Poincaré Conjecture May Be Solved 299
Posted
by
CmdrTaco
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 milliondollar 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 followup that was just released last month."
Y'know (Score:3, Insightful)
Re:Y'know (Score:2, Insightful)
Re:Y'know (Score:2, Informative)
So here is the Google/NYT partner link [nytimes.com]
Re:Y'know (Score:5, Funny)
You're new here, arent you?
Re:Y'know (Score:2)
Re:Y'know (Score:3, Insightful)
Re:Y'know (Score:2)
I was very disappointed in the article. I couldn't get past that bit about "2 dimensional spheres". This is either a piece of topological technical jargon that needed an explanation when it was introduced or the article is so screwed up I can't give it any credence.
Is anyone aware of any better written popular (nontechnical) reporting on this?
Cool. (Score:3, Funny)
Re:Cool. (Score:2, Funny)
Re:Cool. (Score:2)
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.
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.
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 wellthoughtout 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)
The part of the proof where it says "then a miracle occurs..." is being questioned by numerous mathematicians.
Re:What about the Dunwoody paper? (Score:2)
3) ?
4) Profit!
Re:What about the Dunwoody paper? (Score:2, Funny)
I prefer to think of it as
public static void main (String[] args) {
doStuff();
}
Donuts, apples, I'm hungry (Score:2, Funny)
Re:Donuts, apples, I'm hungry (Score:5, Funny)
Women, right???
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...
Re:Explanation (Score:5, Insightful)
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.
Re:sigh (Score:3, Funny)
Re:sigh (Score:2)
It's like kids in school saying they shouldn't have to learn algebra since it is never going to do them any good in real life... like balancing their checkbook, or knowing how to make change.
You're right, we would still be living in caves if people sat around waiting for great, really pratical ideas to show up.
Re:sigh (Score:2)
I was going to bring up that example myself, but Boolean algebra had practical applications at that time.
...in philosophy, and rethorics for example.
But, yes I'm sure ol' George would have been amazed at exactly how useful his little contribution to math turned out to be in entirely different fields less than a century later. And there were no way to predict that at the time.
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:2)
I live in fear that one day advanced mathematicians will discover proof we really don't exist, and we all die in an instant.
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.
"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:2)
And I guess, if you want some application to that, then you might want to read some of Hawkin
Re:Explanation (Score:2)
The conjecture itself is something fairly abstract, but it's widely considered the most important unsolved problem in topology and has so far induced a long list of false claims and proofs, some of which have led to a better understanding of lowdimensional topology. Solving the problem would further increase knowledge about topology and many fields of research in mathematics, geometry, physi
Re:Explanation (Score:5, Funny)
Re:Explanation (Score:5, Funny)
> 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 3sphere 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:Practical Applications? (Score:2)
So in the end, its mathematically impossible for me to catch the girl. And here I thought the problem was my fixation on Star Wars action figures.
Re:Explanation (Score:4, Funny)
Re:Explanation (Score:2)
Re:Explanation (Score:2)
Not being a mathemetician, I think it's a rather silly designation, however.
Re:Explanation (Score:2)
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:2)
The 'plain English' version, despite being much longer, is not a perfect translation. It mentions 'a set of spherelike properties', without defining which properties are included in that set.
On the other hand, 'simply connected' is both shorter and more precise, but most people don't know what it means. However, you can look up very fine definitions at Mathworld [wolfram.com] or the Wikipedia [wikipedia.org].
Re:What's that conjecture again? (Score:2)
Maths seems to be one of the few things (especially at high levels such as this or when you move into R^n dimensions) that is extremely difficult, if not sometimes impossible, to put into plain english at all. How can you possibly describe, for example, a 5dimensional object (or object in R5 space)  you can't make up an analogy or describe some
Re:What's that conjecture again? (Score:2)
I disagree strongly. I don't want to flame or be rude or anything, but the mere thought of a scientific subject matter which can be explained and tought to students, yet can't be put into plain english, is sureally stupid. As I pointed out above, properly used technical terms give a lot of information in few words, but that only works because they first have to
Re:What's that conjecture again? (Score:2)
There is a certain minimum amount of familiarity with the relevant field that is demanded when discussing certain concepts.
Re:What's that conjecture again? (Score:2)
The definition was extremely clear by anyone with a concept of a few simple mathematical concepts. If you dont know those concepts, then the content of the definition has no bearing on you at all anyway; it only affects someone who'd be interested enough to know such words.
