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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!"
Goddamn I love freaky misfit mathematical geniuses. They're even better than their nerdier cousins, the chess geniuses. The ones from Central/Eastern Europe and South Asia always seem to be the most fun.
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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
Most of the freaky genius mathematicians who can do the really wierd stuff are usually (but not always) high MIPS, low I/O types anyway. Spend a week coming up with a partial proof of one percent of a subproof for a much larger problem, no problem. Contemplate going out of the house for bread and milk. See if you can get it delivered, or maybe get someone else to do it (you know, someone you know, someone you won't have to talk to very long...)
$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.
If I were known for proving Poincare Conjecture, I wouldn't give a damn to be known as a Fields medal winner. (They'll give it to him anyway, whether he's there personally to receive it.)
Isn't the answer 42?
Also, Occam's Razor would suggest it to be a ridiculous outcome. If I believed in this I might as well just believe in Santa Clause, the Tooth Fairy, or that crazy man on the moon myth!
Keep in mind though that some of the easiest problems to solve (in say for instance, graph theory) require multiple steps-- and as is stated in the original post, there's a proof for each step.
So it's not really a 1000 some odd page proof for an entire concept, but a 1000 pages of proof s for each individual component.
People who tried to do it on 999 pages or less all failed.
The chinese press distorted the news:e nt_4644754.htm
http://news.xinhuanet.com/english/2006-06/04/cont
Yes. But how do you know any one of those steps you've taken is really correct? Whoa. That makes you think twice, doesn't it?
Hey dung beetle, it's called peer review. Google it.
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.
He realized his time is running out and he wants to solve more problems. Maybe he has started solving another problem and dosen't want any outsiders to disturb him. Didn't he do the same with this problem? Maybe that's why no-one can contact him...
I remember that is was important to string theory, I just don't remember why. I did a search and found nothing. Can anyone elaborate?
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
Quite an interesting character, this Perelman, and his proof could turn out to be a real landmark for mathematics.
I liked this bit:
Whatever he's smoking, I want some!
Soylent Green is peoplicious!
I can get $1,000,000 by answering 15 questions on "Who wants to be a millionare", or even better yet, by giving money to some poor nigerian who can transfer vast sums of money into my account :)
(Am I the only one who read the title "Porncare conjecture proof completed"?)
"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 curse of the gifted is that niggling worry in the back of the mind that if one accepts praise, one may lose his focus, drive or muse, if you will.
Actually, they roll the dung of other animals (e.g., large mammals) into balls, which of course are of spherical nature.
Indeed, if you study the field of topology, you would realize that we've proven that you're actually just a morphed ball of elephant dung.
I think the greatness of the prize isn't the mercenary value people seem to think it holds. The money just shows importance. The prize's value comes from the dialogue and new paths of discovery that are opened up. Remember that in the end Fermat's last theorem (proof of which is what prompted this, at least in part) wasn't important in its result. It was important because the search for a proof resulted in huge new areas of research that are much more fruitful both in the purely abstract mathematical sense and in the practical sense. The fruits of that labor wouldn't have come out without placing such emphasis on the problem. Hilbert's lecture at the beginning of the 20th century was similar. Here was (one of the best minds at the time propising a framework in which to work, goals to look towards. Not even close to all of them have been resolved, but they are smart problems that have led to all sorts of applications and results. It's a goal to work towards. The Clay prize does the same thing. Is the Navier Stokes problem that important? Yes, that's why we have this great initiative for a derivation of classic and not weak solutions, or at least existence. The quest for the solution to the problems and those like it have created real progress. Without this kind of framework, we'd possibly not have the amazing work in PDEs and weak solutions that let us do great composite designs and image processing (to name two areas).
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?
just found a girlfriend? //I keed.
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.
Mathematician: "Now watch as I make this remainder diiisaaappearrr"
Lisa: "But seven goes into twenty-eight four times"
Mathematician: "Uh... this is a magic seven"
Does this make my brain look big?
