Russian May Have Solved Poincare Conjecture

nev4 writes "Reuters (via Yahoo News) reports that Grigori Perelman from St. Petersburg, Russia appears to have solved the Poincare Conjecture. The Poincare Conjecture is one of the 7 Millenium Problems (another is P vs NP, also covered on /. recently). Solving a Millenium Problem carries a reward of $1M, but apparently Perelman isn't interested..." nerdb0t provides some background in the form of this MathWorld page from 2003.

He'd post AC

True math genius and the desire for money (and fame and babes, etc.) seem to be mutually exclusive traits and I think that's rather inspiring (and damned practical).

Take the case of Paul Erdos [wikipedia.org] who was essentially homeless, but published over 1500 papers and is considered one of the all time greats in the field.

Perelman just casually posted his solution out to the web in much the same way that some of the most brilliant posts on /. come form "anonymous cowards" sitting in their offices at MIT. What a god.

Re:He'd post AC

It makes sense. Anyone that brilliant would see how pointless it is to worry about money. When will the rest of us learn? There's more to life than money.

Yeah, it's broadband.

Re:He'd post AC

This observation of Stevyn and the answer to his question "When will the rest of us learn?" is well explained by Maslow's heirarchy of needs [wikipedia.org]. The was Maslow would havd put it is that this guy and other brillian people are 'self actualized' "A musician must make music, the artist must paint, a poet must write, if he is to be ultimately at peace with himself. What a man can be, he must be. This need we may call self-actualisation. (Motivation and Personality, 1954)". This happens after the various esteem needs, love needs, safety needs, and physiological needs are met. I think the average person gets stuck dealing with the "safety needs" (thus easy 9/11 manipulation). And the average reasonably-successful-slashdotter-guy gets stuck with the "esteem needs" stage aiming for Karma.

Only us self-actualized "Anonymous Coward" guys rise above this with insightful and informative posts such as this one without whoring for karma.

Re:He'd post AC

And the average reasonably-successful-slashdotter-guy gets stuck with the "esteem needs" stage aiming for Karma.

But the geeks are all kept equal with hatchet, ax, and 50-point karma cap.

Re:He'd post AC

Anyone that brilliant would see how pointless it is to worry about money. When will the rest of us learn?

Oh please. What is this? The 60s? Apparently the guy is able to find enough time to work on these problems. That kind of freedom is what money buys. If he didn't have enough money to do that then it would suddenly become much more important.

"Money" is not some stack bills in your wallet. It represents some tangible effort that had value, and that value is now stored in a convenient form, ready to be exchanged for something else of value.

Re:He'd post AC

Apparently the guy is able to find enough time to work on these problems. That kind of freedom is what money buys.

It probably would only take $15K in the US to rent a small apartment in a cheap city and buy food for a year, allowing him to work on his problems. I think the point is that this guy may have been able to make a significant contribution to human knowledge and maybe centuries of notoriety with what it cost to live for a few years. Most of the rest of us would have taken the same amount of money and just dumped it into buying an upscale SUV.

Re:He'd post AC

I think what he was saying was that the ONLY way money comes into circulation is through loans. Therefore, although some can pay back there loans, it is physically impossible for the entire country to ever pay back their loans, because not only are we responsible to pay back the loans, but we also have to pay back interest! But the banks only created enough money for the _principle_ of the loan, not for the interest. So, while you and me can pay back our individual loans, it is physically impossible for the whole country to pay off its debt, because the money supply would be gone, and there would be nothing left to pay with.

Let's say that there is a small economy. I am a central bank. Right now, there is no money. Therefore, you take a loan out for $10, and I charge $1 interest. Frank takes out a loan for $10, and I charge him $1 interest. The whole economy has $20 in it, but they owe $22. There's no way this can be paid off. Now, one of you could handle their money better than the other, and get a $1 advantage to pay off their loan, but that would leave only $9 in the economy to pay off a remaining $11 loan. One of you would be fine, but there is no way in this system for everyone to pay back their debts. So, eventually, the banks own nearly everything.

This is why the founders of our country hated central banks, and was one of the primary reasons for the revolutionary war.

Re:He'd post AC

We don't need to "learn" from this, really. it's perfectly OK in our society to take pride in our achievements and to try to gain from them. Unless you're truly self-actualized (as another poster astutely pointed out), we're all subject to certain realities and desires. After all, monetary reward can enhance your ability to do more good. As Hunter S. Thompson once said, "feed the body or the head will die." There's no shame in that. I find it interesting though, that some artists and scientists seem to exist on another plane altogether.

Re:He'd post AC

You're completely correct; I think my comment was mistaken. Without the reward of money at the end of the tunnel, I probably wouldn't be in school now working towards a goal. There is no shame in working for money because it represents a reward for an invaluable effort.