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 4dimentional 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?

Re:What's that conjecture again? (Score:2)
What does that make you? (Score:2)
As a computer scientist, I must inform you that all small integers (anything less than +/ 2^23 for IEEE) are accurately represented, and specifically that 10 is NOT equal to 9.9 (recurring). Yes, (10/3)*3 will yield that value  as will decimal arithmetic to any precision.
Oh well, I guess that makes you a troll.
Re:What's that conjecture again? (Score:2)
Not my fault if I'm stupid (Score:2)
Why is a goodold friendly sphere called a 2dimensional sphere?
Did I miss something? Are we living in Flatland?
I am fascinated by the idea of multiple geometrical dimensions.
Re:Not my fault if I'm stupid (Score:2)
Re:Not my fault if I'm stupid (Score:2, Informative)
Re:Not my fault if I'm stupid (Score:2)
Thanks!
Re:What's that conjecture again? (Score:2)
What is it ? (Score:2, Informative)
Easy, i shall explain
The conjecture that every simply connected closed 3manifold is homeomorphic to the 3sphere. 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 nmanifold is homotopyequivalent to the nsphere iff it is homeomorphic to the nsphere. 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 2dimensional 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 ndimensional figures to a simpler form (like a sphere) and know that the continuity and general structure of the figure will remain the same.
ARGH...here we go again (Score:2)
another thing is interesting to note: there are a LOT of problems in mathematics in which n=1 is trivial, n=2 is hard but straight forward. n>3 is not too hard and usually falls under one proof, and n=3 is EXTRAORDINARILY difficul
Re:ARGH...here we go again (Score:2)
Unfortunately, the reality (sic!) is a little bit more complicated. Terms as surface and volume as you use them don't make sense with topological objects. The problem is that you are implying that a 3manifold is the surface of some
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.
Re:Now THATS Patience... (Score:2)
Re:Now THATS Patience... (Score:2)
The beauty of mathematics is that it doesn't work like natural sciences. Once something is proved, it is forever proven and correct in maths. He will know for sure whether his answer is correct in a few months, most probably.
I can see him now... (Score:2)
1. Mathematics is the language of nature.
2. Everything around us can be represented and understood through numbers.
3. If you graph these numbers, patterns emerge. Therefore: There are patterns everywhere in nature.
Sequel (Score:2, Funny)
Re:Sequel (Score:2)
Poincare Conjecture Solved Ages Ago (Score:5, Funny)
Now I Understand... (Score:5, Funny)
Re: Now I Understand... (Score:2)
Re: Now I Understand... (Score:2, Funny)
Re:Now I Understand... (Score:2)
("Christina had that notsofresh feeling..." Oh, come on. Like I'm the only one who thought that.)
Perl? (Score:5, Funny)
Re:Perl? (Score:2)
Wait for it wait for it.... (Score:5, Insightful)
Why is this modded up? (Score:2)
So what the hell are you talking about... was it supposed to be toungeincheek humor?
Typo... (Score:3, Funny)
It is not "mathematician"
Please make the appropriate corrections.
Proof of Poincare conjecture.... (Score:2, Funny)
(This of course assumes that 3manifolds 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 TaniyamaShimura).
great (Score:2)
Two dimensional sphere! (Score:2)
Do they mean the sphere is 2dimensional?
Re:Two dimensional sphere! (Score:2)
Corollary (the proof made intuitive) (Score:4, Funny)
Conclusion: Perelman's proof is a 3sphere.
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...)
Re:perhaps a lesson in logic (Score:2)
my point was:
1) the first possibility (dunwoody) was already ruled out.
2) even if poincare isn't solved, this work is still a valid contribution to the canon of mathematics.
2 is not a direct response to the parent post. It is a preemptive response to people who say "well, the paper has to be around for 2 years and this doesn't prove anything and why is this even news?!" the typical rabble.
Re:I'm pretty sure this is a dupe (Score:2)
Re:I solved this first!! (Score:2)
Re:Yeah you and me! (Score:2)
Re:And the answer is... (Score:2)
"It was in fact a trick question. Coventry City have never won the FA Cup."
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)
Re:In Squarepoint's own words (Score:2)
Then it's a motto with a rather slanted translation, IMO. "Une idée préconçue" might be more accurately translated as "a preconceived idea", which is a decidedly less pejorative way of putting it than "prejudice".