Correction it was the circumference not area
"To be is to do." --Socrates
"To do is to be." -- Aristotle
"Do-Be-Do-Be-Do..." --Sinatra
The USA, instead, is not subject to problems of abuse of the legal system, as the case of Dmitry Sklyarov demonstrated.
Oh no, not in Sweden! We should send a rescue party before the socialists and insane feminists get to him. He may be taxed to death!
Comment removed based on user account deletion
Remember, the proofs that appear in textbooks are very polished, and use many lemmas that were proved earlier. The 1000 page proof includes proofs of all lemmas needed to complete the full proof, and there may be some redundancy as multiple lemmas may have similar proofs. Compare this to the classification of finite simple groups, which is tens of thousands of pages, and known as "the enormous theorem" (although that proof is needfully more complex, as there are many special cases that need to be dealt with).
However, there is definitely some cause for skepticism, as such a long proof is very hard to check, and other similar announcements in the past have had later holes found (such as Andrew Wiles first announcement of the proof of Fermat's Last Theorem, and the above mentions Enormous Theorem.) On the other hand, bost of those proofs have had (in the case of the enormous theorem, probably had) the holes plugged, and I feel confident that if a gap is found in this proof, it will also be bridged.
David
http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2- 165-172-intro.pdf
I don't think there will be a T-shirt with that proof anytime soon... :o)
Did Picasso appreciate his work? Probably not...at least not as much as others do. Sure it was a great challenge, but math is like magic once you know how to get the answer...all the mystery is gone.
-ac
My proof is only 1 page long, but its a really BIG page.
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).
That's like reading fermat's margin note ... just frustrating.
I am one of many. My idea is not unique, nor do I expect my voice alone to sway you. I speak in a chorus of opinion.
It is said that the Poincare Conjecture proof is one of the most important proofs in Mathematics. But I never managed to understand why. What are the practical consequences of this proof? does it have any real-world applications?
but at least on the positive side he'll have access to great health-care, low-crime, respectful co-citizens and one of the highest standards of living on the planet
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."
According to The Guardian
I understand
[/lie]
If this were really happening, what would you think?
From your sarcasm it seems that you have no idea how free markets work... There is no such thing as innate value, the only value that something has is the demand for that thing.
The demand for comedy is higher than the demand for mathematical proofs. The recompense for either has absolutely nothing to do with merit, even if you believe a mathematical proof has more innate merit than comedy. BTW, if you do believe that, please define for us exactly how a mathematical proof is better (has more value or merit) than comedy.
Deleted
So the other person being tipped is Terrence Tao, anyone else?
'Frustrating' as in 'outright lying' I hope.
Oh yeah we invented a new method of doing something and we did it for weeks but we forget it now.
Add it to the cold fusion and cloned human pile and fuck off.
i keep asking for a "Tragic" modifier, but i can't decide whether it would be +1 or -1.
i speak for myself and those who like what i say.
In Soviet Russia, Poinclare Conjecture proves you!
Something that's intrigued me since I noticed it, relating to hypersphere volumes. In 1D it's 2r, in 2D it's pi.r^2. 3D is 4/3 pi.r^3. The sequence continues: const.pi^2.r^4, const.pi^2.r^5, const.pi^3.r^6, const.pi^3.r^7 (can't remember offhand what the consts are but they can easily be found).
Obviously you're going to get an extra r with each dimension, buy why do you only get another pi every other dimension?
While I'm at it, on a related subject it seems to me there are two possible ways of constructing a 4D hypersphere. Both are similar to two different approaches for constructing a 2D hypersphere (aka circle): (a) take a short line, increase its length to the midpoint, then decrease its length (a bit like integration?); (b) take a straight line from (-r,0) to (+r,0) and spin it 180 degrees around (0,0). The first is somewhat similar to blowing up a balloon then letting it down again, if you take time as the fourth dimension (which I know you can't and all that, but IF you did purely as a mental exercise rather than a rigorous mathematical proof), and the resulting cross section would be a single sphere. The second takes two spheres and spins them round 180 degrees to form a sort of torus whose cross section would of course be two spheres. Which is correct and why?
