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This article has been mentioned by a media organization: "Perelman's Poincaré Proof Back/ the News". MathForge.net. September 6, 2004.

Poincaré conjecture received a peer review by Misplaced Pages editors, which is now archived. It may contain ideas you can use to improve this article.

Untitled

For previous discussions see: 1st archive of this talk page

Pronounciation

From the article: Poincaré conjecture (/pwɛn.kɑːˈreɪ/ pwen-kar-AY;

Seriously???? There is only one way to pronounce that name. I can't see myself putting a bad pronounciation on an English name in a French article. How is that relevant? — Preceding unsigned comment added by 82.244.149.143 (talk) 08:34, 2 May 2014 (UTC)

Please explain to the layperson why this is such a big deal

I agree with the above request... I would be very grateful if someone could write a "simplified overview" of this article for readers who are not mathematical experts. I am reasonably well educated, but the content of the article is WAAAAY over my head. I guess this could be taken as a compliment to the author, but for 99% of the world population - including young people who may just be developing an interest in mathematics - the article is far too scholarly. I am not asking that the article be 'dumbed down', but rather that a new section be added that summarizes the article such that a non-mathematician can comprehend what the general concept is. —Preceding unsigned comment added by 24.108.14.221 (talk) 06:03, 9 July 2010 (UTC)

I really want to be interested, because I have a somewhat mathematical bent and because people are making such a big deal about this. However, the last math-related class I took was calculus, 10 years ago. I cannot make head or tails of this article or why it is such a big deal. It sounds so arcane, like an intellectual puzzle that has no application to real life. I don't doubt its importance, but please someone who knows, explain what practical ramifications this conjecture will have so that laypeople like myself can be in on the big deal too.

I got a better idea: stop acting like somebody owes you something. I bet that there's something YOU understand that not everyone else here does. Why aren't you spilling it, elitist? 68.121.164.157 18:44, 23 August 2006 (UTC)

= I agree! Not to be cheeky, now it's proved -- so what? --24.249.108.133 08:30, 23 August 2006 (UTC)

Not to be cheeky? Please! Such insincerity gets you no love. 68.121.164.157 18:44, 23 August 2006 (UTC)

Hi 68.121.164.157, is that your real name? Perhaps you should address the clearly legitimate commentaries made above instead of throwing the toys out of the pram. This is after all a general interest encyclopedia and as someone with a small amount of mathematics in my history I don't understand this article! If you have no interest in writing an article that is accessible to the general public then your edits have no place on Misplaced Pages. Mglovesfun 17:41, 27 August 2006 (UTC)

Just thought i'd try and understand this.....and failed! Is there anyone out there who can write about maths in a way that the lay person can understand? I know how difficult it is describing complicated theories in simple ways..however there must be diagram or two that will show what is described graphically....a picture speaks a thousand words! Theball90 13:53, 19 March 2007 (UTC)

why not do it for the sake of DOING IT Confront (talk) 02:56, 6 March 2009 (UTC)

what does the final step mean?

see the lastest link in the introduction page.

Is it obvious from Thurston's geometrization conjecture to Poincaré conjecture or there really still need a 300-pages proof? just want know the importance of the china scientist work.—Preceding unsigned comment added by mathematic (talk • contribs)

The paper isn't about going from geometrization to Poincare: that, as you suspect, is trivial and requires at most a short paragraph (see elliptization conjecture for short explanation). I haven't looked at the article, but I read the abstract. It sounds like they have basically a complete write up of Perelman's work, with probably details that he didn't fully explain, and probably cast and reformulated somewhat differently. The Xinhua article is very poorly written, and probably the headline and focus of the story gives a misleading impression of what the two mathematicians did. --C S (Talk) 08:59, 4 June 2006 (UTC)

My understanding is everything had its beginning and finishing. It verything important to clarify who finsh it. It's very important too, unless someone can claim Cao-Zhu is wrong. —Preceding unsigned comment added by Moreton bay bug (talk • contribs)

Well, in this case, it's more complicated than you think. There are other people, who have already come forth and said they have checked Perelman's proof of the Poincaré conjecture; nobody claims they are mistaken. There are even a few books in the works that will come out in the near future. So Cao-Zhu didn't really finish it. Most specialists have already been convinced before their paper. It's good to be even more certain by having different groups of people come forth with their papers. But the line of investigation definitely does not go directly from Perelman to Cao-Zhu. We should probably include a bunch of other people, and a lot of people would consider some stage in between Perelman and Cao-Zhu to be the finish. Some people wouldn't even say it's finished until Perelman gets his Fields Medal. In which case, they probably would just stop with Perelman. I think you will have to try and understand that it's the mathematicians who will to a large extent get to say who "finished" it. And in mathematics, stuff like this happens all the time with big theorems, where some group of people help to clarify a proof, but they don't get much of the credit. --C S (Talk) 02:35, 6 June 2006 (UTC)

Yes, it is more complicate (like political mine field) than I think, since you did not read the article. If we narrow it down to a samll group of mathamatician who can have a say about it, then it is Fields Medalist Shing-Tung Yau said Cao-Zhu's work put a finishing touch on the Poincare-Conjecture. There are mathematician claim that Hamilton contribute about 50%, Perelman 25%, and Cao-Zhu 25%. I would very much like to hear from other eminent mathematicians jump out and to say something about it in next few months. —The preceding unsigned comment was added by Moreton bay bug (talk • contribs) 04:34, June 6, 2006.

