeman, who is my PhD grandsupervisor - my supervisor was one of his students). The 4-dimensional case is weird, and there are three different forms to consider - the `piecewise linear' and `topological' cases have been proved, but the `smooth' case is still unproven.
As I understand it, what Perelman claims to have done is prove Thurston's Geometrisation Conjecture, which implies the Poincaré Conjecture as a special case - rather lik
You're right because you can't see anything practical coming from it? *That* makes you right? Someone needs to take some logic courses.
No, Branges needs to prove that the counterexample does not apply to his theory.
It's not a court, Branges doesn't need to do anything - someone needs to prove it one way or another for the science to progress.
Future Wiki -- If you don't think about the future, you cannot have one.
What's often overlooked in Maslow's heirarchy of needs is the fact that it is a heirarchy. In other words, it's all well and good to be self-actualized, but you need to have your rent and food bills covered first . You can't just skip from "poor starving genius huddled in an alley scrawling your brilliance in feces on the walls" to "self-actualized."
Yeah, I skimmed his paper, and noticed that as well. Apparently, "apology" in this context means a proof that has not yet been subjected to peer review, but which the author is deeply convinced is correct. Pasting some output from a dict apology, it seems:
Clearly, de Branges is using the term in this sense.
It doesn't really matter though, because if you actually read his paper, the first third is all incomprehensible background nonsense about the nature of the problem, while the last third is all incomprehensible arrogant nonsense about what he wants to do with his prize money. The actual meat of the paper is buried somewhere in the middle, but it's like that's all just an afterthought to the guy's mad ravings about his place in history and his imminent wealth.
He couldn't be more different than the person that seems to have solved the Poincare conjecture. Where Perelman is silent behind a paper that seems to concretely prove not just the problem at hand, but a whole broader class of problems, de Branges has this ridiculous paper that goes on and on about what a big shot he is, while stomping around his university like a little tinpot Napoleon. I'm no math whiz, but hot air isn't always hard to recognize...
DO NOT LEAVE IT IS NOT REAL
This is a useful quote, but don't take that "require" too literally. Yes, you are perfectly justified not to believe that the conjecture is solved, until you read about it in Science - this is reasonable. What is not reasonable is to think that de Branges is somehow required by someone or something to provide extra proof or something else. He doesn't have to if he doesn't want to.
Just like Perelman, de Branges can just sit and wait. Hopefully, someone will eventually read the proof thouroughly and see whether there are any significant errors. And then everyone will benefit from knowning whether the proof exists and if yes, what it is and that the conjecture is true.
Future Wiki -- If you don't think about the future, you cannot have one.