IANAM, obviously. Got A-level maths in 1986 though.
BlueZ3: Heck, perhaps he just enjoys math for its own sake and doesn't want to deal with all the side-effects of notoriety.
Quite.
Perhaps he just hates parties. It's not like he'd be the first mathematician to do so. I and many other Slashdotters can sympathise with this, surely.
Andrew Oakley - www.aoakley.com
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
Sounds like he's hiding from an ex-wife to me. In fact, that's probably how she became an ex-wife.
http://www.nytimes.com/2006/08/15/science/15math.h tml?ex=1313294400&en=ad11dd7003387acf&ei=5090&part ner=rssuserland&emc=rss
Yeah right. You tested the formula for one month. And you "found no conjecture" (whatever that means). And you don't have any paper records for it. And three people forgot anything about what the formula looks like. Anything else?
The AACS key is NOT 0xF606EEFD628B1CA427BEA93A9CA9773F
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.
Those 1000 pages do not prove anything from first principles. It is not a text book or anything like that: it is just a very, very long paper. You have to be an expert in the subject to be able to get past the first couple of pages of the introduction.
m0nstr42.blogspot.com
You are confusing "capital" with "currency." It is trivially true that governments print currency, but that doesn't mean they are creating productive capital, not by a long shot. In fact, if they print currency unwisely, they can destroy capital.
Maybe it would help you to think of intellectual capital as a source of "wealth" rather than a source of "money". Money is just a convenient symbol for wealth and purchasing power and one's standard of living. It depends for its value on the productive capacity and fiscal and monetary prudence of the nation that issues it. And the productive capacity of all nations is increased dramatically by scientific knowledge, especially those who are best able to "capitalize" on it (i.e. free-market liberal democracies.)
It is no exaggeration to say that without the intellectual capital bequeathed to us by generations of mathematicians and other innovators, life would end for many of us and be made far more miserable for the rest. Food arrives on your table speedily and cheaply, because of innovations in agriculture, transportation and communications that depend on advanced mathematical and scientific knowledge.
-ccm
Too much Law; not enough Order.
(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
Moderation -1
30% Flamebait
30% Troll
30% Interesting
There's nothing remotely Flamebait or Troll in that message. TrollMods spew cosmoturf to suppress discussion of growing Russian backsliding towards tyranny. Slashdot's mod system really is disgusting sometimes when it's abused by political operatives.
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In Soviet Russia, the math proves YOU.
Cool! Amazing Toys.
If Perelman is truly a genius on par with Albert Einstein, he faces two problems in Russia.
First, the Russian government will want to tap into his genius to improve its weapons systems.
???
So, I hope that Mr. Grigori Perelman is okay. If he can read this message, then, I wish that he would, at least, post a message on SlashDot so that we know that he is all right.
That's possibly the dumbest thing I've read in my life.
Perelman really should come to USA. Here, he can work on neat projects like the new hyperdrive for space travel. If this hyperdrive is ever to succeed, we will need the enormous intellect of Perelman to work out the hairy mathematics.
You're an embarrassment on so many levels I don't know where to start.
Finally! Now I can sleep at night.
Mazel Tov,
Goodness, this is absolutely great news. John Von , and Oscar Morgenstern with their Theory of Games and now an amazing proof! Combined with all the contributions in science and medicine, people of the Jewish faith are helping tremendously to create a better world.
Not to mention they are on a decent 3th place on this chart(ratio of people that belive in human evolution)
16,777,216 comments ought to be enough for any forum!
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.
Quite right. I was merely trying to illustrate that a proof can be extremely long, with many steps. And yet, the results can be widely accepted. I suppose my point would have been clearer had I replied to this post instead.