They didn't put the finishing touches on anything. They published a summary of work that Perelman had already completed. At best, they filled in details. Then some Chinese newspaper published a nontechnical article with a headline that made it look like they did something unique and important, and that article went to slashdot. This happens all the time with popular press and slashdot. They put sensationalistic headlines on. This doesn't change the fact that Perelman released his results in 2003, and that these two guys are just 2 out of dozens who have been vetting the work ever since. Cao and Zhu have vetted Perelman's work, but that doesn't mean much. -lethe 05:34, 6 June 2006 (UTC)

Yau's comments seem to me to be pretty carefully phrased, and I think calling the Cao-Zhu work a "finishing touch" is fair and a nice move on his part. I am indeed aware there are some politics at play here. However, that seems to be the best quote we have at the moment. I agree it'd be nice to have some more comments from some eminent experts on the subject. In any case, I don't believe Yau's comments justify putting such prominent mention of the Cao-Zhu paper in the lead section.

There are mathematician claim that Hamilton contribute about 50%, Perelman 25%, and Cao-Zhu 25%. Interesting, where did you get this? I've very suspicious that anyone reliable would make such a coarse and misleading statement. Those percentages leave out that guy that made geometrization a likely reality instead of a pipe-dream. Those percentages also put Cao-Zhu and Perelman on an equal footing, which really is preposterous. No way, Yau or anybody that knows about the subject would agree to something like that. --C S (Talk) 06:11, 6 June 2006 (UTC)

I definitely agree with 'Moreton bay bug': the "finishing strike on a global collaborative work" needs to be mentioned in the intro. Every science is built on other's excellent work. By mentioning the people who finished up doesn't mean to ignore previous work. Researchers stand on giants' shoulders; but giants didn't finish up their own work. I don't think the final step is so trivial, otherwise it shouldn't take so many hours, from last Sep to Mar, to convince the faculties at Harvard. If the final work is trivial, I believe Perelman should have finished/published it by himself well before 2006. —Preceding unsigned comment added by AnRtist (talk • contribs)

Mentioning them in the intro does mean to ignore previous work, because others who have contributed more (Thurston, Hamilton) are not mentioned. So I support Chan-Ho and Lethe on this. Of course it takes many hours to go through the proof, because it is a long proof (for instance, it takes me about a week if I have to go carefully through a 20-page paper on a subject that I understand well). Why Perelman has not published it himself is a bit of a mystery to me. I certainly believe him to be capable of doing it, but his priorities are different from those of most mathematicians. One story I've heard is that somebody remarked that Perelman should publish his work to be considered for a Fields medal, and that Perelman replies that that's why he hasn't published it. -- Jitse Niesen (talk) 13:13, 6 June 2006 (UTC)

I agree with Chan, Lethe and Jitse. We need to wait until the mathematics community tells us (more clearly than they have so far) the significance of Cao and Zhu's work. —The preceding unsigned comment was added by Paul August (talk • contribs) .

This is a brief account of what Prof. Hamilton said about Poincare Conjecture in Beijing, 06/13/06: ..... Cao huai-dong and Zhu Xiping have recently given a complete and detailed account of the proof of Poincare conjecture based on the work of Perelman and earlier work of others. It’s very nice to have such an account written by two outstanding people in the field of Ricci flow. They also introduced ideas of their own which makes the proof easier to understand. This includes a new proof for the uniqueness of solutions on complete manifolds, and different idea for doing the backwards blowup in time and proof of the canonical neiborhood theorem based on results of Zhu and Chen on expanding solutions. They fully acknowledge Perelman’s role in the completion of the proof of Poincare conjecture and likewise Perelman has acknowledged the work of previous researchers on which it’s based. ..... I’m here in Beijing discussing the details of the proof with professor Cao Huai-dong and I’ll talk about that work with Huisken and Ilmanen when I got in Zurich next week. We want to be complete certain that everything in the proof is beyond question before making a xxxxal announcement, because many researchers will base their work on it.

Great! Thank you for posting this. Very informative. --C S (Talk) 00:32, 15 June 2006 (UTC)

Interesting link

I found a very interesting link related to the Poincare's conjecture: . However I don't know if it is a hoax or not. Can someone verify their proof? --Matikkapoika 19:59, 4 June 2006 (UTC)

This is the same thing I responded to in the previous section. No hoax. Just, well, a very interesting point of view that I think would not be universally agreed to. It's an even worse article than the previous one mentioned...the quotes are kind of misleading, and I wonder if the journalist did a very selective job of quoting. Both articles do not do a good job of explaining the different contributions (and kind of contributions). One particularly bad thing is that this article's emphasis on Cao and Zhu's contributions comes at the expense of lessening Perelman's contributions; however, it is clear that is is really Perelman's work that is significant, not theirs. Of course, they have done something important here, which is to clarify Perelman's work...but other groups of people have and are doing this. This team was apparently just one of the first to complete the task. Perelman will probably win a Fields medal for what he did. --C S (Talk) 22:19, 4 June 2006 (UTC)

major editing needed

According to slashdot, it's been proven. To a certain degree anyways. Current tag up. --Rake 08:51, 5 June 2006 (UTC)

URL: http://news.xinhuanet.com/english/2006-06/04/content_4644754.htm

This is the same link that has already been posted above. Read Chan-Ho's reply there. --Zundark 09:08, 5 June 2006 (UTC)

Despite it being on slashdot, this isn't really news, and major editing is not needed. -lethe 09:10, 5 June 2006 (UTC)

Consider renaming this article

Now that the conjecture has been completely proven, the article may bear renaming to something like "Poincaré theorem." --70.7.217.227

No, because that's not what it's currently called. It may need to be moved later (but not necessarily to "Poincaré theorem", as theorems are usually named after the people who prove them). --Zundark 12:27, 5 June 2006 (UTC)

Heck, Fermat's last theorem still hasn't been moved to "Wiles's theorem". —Keenan Pepper 13:42, 5 June 2006 (UTC)

