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Poincaré Conjecture May Be Solved
Posted by
CmdrTaco
on Tue Apr 15, 2003 08:42 AM
Flamerule writes "The New York Times is now reporting that Dr. Grigori (Grisha) Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg, appears to have solved the famous Poincaré Conjecture, one of the Clay Institute's million-dollar Millennium Prize problems. I first noticed a short blurb about this at the MathWorld homepage last week, but Google searches have revealed almost nothing but the date and times of some of his lectures this month, including a packed session at MIT (photos), in which he reportedly presented material that proves the Conjecture. More specifically, the relevant material comes from a paper ("The entropy formula for the Ricci flow and its geometric applications") from last November, and a follow-up that was just released last month."
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Poincare Conjecture Proof Completed 222 comments
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!"
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Poincaré Conjecture May Be Solved
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Y'know (Score:3, Insightful)
(http://slashdot.org/)
Re:Y'know (Score:5, Funny)
You're new here, arent you?
Cool. (Score:3, Funny)
What about the Dunwoody paper? (Score:5, Interesting)
(http://www.fscked.co.uk/ | Last Journal: Saturday May 03 2003, @11:59AM)
The link to mathworld.wolfram.com [wolfram.com] from the post says:
So, why the excitment about this later Perelman paper? Has the Dunwoody paper been debunked?
Re:What about the Dunwoody paper? (Score:5, Informative)
It seems as if he missed a step and couldn't figure it out.
Re:What about the Dunwoody paper? (Score:5, Informative)
(http://taltos.pha.jhu.edu/~tamas/)
From the site:
It is unclear as of this writing if Dunwoody's proof will last even a fraction of that duration.
In fact, it appears that the purported proof has already been found lacking, judging by the facts that (1) the abstract begins, "We give a prospective [italics added] proof of the Poincaré Conjecture" and (2) the revised April 11 version of the preprint contains a small but significant change in title from "A Proof of the Poincaré Conjecture" to "A Proof of the Poincaré Conjecture?" In particular, a critical step in the paper appears to remain unproven, and Dunwoody himself does not see how to fill in the missing proof.
Re:What about the Dunwoody paper? (Score:5, Informative)
(http://www.missouri.edu/~kingjw)
A gap or three in the proof were found within days, and a mathematician friend of mine reported that it didn't look like solutions to these problems were immediately forthcoming.
The excitement about this paper comes from the fact that the guy who did the work has come up with impressive results in the past, builds on important and cutting edge work, and seems to have a really thorough command of the potential difficulties. (In other words, when he is asked questions about the tricky points, he immediately responds with what look like strong and well-thought-out answers.) For that matter, his work claims to prove a more general conjecture of which Poincare is a special case, and so this work could have more general significance to many other problems, even if there turns out to be a glitch or two in this iteration of the proof.
It's a very hard problem, and this answer could be wrong, too. But there's a big difference between tossing a paper up on a preprint server and giving a lecture at MIT where nobody can (yet) touch you. :-)
Re:What about the Dunwoody paper? (Score:5, Funny)
(Last Journal: Monday November 21 2005, @11:45AM)
The part of the proof where it says "then a miracle occurs..." is being questioned by numerous mathematicians.
Donuts, apples, I'm hungry (Score:2, Funny)
(Last Journal: Wednesday August 27 2003, @02:48PM)
Re:Donuts, apples, I'm hungry (Score:5, Funny)
(http://www.theld.net/)
Women, right???
Explanation (Score:5, Informative)
Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?
Re:Explanation (Score:4, Funny)
Has Fermat's Last Theorem actually been used in practical applications? I don't think so...
Re:Explanation (Score:5, Insightful)
(http://slashdot.org/)
sigh (Score:5, Insightful)
(http://slashdot.org/)
Has Fermat's Last Theorem actually been used in practical applications? I don't think so...
If everyone thought like you we'd still be living in caves.
Just because practical applications aren't totally obvious for a layman (or even a matematician) doesn't mean this will never be of practical use.
Even if no practical applications are ever found, this proof (if it survives peer review) may well pave the way for something else that is immensly useful.
There's just no way to tell right now.
Re:Explanation and George Boole (Score:5, Insightful)
Give maths time and it will applicable to your everyday life. What has been going on for the past 3,000 years?
Re:Explanation (Score:4, Informative)
(about:mozilla)
http://www22.pair.com/csdc/car/carhomep.htm [pair.com]
Granted, none of this is stuff I would expect a gas station attendant to be playing with, but it's apparently increasingly important for researchers and engineers.
"Useless" mathematics that we use (Score:5, Insightful)
Galois fields are used for checksum algorithms, something I'm sure Galois never thought of.
