Slashdot Log In
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."
Related Stories
[+]
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!"
This discussion has been archived.
No new comments can be posted.
The Fine Print: The following comments are owned by whoever posted them. We are not responsible for them in any way.
Full
Abbreviated
Hidden
Loading... please wait.
Y'know (Score:3, Insightful)
Re:Y'know (Score:2, Insightful)
Re:Y'know (Score:5, Funny)
You're new here, arent you?
Parent
Re:Y'know (Score:3, Insightful)
Cool. (Score:3, Funny)
What about the Dunwoody paper? (Score:5, Interesting)
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.
Parent
Re:What about the Dunwoody paper? (Score:2)
Here's his (potential) proof. [soton.ac.uk]
Re:What about the Dunwoody paper? (Score:5, Informative)
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.
Parent
Re:What about the Dunwoody paper? (Score:5, Informative)
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. :-)
Parent
Re:What about the Dunwoody paper? (Score:5, Funny)
The part of the proof where it says "then a miracle occurs..." is being questioned by numerous mathematicians.
Parent
Donuts, apples, I'm hungry (Score:2, Funny)
Re:Donuts, apples, I'm hungry (Score:5, Funny)
Women, right???
Parent
Re:Donuts, apples, I'm hungry (Score:3, Interesting)
Re:Donuts, apples, I'm hungry (Score:3, Insightful)
Have you ever drunk something, started to laugh, and have the stuff come out your nose? That proves that nose and mouth are connected, and the topology of a person is therefore more complicated than a torus. Because of the two holes in your nose, we're talking at least genus 3. I think the ears are connected to the nose/mouth system too, which would make it genus 5.
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...
Parent
Re:Explanation (Score:5, Insightful)
Parent
sigh (Score:5, Insightful)
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.
Parent
Re:sigh (Score:3, Funny)
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?
Parent
Re:Explanation (Score:4, Informative)
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.
Parent
"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?
Parent
Re:Explanation (Score:5, Funny)
Parent
Re:Explanation (Score:5, Funny)
> applications there might be of this?
An application would be to make better doughnuts, I suppose.
Parent
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
Parent
Re:Explanation (Score:4, Funny)
Parent
Re:Explanation (Score:3, Informative)
Hmmm, a lot of work in mathematics may not have immediate applications or uses. But down the line, they just might get used.
As many posters have already mentioned, Boolean algebra is one such case, and another example would be the work done by Fourier [mathphysics.com] - particularly his integral transforms and series.
I mean, today these are used so much in DSP and the like, I doubt Fourier had these in mind
Re:Explanation (Score:3, Funny)
Google Partner Link (Score:3, Informative)
Explanation (Score:2, Informative)
What's that conjecture again? (Score:5, Informative)
Re:What's that conjecture again? (Score:2, Funny)
Well why didn't you just say so in the first place. It's so simple when you put it in plain english
[/sarcasm]
Re:What's that conjecture again? (Score:5, Informative)
[/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?
-
Parent
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
Re:What is it ? (Translation to make it easier) (Score:5, Informative)
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.
Parent
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)
Perl? (Score:5, Funny)
Parent
Wait for it wait for it.... (Score:5, Insightful)
Typo... (Score:3, Funny)
It is not "mathematician"
Please make the appropriate corrections.
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).
Corollary (the proof made intuitive) (Score:4, Funny)
Conclusion: Perelman's proof is a 3-sphere.
Proof: Apply the Theorem to its proof.
Re:What the heck is this? (Score:2)
You don't need to ask Slashdot. Google is your friend.
Nope. (Score:4, Interesting)
Here's to Perelman.
regardless, as the article suggests, even if it doesn't solve the poincare conjecture, the work will hopefully remove anaomalies in Ricci flows. Which is exciting if you are a mathematician and not very interesting at all if you are at a coctail party (unless you are three sheets to the wind, and then the mathematicians around you can talk about the topographic properties of those sheets...)
Parent
Re:Oh no.. (Score:3, Insightful)
Math is one of those disciplines where you just can *not* skim the problem and expect to understand it... you have to load into memory every word that is in the text (like 'manifold' etc), and create a working instance of that object in your brain...
It's basically like launching
In Squarepoint's own words (Score:3, Insightful)