It's not uncommon for a distinguished conjecture to keep its name, even after being proven. If people start calling the Poincaré conjecture something other than "Poincaré conjecture", then we will rename the article after an appropriate amount of time. --C S (Talk) 02:21, 6 June 2006 (UTC)

I second C S's opinion. Given the controversy surrounding it, and the fact that we've not heard any definitive pronouncement from the ICM (have we?), it may be too premature to say that "the conjecture has been completely proven", notwithstanding the fact that Morgan and others — and I say this with all due respect to their expertise and opinions — have stated that it has been done. As I see it, the best milestone for now is probably when the Clay Institute starts deliberations on giving prizes, two years from now, since that is when a reasonable amount of scrutiny has taken place, and more experts can reasonably stick their heads out, so to speak, and literally put the money where their mouth is. -Kidiawipe

I'm not sure when that last post was made, but now it's accepted as proved. The word 'conjecture' is misleading... it gives the impression than it's an unproven guess. I don't agree that theorems are usually named after the people that prove them. The nomenclature should be done on a case-by-case basis based on who is most deserving of it. I think Poincaré was the real genius here because he came up with the theorem. He pretty much new it was true but wasn't pedantic enough to prove it rigorously. Proving it may have been even harder than coming up with it itself, so one might think of calling it the Poincaré-Perelman theorem. But then what about all those hard-working mathematicians that proved the important lemmas along the way which were necessary for Perelman to succeed ? Some of those lemmas themselves may have been harder to prove than the part that Perelman played. I say it becomes called the "Poincaré Theorem", and "Perelman's Proof" ... Don't wait for someone else to wake up and realize that it's no longer a conjecture ... just change the title ... afterall, in music, the term "whistle register" was invented by wikipedians. We have the license to design truth if enough of us agree it's ethically sound. Dr. Universe (talk) 16:40, 19 January 2010 (UTC)

The most likely result in the real world will be that the result is eventually called "Perelman's theorem", since he is the person who proved it. Similarly, what was once Hilbert's 10th problem is now Matiyasevich's theorem or the MDRP theorem; it is never "Hilbert's theorem". But in any case our role is not to invent new terminology here; we should simply follow the terminology used in the mathematical literature. — Carl (CBM · talk) 16:45, 19 January 2010 (UTC)

But it's not Perelman's Theorem though! (the theorem is not HIS, it's Poincaré's , just because some poeple were clumsy and misnamed theorems in the past doesn't mean we have to be. Hilbert came up with the theorem from scratch.. Matiyasevich simply stood on the shoulders of many many giants and actually contributed very little to our understanding of the theorem being true. 100% of creating the theorem was due to Hilbert, while the task of proving it was shared by many many many people.

Right now the terminology in the mathematical literature is misleading, it calls Poincaré's theorem a "conjecture". But in order to disseminate the fact that this nomenclature is misleading, according to Carl, we have to come up with a mathematical result related to the theorem that's worthy enough to publish in the "mathematical literature" , just to attempt to change what people call it. Even then we'll not be sure what will happen, if 2 other people publish articles on the same day calling it something else we're screwed.

Why don't we come to an agreement here before it's too late. If in the future, the trend in the literature tends towards calling it something else, this can always change. But if someone (like the one who decided to call Hilbert's 10th problem the "Matiyasevich Theorem") decides to call it something unfairly, and people follow that trend, the chances of changing it after that will be very slim Dr. Universe (talk) 16:40, 22 January 2010 (UTC)

To be blunt: It isn't Poincaré's theorem because Poincaré didn't prove it. The theorem is due to Perelman, solving a conjecture of Poincaré. So I would personally disagree with us calling this page "Poincaré's theorem". However, my personal opinion is not the determining factor; the way that we name things on Misplaced Pages is by simply adopting the names used by published sources. In this particular case, I'm sure that the reason published sources don't call it "Poincaré's theorem" is because mathematicians almost always name theorems after the person who proves them, not after a person who doesn't prove them. — Carl (CBM · talk) 20:45, 22 January 2010 (UTC)

And Fermat's Last Theorem was proven by Andrew Wiles. Both precedent & logic support either arguement, but I think that Perelman would likely rather not have the theorem named after himself. —Preceding unsigned comment added by 63.111.9.245 (talk) 14:33, 2 July 2010 (UTC)

Mentioning Cao and Zhu in intro

I believe it is quite wrong to give undue emphasis to their work by mentioning them in the introduction. They are mentioned in the main body of the article, which is quite enough, given that their work is not developed in isolation, but relies on other teams of mathematicians who have worked out the details of Perelman's work. See also my previous comments in the prior sections. --C S (Talk) 02:08, 6 June 2006 (UTC)

Upcoming announcements at ICM 2006

Somebody has asked that a citation be given for the claim in the lead that a consensus of experts has concluded Perelman's proof proves the Poincare conjecture. In response to this, Jitse has included . I didn't find this completely satisfactory, as it, in isolation, only indicates that experts are very very confident, but nobody has stuck their necks out saying it is correct. So I replaced the link with this one . This is an announcement by the ICM organizers that they expect Hamilton and Morgan to announce in their talks that the conjecture is proven. Clearly, they would not say this unless they know there is a strong consensus on the correctness of Perelman's work. Not only that, but at the end of the announcement, there is a quote from an expert explaining that experts have verified enough of Perelman's work to establish the Poincare conjecture.