Fourier transforms are used for image compression (JPEG).
Who knows what Poincaré's topology might be used for in the future?
Re:Explanation (Score:5, Funny)
(http://chrisbbehrens.blogspot.com/)
Re:Explanation (Score:5, Funny)
(http://www.pykota.com/)
> applications there might be of this?
An application would be to make better doughnuts, I suppose.
Practical Applications? (Score:4, Insightful)
> there might be of this? Or is it strictly an abstract concept?
Speaking as a layman, the practical application of these sorts of proofs is that you can use them to prove equivalent, more practical questions.
One of the references in another comment explained that this conjecture has been proved for all other dimensions, and this 3-sphere seems to be a special case, as far as proof is concerned.
If the Poincare' conjecture were proved, then the general case could be solved. After that, "simply" proving that another hard problem is equivalent to the Poincare' conjecture is enough to prove that other problem.
Now, I've heard the problem described with a lasso instead of with a rubber band. I can imagine times when I'd really like to know when my lasso is going to close around something or if it's just going to slip off
Re:Explanation (Score:4, Funny)
(http://www.glpwd.com/)
Google Partner Link (Score:3, Informative)
Explanation (Score:2, Informative)
(Last Journal: Tuesday March 04 2003, @02:43AM)
What's that conjecture again? (Score:5, Informative)
(http://err.antville.org/ | Last Journal: Friday February 21 2003, @04:53PM)
Re:What's that conjecture again? (Score:5, Informative)
(http://slashdot.org/)
[/sarcasm]
Ok, try this:
We long ago proved that an ordinary sphere is the only shape in 3 dimentions with no holes in it.
Note that the "shape" is "made of clay". You are allowed to stretch it, squish it, and bend it all you want. You aren't allowed to cut it or put a hole in it. And you can't "meld" parts togther.
A coffee cup is the same "shape" as a donut because you can smoothly "flow" the cup part into the handle and you get a donut.
What they just proved is that a 4-dimentional sphere is the only shape with no holes in it.
So what? Well if you have some wierd complex 4 dimentional "thing" and you know it doesn't have any holes in it then you now know it has to be equal to a sphere. It SEEMS obvious, but it was still important to prove. It is important for many other proofs.
Better?
-
A solution was repoted one year ago... (Score:1, Redundant)
What is it ? (Score:2, Informative)
Easy, i shall explain
The conjecture that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincaré (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n = 3.
The Poincaré conjecture has proved a thorny problem ever since it was first proposed, and its study has led not only to many false proofs, but also to a deepening in the understanding of the topology of manifolds (Milnor). One of the first incorrect proofs was due to Poincaré himself (1953, p. 370), stated four years prior to formulation of his conjecture, and to which Poincaré subsequently found a counterexample. In 1934, Whitehead (1962, pp. 21-50) proposed another theorem which proved to be incorrect, then discovered a counterexample (the Whitehead link) to his own theorem.
The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical, n = 3 (the original conjecture) remains open, n = 4 was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), n = 5 by Zeeman (1961), n = 6 by Stallings (1962), and by Smale in 1961. Smale subsequently extended his proof to include
you see ?, its all quite clear if you think about it
Re:What is it ? (Translation to make it easier) (Score:5, Informative)
(Last Journal: Saturday September 04 2004, @10:35AM)
basically all the poincare conjecture says is that if you have a 3 dimensional figure which is closed (therefore, it it bounded (doesn't go off to infinity in either direction), and doesn't have any "holes" in it (like a donut)) then you can take every point and map it to a point in an equivalent sphere without losing continuity (therefore, everypoint will have the same "neighbourhood" of points as it had in the initial shape.)
ie. You can map a cube into a sphere, or a dodecahedron, or a weird globlike thing that doesn't fold back on itself, or a whole piece of paper (without holes), or a pencil, or a lot of different figures.
As well, this conjecture also handles figures with holes in them (like donuts), and maps them all to simpler figures.
It's a very simple concept, but has been incredibly hard to prove, and what makes this conjecture even more frustrating is the fact that 1 and 2-dimensional forms of this conjecture were incredibly easy to prove, as well as 4 and up have been solved, and were reasonably easy as well. Yet for some reason the 3 dimensional version does not lend itself easily to a simple proof.
Everyone generally believes this is true, but no one has been able to prove or disprove it.
If proven, this is an important aspect of topology, because then we can map all n-dimensional figures to a simpler form (like a sphere) and know that the continuity and general structure of the figure will remain the same.
Quick! (Score:1)
(http://compustore.com/ | Last Journal: Thursday May 12 2005, @08:06PM)
--Joey
I'm pretty sure this is a dupe (Score:1)
(Last Journal: Sunday November 11, @09:31AM)
Now THATS Patience... (Score:4, Interesting)
"However, according to the rules of the Clay Institute, the paper must survive two years of academic scrutiny before the prize can be collected."