Don't you hate meta-sigs?
Whoa. You're right. We should probably have lots of other people look at it twice. This will revolutionize the academic world. Congratulations, sir.
If you think Occam's Razor applies here, you have a serious misunderstanding of mathematics or Occam's Razor. In either case, it's simply a common sense suggestion written down hundreds of years ago. It's not an axiom of the universe, nor is it even necessarily an axiom of science.
Do you see those two -1, Troll moderations? They happened for a reason. I could argue that it was clueless moderators who had no idea of the deep relevance inherent in my top post. And were I drunk enough like last night, I might be willing to. But, let's face it. Muh posts were trolls. Shitty lame-ass trolls.
Bad maynard! bad!
Does anyone have a version of the proof that can be verified using the matamath proof explorer?
I metamoderated every time I got offered for months. I always metamod'ed everything on which I could form an opinion, though I skipped a few percent of the questions. I don't know the effect, but I did get modbombed last Fall, for weeks. Despite responding to what I call "TrollMods" with brief reasons why I thought the mod was unfair, and often with specific suggestions to the Slashdot operators for better meta/moderation accountability. And without responding to every negative mod with whining. But I got modbombed pretty hard, and decided metamoderation wasn't giving the return in "moderation moderation" on my time investment that I needed to justify participating.
If Slashdot included some accountability in its moderation beyond metamod'ing, which can clearly be gamed in enough ways that the mod system is widely abused to suppress disliked content, I'd participate again. Like if negative mods required a reason which the mod'ed poster could view, if only to learn what pisses people off. Or if there were a web of trust that made moderation less absolute, but rather weighted moderation by the reader's agreement with different moderators over time.
I don't know how you know that your metamods kicked people from mod'ing. I'd love to see the results of my metamods, or even just the results of metamods on my mods.
Meanwhile, I'll continue to post as I actually feel. Somehow, despite the attacks, I've consistently got enough karma to post as much as I like, except once in a while when I'm probably wasting too much time posting anyway.
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My efforts have resulted in some moderators permanently losing the privilege of being a moderator.
Moderation 0
50% Troll
50% Insightful
TrollMods certainly are diligent. But they don't have enough power yet to totally control which stories are read by people with default settings.
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You claim that "In the annals of history, the people who change the world generally take credit for their work"
It would be more accurate to say "people take credit for world-changing ideas." I don't think it follows that the majority of people whose ideas were world-changing are credited (think of all the inventions from the 19th century where we're unsure of who actually did it "first") or necessarily interested in credit. And I believe you're drawing an especially unreasonable conclusion to assert that people who aren't seeking the limelight (either exclusively or as a byproduct) are "remarkable." Maybe it's become remarkable in the 15-minutes-of-fame world that the majority of slashdotters live in, but it doesn't seem that surprising to me.
In my experience, the people who make the biggest difference often don't want a big deal made of what they do. That's either because of their social proclivity (shyness) or because they're genuinely modest. Neither of those traits should be, I think, unexpected in a mathematician. I recognize that there are some folks who get a thrill out of the publicity, but most of the mathematicians that I know (and my wife is one) are not really that interested in acclaim.
All that aside, you completely missed (whoosh) my point that Russia is not the West. Which means that perhaps this person is more interested in what his fellow Russian mathematicians (perhaps coworkers and friends) think than in where he sits in the estimation of some American researchers.
Interested in a Flash-based MAME front end? Visit mame.danzbb.com
Actually I'm just a morphed donut. Or coffee mug.
Please, for the good of Humanity, vote Obama.
Every thing about Grigori Perleman is so fascinating. Brilliant mind but doesn't care about money and fame. He is 40 years old with zero bank account and no job and depends on his mother for living. Gathers mushrooms for enjoyment. Is he human ?. But I really wish he has lot of sex with lot of beautiful young Russian women and have lots of children to pass his brilliance. This world needs several Grigori Perlemans.....