This is all old news in a way, as people have been saying this a while; in fact, Morgan has already stated that Perelman's proof of PC is correct at a conference I attended last October. He stated this at the end of a talk, and even said he had verified much of it (except for standard well-known stuff) himself. So I'm confident that Morgan will just reiterate this at the ICM. In any case, I think the ICM article (plus the Cao-Zhu paper which has been vetted by Yau) is certainly sufficient justification for the challenged statement. We will definitely have much better sources after the ICM, but we have enough for now. --C S (Talk) 11:58, 11 June 2006 (UTC)

Peer review

A little while ago, I placed a request for peer review of this article. One clear problem is that people are not able to understand the article. This is already quite clear from comments made on Slashdot whenever PC gets in the news. I think the most fundamental question is, what is the intended audience? Once that is resolved, it should be much easier to know what should be included or not.

Also, I think it should be quite feasible to include short description (modulo the technical details) of how Perelman's proof works. Should the article include this?

When I get a chance, I will create a proposal version at Poincaré_conjecture/rewrite; people are welcome to beat me to it.  :-) Until then, I would like to open up a discussion here and see what your thoughts are on how this article should look. --C S (Talk) 12:40, 11 June 2006 (UTC)

Well I can say that it's shameful that the article does not contain a quick-n-dirty description of what it means for a space to have trivial fundamental group before it states the conjecture. -lethe 16:10, 11 June 2006 (UTC)

As to the question of who the indended audience is, I'm afraid the inconvinient answer is, just about everyone. Paul August ☎ 16:40, 11 June 2006 (UTC)

Also it should include, probably in the first sentence, something like: "The only bounded three-dimensional space without holes is "essentially" a sphere." ( see for a good intuitive description of the conjecture. Paul August ☎ 16:54, 11 June 2006 (UTC)

I edited the intro in an attempt to make it a bit more accessible to non-technical audiences.--agr 19:54, 11 June 2006 (UTC)

If you guys are going to rewrite it, and just in general, please watch your grammar and sentence layout as I've noticed several implications which are not correct - for example due to paragraph layout the suggestion that Perelman has refused anything from the Clay people (fixed). Unfortunately while I have done highschool and limited Uni maths I am still at a bit of a loss to understand this article, even after reading other background pages, so I have been a bit worried about correcting things. --eps 1143, 23rd August, 2006 UTC

This article is nearly impossible to understand without a familiarity with rather advanced mathematics. Following the links tends to turn up other articles on math topics that are equally difficult to grasp. It needs to be translated for the layperson; as it's written now it's only of use to those who already know the lingo. 74.99.167.47 07:01, 23 December 2006 (UTC)

Statement in higher dimensions

I think there is an problem with the statement of the result in higher dimensions:

"Every closed n-manifold which is homotopy equivalent to the n-sphere is homeomorphic to the n-sphere."

A better statement might be:

Every closed n-manifold which has the same homotopy groups as the n-sphere is homeomorphic to the n-sphere.

As I understand it, and as Misplaced Pages defines it, the term "homotopy equivalent" usually means there exists a smooth deformation (homotpy) between the two objects. The same (mis)use of of "homotopy equivalent" occurs in Homotopy sphere. Am I missing something?--agr 15:18, 12 June 2006 (UTC)

Homotopy equivalent means isomorphic in the homotopy category of topological spaces. In other words, there are maps between the two spaces whose products are homotopic to the identity. Two spaces can be homotopy equivalent without either one being a deformation retract of the other. Anyway, I don't see anything in the sentence you cite which seems to rely on the particulars of the definition of "homotopy equivalent", so I'm not sure what your complaint is, can you clarify? And as far as I know, if two spaces have the same homotopy groups, they are homotopy equivalent, so your proposed sentence is equivalent to the existing one. -lethe 15:38, 12 June 2006 (UTC)

The problem is your last sentence. It's been a while since i studied this stuff, but I don't think its true. (The converse clearly is true, two spaces that are homotopy equivalent do have the same homotopy groups.) If I remember right there are counter examples in knot theory.--agr 16:41, 12 June 2006 (UTC)

OK, well I'll see if I can find a reference for that sentence (or its counterexample). Let's get to the bottom of this! -lethe 12:41, 13 June 2006 (UTC)

In general, it is false that two spaces with isomorphic homotopy groups are homotopy equivalent; in fact, it is false even for closed manifolds of dimension at least 3. For example, in dimension 3 the lens spaces L(5,1) and L(5,2) have isomorphic homotopy groups but are not homotopy equivalent. For a more delicate answer, the 3-dimensional lens spaces L(7,1) and L(7,3) are homotopy equivalent but not homeomorphic, and neither of these is homotopy equivalent to the lens space L(7,2) - yet all three of these spaces have isomorphic homotopy groups. These are results first proved by Kurt Reidemeister decades ago. On the other hand, there is the (J.H.C.) Whitehead theorem which says that if there is a map between two spaces which have the homotopy type of finite CW-complexes (closed manifolds are examples of such spaces) and if the map induces an isomorphism between homotopy groups (with universal coefficients, in the case of non-simply connected spaces), then the map is a homotopy equivalence. As corollary to this, if n > 1, any simply connected closed n-manifold whose homology groups in degrees < n are zero is homotopy equivalent to the n-sphere. To see this, take such a manifold, collapse the exterior and boundary of a small regularly embedded n-ball to a point, obtaining an n-sphere, then note that the collapsing map is a degree 1 map and hence satisfies the hypotheses of the Whitehead theorem. Chuck 14:49, 25 January 2007 (UTC)

I don't believe the second statement is any better. There is no misuse of terminology in the first statement. Homotopy equivalence is a fundamental notion, more basic even than homotopy groups. The phrasing in terms of homotopy groups strikes me as clumsy. Additionally, the first statement is the usual phrasing of the generalized Poincare conjecture, e.g., Smales' Annals paper.