So, all told, Perelman is going to wait a total of 10 years from the time he started to work on the solution to the Conjecture, to the time where the scientific community lets him know if his answer is correct. Wow.
Sequel (Score:2, Funny)
Poincare Conjecture Solved Ages Ago (Score:5, Funny)
Now I Understand... (Score:5, Funny)
(http://slashdot.org/~masq/journal/ | Last Journal: Friday May 02 2003, @07:04AM)
Perl? (Score:5, Funny)
A packed session at MIT indeed... (Score:1, Interesting)
squarepoint (Score:1, Funny)
are you some sort of unamerican antipatriot ?
better change his name to "squarepoint" before this site gets banned...
Wait for it wait for it.... (Score:5, Insightful)
(http://www.gnu.org/)
Typo... (Score:3, Funny)
It is not "mathematician"
Please make the appropriate corrections.
Proof of Poincare conjecture.... (Score:2, Funny)
(Last Journal: Saturday February 15 2003, @10:04AM)
(This of course assumes that 3-manifolds are malleable.)
this can't be (Score:3, Funny)
1. When Wolfram and Hart were all killed by the Beast, Wolfram was in the house.
2. Wolfram is human and isn't as smart as the papers say.
3. He stepped up to MCHawking and is now hanging from a tree with a sign pinned to him that reads: WHACK EMCEE.
Actually, Perelman is claiming much more... (Score:5, Informative)
Anyway, if true, this is kind of like Wiles proof of Fermat's Last Theorem -- proving an old conjecture by proving a more general (and more modern) one (in Wiles case, it was proving part of Taniyama-Shimura).
great (Score:2)
(http://slashdot.org/ | Last Journal: Thursday February 21 2002, @04:37PM)
what?Only 1milion $ ??? (Score:1)
Moral of the story..
Stick to porn business, much better buck..and you dont have to think so much
Two dimensional sphere! (Score:2)
Do they mean the sphere is 2-dimensional?
4 colour problem (Score:1)
Access Denied! (Score:1)
(http://spurl.net/)
How Eric can keep from writing "Fuck CRC Press" on every page, I do not know.
(BTW, what a nice way to discover my ISP has a transparent proxy.)
A topologist's perspective (Score:1, Informative)
First, what is a manifold? Well, take a bunch of tetrahedra, and start gluing their sides together in pairs. Start with finitely many, pair the sides up, and say how the sides are matched. What you have at the end will be a closed 3-manifold, providing every face gets glued to exactly one other face, and providing some number you can calculate called the "Euler characteristic" is equal to zero. The Euler characteristic is just
the number of vertices, minus the number of edges, plus the number of triangles, minus the number of tetrahedra *after* you have glued it all up. You now have your manifold M.
Simply connected has "something" to do with holes, but I think it's easy enough to say exactly what it is. Think of the unit circle in the plane, the set of points which are distance 1 from the center of the plane. Then think of a continuous function from the circle to the manifold. That is, for every point in the circle, you get a corresponding point in M such that when you vary the point you choose in the circle continuously, the corresponding point in M moves continuously. Since a circle is 1-dimensional, we can wiggle the image a little bit so that it doesn't cross itself; we call this image a *knot*. "Simply-connected" means that every continuous map from a circle to a knot in M extends to a continuous map of the unit disk. So the knot "bounds" a disk in M (which *is* allowed to intersect itself, and probably has to) which gives a way of shrinking the knot down to a point continuously.
The Poincare conjecture claims that the only 3-manifold M with this property is the 3-sphere.
Well, what is the 3-sphere? Take two solid balls and completely glue their boundaries together. What you get is the 3-sphere. Another description is as the set of points in 4-dimensional space which are distance 1 from the origin, just like the circle was the set of points in 2-dimensional space (i.e. the plane) at distance 1 from the origin. How would you prove such a thing? Well, one way is to use a criterion of Bing, who showed that any closed manifold with the following property is S^3: Bing's property says "every knot in M is contained in a solid ball in M". There are other criteria, but part of the problem is that they are very hard to check or verify, and the hypothesis (that S^3 is simply connected) is hard to use.
So, what does Perelman do? He actually proves not just the Poincare Conjecture but a much stronger conjecture called Thurston's Geometrization Conjecture. Unlike the Poincare Conjecture, which is a conjecture just about S^3, Thurston's conjecture is a conjecture about *every* 3-manifold. It says, roughly speaking, that every closed 3-manifold which is *irreducible* (i.e. every sphere bounds a ball) can be cut up into a finite nu