It is not true, in general, that same homotopy groups imply homotopy equivalence. For example, consider three dimensional lens spaces. There are non-homotopy equivalent lens spaces of the same fundamental group; their higher homotopy groups are necessarily the same since all three dimensional lens spaces are covered by the 3-sphere.

This is why some trick like Whitehead's theorem is required to show homotopy equivalence (if possible) for spaces with the same homotopy groups.

There can be no such examples for knot complements. All knot complements are Eilenberg-MacLane spaces, K(G,1)s. So if they have isomorphic fundamental groups, then they are homotopy equivalent (if the homotopy equivalence preserves peripheral structure, then we can even get homeomorphic by Haken-ness).

In the case of a closed n-manifold with the same homotopy groups as an n-sphere, that would imply homotopy equivalence, but that takes a little work; taking this into consideration, and considering that the current statement is the usual, clean one, I would recommend not changing it. --C S (Talk) 20:56, 14 June 2006 (UTC)

Thanks for the clarification. I think this material should be worked into this article and the one on homotopy equivalence. It seems to me both statements of the problem have a place. The program of algebraic topology is to characterize topological spaces by means of algebraic invariants. That now seems to have been accomplished for spheres and that should be clearly stated in the article, not left to inference. --agr 11:28, 16 June 2006 (UTC)

Non-IPA pronunciation

What do people have against including a pronunciation that can be understood by people who do not understand IPA? If someone wants to know how to pronounce it, and doesn't know IPA, then chances are they aren't going to take the time to learn IPA, they will just pronounce it incorrectly or look somewhere else. One person said that it is ulgy -- that must be their own personal view, I think many people would find the IPA pronunciation with its "funny looking" characters ulgy. If it is of "dubious quality" then improve it, but even in the form it was in it is better than nothing (if the person doesn't know IPA, then there is no pronunciation information in the article for them). Qutezuce 00:32, 20 June 2006 (UTC)

IPA is standard on Misplaced Pages. Silly rhyming pronunciation guides are also somewhat standard, but I don't think we need to have every different pronunciation guide in this article. People who want to learn more about Henri (including other transcriptions of pronunciation) should view his article. -lethe 00:42, 20 June 2006 (UTC)

Maybe I'm missing it, but I see the same pronunciation guides at Henri Poincaré as I do in this article. Qutezuce 01:17, 20 June 2006 (UTC)

Case of n=1

The article states that "the case of n=1 is easy", but isn't n=1 actually totally different? The homotopy group of n=1 is Z, not 1, so it's not really the same thing you're proving, right? MGolden 06:47, 16 August 2006 (UTC)

I've moved other dimension to their own page; if you say "every homotopy sphere is homeomorphic to the standard sphere", then the n=1 case is the same. Nbarth 22:35, 20 December 2006 (UTC)

citation for yau's comment

http://www.theepochtimes.com/news/6-6-6/42408.html http://blogs.guardian.co.uk/technology/archives/2006/06/06/has_poincares_conjecture_been_solved_the_conjecture_continues.html http://news.xinhuanet.com/english/2006-06/04/content_4644754.htm

bbc

The BBC just had someone on that explained it this way. If you have a rubberband and you make the rubberband smaller and smaller then if the rubber band becomes a point then the space is a sphere. Not exactly what we have in the article. --Gbleem 07:40, 16 August 2006 (UTC)

Maybe BBC should read Misplaced Pages more often. Nature does. I got some emails from them when they were preparing for an article in the issue that appeared earlier this month. --Francis Schonken 07:48, 16 August 2006 (UTC)

unsolved category

If this is now solved, shouldn't it be removed from the unsolved category?

Application or meaning?

Does the solving of this conjecture have any practical applications beyond the world of topology? I know that Fermat's Last Theorem was more of a mathematical novelty than a serious problem, and I'm not clear whether this issue is any different. It's great that someone solved it, but what does it mean to folks outside the world of abstract mathematics? I think the article could really use a section on this topic. | Mr. Darcy talk 01:32, 23 August 2006 (UTC)

It doesn't have immediate applications. As with the Fermat theorem , it says no counterexamples exist, when we already knew that they would be hard-to-find. Charles Matthews 09:24, 23 August 2006 (UTC)

Thanks. Then in my opinion, adding something to the introduction that says that the solution has no practical applications would be worthwhile. I think that's going to be the primary question of any lay reader who comes across this page. | Mr. Darcy talk 15:19, 23 August 2006 (UTC)

We would end up adding that to 95% of mathematics pages. Not a great idea, and also not 'encyclopedic', either. Mathematics is unreasonably effective, as Wigner says; but not if you throw out everything that doesn't have immediate paybacks. Charles Matthews 19:14, 23 August 2006 (UTC)

It's also important to remember that many results in mathematics are not applied practically for decades. The oft-cited example of Riemannian geometry is a case in point; it was instumental in Einstein's formulation of general relativity, but was developed by Riemann a half-century before. Just because Perelman's proof has no immediate practical applications does not mean it will not in the future. Yill577 02:25, 29 August 2006 (UTC)

The Sydney Morning Herald referenced us!

"Perelman's achievement has been to solve the Poincare Conjecture, which, says Misplaced Pages, has been one of the most well-known - and most difficult - open problems in mathematics since it was first posed by Frenchman Henri Poincare in 1904. "

Katherine Kizilos (August 26, 2006). "When being a genius just doesn't add up". Sydney Morning Herald.

Ta bu shi da yu 14:51, 26 August 2006 (UTC)

Move

Can't we move it to Perelman's theorem? I created that article to redirect here, but it ought to be the reverse. --Ysangkok 10:28, 2 September 2006 (UTC)

Currently, no. The common name is still Poincaré conjecture. I don't even think the scientific community has decided unanimously what name to give it after the theorem label applies. Have you any examples of reliable sources that present a new name in a <epithet>+"theorem" format? That might be a first step. --Francis Schonken 11:00, 2 September 2006 (UTC)

And some other considerations,

• From what I understand of it, it might be more likely that Thurston's geometrization conjecture would be rebaptised Perelman theorem than that the Poincaré conjecture would be rebaptised to that name (but again, I haven't seen any proof yet that the scientific community – nor "popular culture" for that matter – proceeded with any of these renamings);
• The current "In other dimensions" section of the article would be inappropriate on a Perelman theorem page, while the theorems described in that section were proved by Smale (for n ≥ 5) and Freedman (for n = 4). This section, with its current content, is, and will continue to be, appropriate on the Poincaré conjecture page. --Francis Schonken 11:38, 2 September 2006 (UTC)

Yes, the Dec. 2006 issue of the "Discover" magazine states: "Just this August, Russian mathematician Grigori Perelman won a Fields Medal... for proving the Poincaré conjecture (renamed the Poincaré theorem as a result)." While this magazine has been known to make some errors, I would believe that it can be considered a reliable source.-Hairchrm 23:30, 19 November 2006 (UTC)

Mathematical theorems are not just renamed by fiat, so the magazine's claim that the Poincaré conjecture has been renamed the Poincaré theorem is meaningless. (And note that Ysangkok was suggesting renaming it to Perelman's theorem, not the Poincaré theorem.) The article should stay where it is for now. It may eventually be appropriate to move it to Poincaré–Perelman theorem (or whatever), but not this year, and perhaps not this decade. --Zundark 09:00, 20 November 2006 (UTC)

The writer is apparently using some "principle of renaming" that I am not aware of. Nobody calls it the "Poincare theorem". Claims only found in popular science magazines should be viewed with caution as they frequently make misleading comments and omissions. --C S (Talk) 21:18, 21 November 2006 (UTC)

Point well taken, it will stay. Thanks for the advice-Hairchrm 21:05, 25 November 2006 (UTC)

There is already another Poincaré Theorem: http://mathworld.wolfram.com/PoincaresTheorem.html Barraki 00:02, 24 January 2007 (UTC)

Comment by Sukumaran (203.115.13.78)

Moved here from Article page by Francis Schonken 11:56, 11 October 2006 (UTC)

As for homotopy sphere, there is a more smooth-structured true example for N=3. For the want of self-effacement I do not wish to submit anything on this page, though I may venture to state the fact a 3-d sphere in reality holds out the best real-world(of course given there is a 5th dimension in the knnown universe)argtument for the ultimate proof of the poincare's conjecture. Anything contrary to this does not have a homotopic relevance whatsoever for there is no corroboration or proof for a dynamic homotopy sphere even within the extended analysis of the conjecture.--203.115.13.78 11:38, 11 October 2006 (UTC)V. Sukumaran.

"The article implies that ..."

Instead of reverting eachother, let's have a discussion about whether to include the sentence

"The article implies that Yau was intent on being associated, directly or indirectly, with the proof, and pressured the journal's editors to accept Zhu and Cao's paper on unusually short notice."

It seems to me that, if this article include a section on Manifold Destiny, then it should give a summary of Manifold Destiny and the above sentence is a fair summary. So I believe that either the sentence should go in, or (if it is decided that the New Yorker article is not a very relevant or reliable source for the purposes of this Misplaced Pages article) the section should be removed or reduced to one sentence. Removing only this one sentence seems to be a bit inconsistent. -- Jitse Niesen (talk) 13:24, 17 October 2006 (UTC)

The trouble with getting into details about Yau here, is that essentially Yau is irrelevant to the Poincaré conjecture. He may not be irrelevant to discussion of the controversy, but the controversy is not about proving the main question in geometric topology, it is about academic politics and (really) about mathematicians being naive and trusting when talking to journalists. Charles Matthews 16:56, 17 October 2006 (UTC)

I agree, but in that case I think that Manifold Destiny gets too much attention. So I reduced it. -- Jitse Niesen (talk) 15:40, 26 October 2006 (UTC)

Removing all that irrelevant drivel was a big improvement. R.e.b. 15:26, 29 October 2006 (UTC)

Moved other dimensions to GPC article

I've moved the other dimensions to another article, as they are very different. Nbarth 22:38, 20 December 2006 (UTC)

from the article: lacks any boundary (a closed 3-manifold).

I thinks you mean it has a well defined boundry. —Preceding unsigned comment added by 193.136.128.7 (talk • contribs)

No. --C S (Talk) 17:49, 22 December 2006 (UTC)

Conjecture vs. Theorem -- not about moving

Although I support not moving the article, the beginning still states "In mathematics, the Poincaré conjecture (IPA: ) is a conjecture...". Surely the end of that sentence should now be 'theorem'? Just wanted to check whether there was a reason why this hadn't already been changed. Also, the third paragraph seems to indicate his proof hasn't been verified, but if Perelman has been offered the Fields' medal, doesn't that mean it's been verified? DavidHouse 21:31, 26 December 2006 (UTC)

Proof writen 2003, verified 2006. I think some paragraphs are not up-to-date. Barraki 00:03, 24 January 2007 (UTC)

Misleading caption

The caption under the picture of homotopying a loop in the two-sphere in the beginning of the article is misleading. Of course, we do not know that the manifold in question is a three-sphere (that's the whole point of the conjecture), I think it could be very confusing. Arcfrk 05:43, 4 May 2007 (UTC)

I concur, it's wrong!24.58.63.18 (talk) 04:55, 16 March 2009 (UTC)

Book by Szpiro

the popular book - mentioned in the article - is entitled "Poincare's Prize" (not Perelman's Prize). Could someone correct that? --89.139.118.120

I've corrected it. --Zundark 08:50, 2 July 2007 (UTC)

Hamilton's program

"Hamilton's program was started in his 1982 paper in which he introduced the Ricci flow on a (...)" ist that date correct? —Preceding unsigned comment added by 201.62.193.100 (talk) 06:18, 26 May 2008 (UTC)

A link to a new article on "Smooth Poincaré Conjecture" would be useful

Or just a link/redirect to Gluck Twists of exotic spheres?

CHOOR MONSTER's VANDALISM AGAINST PRASTARO's TALK ON THE 'SMOOTH POINCARE CONJECTURE'

There is a clear repeated vandalism in the Choor monster's action against Prástaro's talk on his proof of the smooth Poincare' conjecture published in

A. Prástaro, Exotic heat PDE's, Commun. Math. Anal. 10(1)(2011), 64-81. arXiv: 1006.4483.

A. Prástaro, Exotic heat PDE's.II. arXiv: 1009.1176. Essays in Mathematics and its Applications. (Dedicated to Stephen Smale.) (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York Dordrecht London (2012), 369--419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0 (Online). DOI: 10.1007/978-3-642-28821-0.

A. Prástaro, Extended crystal PDE's, arXiv: 0811.3693. Mathematics Without Boundaries: Surveys in Pure Mathematics.> (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York Dordrecht London (2014), 415--481. ISBN 978-1-4939-1106-6 (Online) 978-1-4939-1105-9 (Print) . DOI: 10.1007/978-1-4939-1106-6.

R. Agarwal & A. Prástaro, Geometry of PDE's. III(II): Webs on PDE's and integral bordism groups. Applications to Riemannian geometry PDE's, Adv. Math. Sci. Appl. 17(1)(2007), 267-285

A. Prástaro, Exotic PDE's. arXiv.1101.0283. Mathematics Without Boundaries: Surveys in Interdisciplinary Research. (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York Dordrecht London (2014), 471--532. ISBN 978-1-4939-1124-0 (print) 978-1-4939-1124-0 (eBook). DOI: 10.1007/978-1-4939-1124-0.

For more details see also the following link: PROOF OF THE 4-DIMENSIONAL SMOOTH POINCARÉ CONJECTURE IN PDE's ALGEBRAIC TOPOLOGY

Extensions of these results in the category of quantum supermanifolds, are also obtained by A. Prástaro in the following papers:

A. Prástaro, (Co)bordism groups in quantum super PDE's. I: Quantum supermanifolds, Nonlinear Anal. Real World Appl. 8(2)(2007), 505-536. DOI: 10.1016/j.nonrwa.2005.12.008.

A. Prástaro, Quantum exotic PDE's. Nonlinear Analysis. Real World Appl. 14(2)(2013), 893-928. DOI: 10.1016/j.nonrwa.2012.04.001. arXiv: 1106.0862.

A. Prástaro, Strong reactions in quantum super PDE's. I: Quantum hypercomplex exotic super PDE's, arXiv: 1205.2894.

'Choor monster' justifies his action claiming that there is not world consensus on these results ... This statement is really ridiculous since Prástaro's results are published on international research mathematical journals or book series. I do not know who is this 'Choor monster', but surely it is not possible to delete author's serious scientific talks when there is someone that dislikes them. I do not think that this should agree with the meaning of Misplaced Pages-talks.

(Agostino.prastaro (talk) 19:08, 2 June 2015 (UTC))

Science cover image

I removed a cover image of Science from this article. In general we avoid using non-free images in articles, by setting high standards for their use. In particular, WP:NFCC#1 and WP:NFCC#8 are relevant here. #1 asks, "Could the subject be adequately conveyed by text without using the non-free content at all?". I think that the answer is yes – the only point of the image was to show that this was a cover story on Science, a fact which can be conveyed via text without the image. In fact this was already in the article's text. — Carl (CBM · talk) 21:15, 29 December 2009 (UTC)

I disagree. It also provides a nice illustration of the main idea of the proof. So I'm returning it. Maxal (talk) 15:25, 30 December 2009 (UTC)

We could create a free image to illustrate the main idea of the proof; see WP:NFCC#1. I don't see any argument why we need to use a non-free image here, and the policy on non-free images (WP:NFCC) is intended to be very restrictive. — Carl (CBM · talk) 15:59, 30 December 2009 (UTC)

The policy is restrictive but this image well satisfies the fair-use policy of using non-free images and perfectly fits the context of the article (both the image and text on the cover is very relevant to the article). So, I don't see why there is a need to avoid using it. Second, there is no suitable equivalent (simple and nice) illustration at the moment. Maxal (talk) 16:11, 30 December 2009 (UTC)

The problem is that the Science cover does not satisfy the WP:NFCC policy; fair use is not a relevant consideration. Just because a non-free image fits into the overall context of the article is not a sufficient criterion for using it. In any case, I created an image of Ricci flow and replaced the cover with that. However, the pre-existence of a replacement is not relevant to NFCC; what matters is whether a free illustration could be created. — Carl (CBM · talk) 17:21, 30 December 2009 (UTC)

Declining millenium prize

Section from wiki article is incorrect.

Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on 18 March, 2010 but declined the prize money stating, "I'm not interested in money or fame. I don't want to be on display like an animal in a zoo."

This italicized quote - was made in 2006 in reference to the Fields Award, not in reference to the 2010 Millennium prize.

He still hasn't decided on the Milennium Prize.

NEW INFORMATION 7/1/2010 The Clay Mathematics Institute (awarding body for the Millennium Prize) made the following announcement: "Dr. Perelman has subsequently informed us that he has decided not to accept the one million dollar prize. In the fall of 2010, CMI will make an announcement of how the prize money will be used to benefit mathematics."

http://www.claymath.org/ —Preceding unsigned comment added by Abter1 (talk • contribs) 16:09, 1 July 2010 (UTC)

http://news.yahoo.com/s/ap/20100329/ap_on_sc/us_sci_math_genius

Even the listed supporting article from the Mail Online states:

Four years ago, the maths genius failed to turn up to receive his prestigious Fields Medal from the International Mathematical Union for solving the problem.

At the time he stated: 'I'm not interested in money or fame. I don't want to be on display like an animal in a zoo.

Coffee Cup / Doughnut problem?

Isn't this just a variant of the old joke about how topologists can't tell the difference between a coffee cup and a doughnut .. with the Poincare conjecture applying to any shape that is topologically simpler than a coffee cup or doughnut? Zaphraud (talk) 15:06, 2 July 2010 (UTC)

A "coffee cup" is not simply connected, and so the Poincaré conjecture has nothing to say about it. Moreover, a "coffee cup" in three-space is a three-manifold with boundary, and so the Poincaré conjecture has nothing to say about it. (Every point has a neighbourhood that is either homeomorphic to an open ball, i.e. ${\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}:x^{2}+y^{2}+z^{2}<1\}}$ or to an upper half-space, i.e. ${\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}:z\geq 0\}}$.) The Poincaré conjecture relates to abstract manifolds, as well as submanifolds. It's a very general and powerful piece of kit. All it talks about is simply connected, closed 3-manifolds. If the space you have does not meet all of these hypotheses, e.g. a "coffee cup" or a doughnut, then the Poincaré conjecture has nothing to say. Please see the article about necessary and sufficient conditions. — Fly by Night (talk) 01:24, 29 July 2011 (UTC)

why a separate "Solution" article?

I propose merging (re-merging?) Solution of the Poincaré conjecture here because more than half of that article consists of statement of the problem (in more detail than this article has!), and what remains is not much longer than this article's "Solution" section. —Tamfang (talk) 04:15, 7 February 2011 (UTC)

I absolutely agree with Tamfang. As with other articles in Misplaced Pages, the same article is composed of both the statement of the problem and its respective solution. If that means that the article might be a bit longer, so be it. An example, not an mathematical article, but an historical one, is the Cuban Missile Crisis article. And if you think about it, telling the problem and its solution is telling its history (intro, development, and conclusion). So, sysop, merge solution with the article, and make solution redirect to the article. Mrmagoo2006 (talk) 01:47, 15 April 2011 (UTC)

It's not a job for a "sysop", it's a job for us editors, assuming that there are no serious objections to such a move. —Tamfang (talk) 03:17, 15 April 2011 (UTC)

I just wanted to mention that I really like the explanation of the conjecture's requirements at Solution of the Poincaré conjecture, which in general seems to be written at about the correct level of accessibility. They're generally well-written and gloss over just enough (though perhaps "homeomorphism" could be recast in terms of limits--points getting closer to a fixed point on the first object get closer to the corresponding point on the second object, and vice versa, with the typical example of something that's not a homeomorphism, f(x) = (cos x, sin x) for 0 <= x < 2pi). 24.220.188.43 (talk) 00:08, 28 April 2011 (UTC)

I couldn't agree more. Shall we start the move? MrJosiahT (talk) 01:09, 20 July 2011 (UTC)

I found the task not as easy as I imagined. —Tamfang (talk) 07:57, 22 August 2011 (UTC)

Since Solution of the Poincaré conjecture redirects here, the two articles were obviously successfully merged. Thank you. 90.180.192.165 (talk) 23:58, 3 July 2013 (UTC)

new book published

There's a new book published about the Russian mathematician that has solved this conjecture entitled "Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century" Perhaps this should be added as a reference to this article? —Preceding unsigned comment added by 217.154.240.195 (talk) 13:18, 21 March 2011 (UTC)

• First, the book is not new, it was published in 2009. Second, it is mentioned in Grigori Perelman for a long time already. Maxal (talk) 11:22, 28 April 2011 (UTC)

Explanation of terms

Didn't there use to be a section in this article that explained what the terms in the formulation of the conjecture mean? Simply connected, manifold, homeomorphism... Why was it removed? Jack Daw (talk) 04:27, 31 January 2012 (UTC)

The "history" tool would let you look up previous versions. Could you give a link to the version you are talking about? — Carl (CBM · talk) 12:02, 31 January 2012 (UTC)

Solution Section

This section reads for me in a somewhat unenyclopaedic manner. Specific examples being the phrase "something called", which is totally unnecessary, and the explanation of eignvalues, rather than merely linking to the article.

In the explanation of shape of the singularities, it says "essentially three-dimensional cylinders made out of spheres stretched out along a line". "essentially" seems unnecessary. Would a sphere stretched along a line, in this context, be four dimensional, rather than three? The whole article appears to be about the four dimensional 3-sphere.

--Atomic Fusion (talk) 04:42, 25 October 2013 (UTC)

Tina S. Chang does not appear in references

The body of the article contains the following sentence, as at 5th. February 2015;

"This process is described in the fictional work by Tina S. Chang cited below."

Not only is no work by Tina S. Chang referenced, but what is a 'fictional' article doing being cited as a reference? — Preceding unsigned comment added by 122.61.14.110 (talk) 11:01, 5 February 2015 (UTC)

Tina S. Chang was removed from the External links section here

Indeed there's no reason left to keep the Tina S. Chang related sentence in the body. --Francis Schonken (talk) 14:03, 5 February 2015 (UTC)

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