However, I've seen many intelligent people work hard without stopping because it was the right thing to do, not because of the monetary gain. That is what I'd hope to highlight.

Re:He'd post AC

You're completely correct; I think my comment was mistaken.

Woah, that's weird! I thought I was reading Slashdot but it must actually be some other site.

Re:He'd post AC

There's more to life than money.

Yes, but he could reinvest the money into rubber bands and apples and solve thousands of Poincaré conjectures at once and thus gather even more money to buy apples for the hungry children in the world and rubber bands for their trousers. Well, if this business model isn't patented yet, of course...

Re:He'd post AC

Well.. I think it's kind of a general thing for all good Science too.

Einstein's original paper on Special relativity was named "On the electrodymanics of moving bodies".. It was not named "Revolutionary new discovery by me, Albert Einstein which will revolutionize the world of physics".

I guess there are several reasons for this.. one is simply manners. Boasting is unpolite. Scientific papers rarely have exciting titles, even when the results are exciting.

The second is of course, that a good scientist realizes the if a result may be revolutionary. A good scientist also always leaves room for doubt.

So the natural behaviour would of course to be careful and discreet, and not go confidently telling the world of your revolution until it has been verified. Otherwise, you'll end up with a lot of egg on your face.

Conversely, most scientists are highly sceptical of 'revolutionary' results which are announced in the press before being published. In fact, most pseudoscientists are very good at publicizing themselves and their 'revolutions', probably because they are totally convinced of their own theories, and are lacking the 'self-doubt' bit.

Re:He'd post AC

Not really, since the field of electrodynamics was only in its infancy at that time, a few years after the publication of maxwell's theorems. And it was almost exclusively applied to fixed bodies rather than moving bodies...

So it would be like publishing a paper called "on datastructures" if you were the person that invented datastructures....

Re:He'd post AC

I guess there are several reasons for this.. one is simply manners. Boasting is unpolite. Scientific papers rarely have exciting titles, even when the results are exciting. The second is of course, that a good scientist realizes the if a result may be revolutionary. A good scientist also always leaves room for doubt.

Contrast this lack of fanfare with another recent publication, Stephen Wolfram's A New Kind of Science [amazon.com]. This 'new' science seems to have been met with mixed reviews at best, and not the paradigm shift that the author seems to have been hoping for. Of course only time will tell who is right... But in the event that Perelman's is incorrect, his humility and lack of hubris regarding his solution definitely earns him my respect, and undoubtedly that of many others in the field.

The "free" internet bubble never burst

But there's a snag. He has simply posted his results on the Internet and left his peers to work out for themselves whether he is right -- something they are still struggling to do.

"There is good reason to believe that Perelman's approach is correct. But the trouble is, he won't talk to anybody about it and has shown no interest in the money," said Keith Devlin, Professor of Mathematics at Stanford University in California.

I'm always amazed how much free stuff is on the internet. Free million dollar solutions! Good luck with em!

Math?

1,000,000 USD is about equal to 560,000 GBP, not 5.6 million GBP.

Re:Math?

In that case it should be 5.6*i, since we all know the WMDs were imaginary.

Re:Math?

I think you misspelled sourted.

Look at his method for solving this!!!

He's trying to integrate homeomorphic convergence using a Baxter-Bates supermodality, which Krause clearly explained is impossible for T(s) in a non-linear progression. Fantastic thought process on this complex differential geometric problem.

Just kidding! I have no clue what the hell this is. I got lost after the word conjecture.

Re:Look at his method for solving this!!!

And if you hadn't added that last paragraph, you'd be +3, Informative by now.

So tell the truth...

You just made all that up, didn't you?

Damn...

I read all the links, and I'm pretty sure they were all in english, but I didn't understand a word of it. No wonder all the mathematicians are nuts.

(I wonder if this is what some of my non-engineering clients think of my work sometimes)

Yes but...

His answer to the problem was "42".

- Greg

Re:Yes but...

Makes sense, as I have no idea what the question is.

Re:Yes but...

Makes sense, as I have no idea what the question is.

Hm... Let's see what the article tells us about it:

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

Ah. Poincaré understood to ask a simple question like "what is six multiplied by seven" in such a profoundly stupid way that it puzzled the world ever since if and why the answer was 42...

$1 million USD?

From the article:

A reclusive Russian may have solved one of the world's toughest mathematics problems and stands to win $1 million (560 million pounds) -- but he doesn't appear to care.

Heh. Last I checked, $1 million dollars was not quite equal to 560 million (British) pounds. (560 thousand, sure ...)

In an article on mathematics. Of all things.

Re:$1 million USD?

That's a British million. A million is only 10^3 over there.

The Whocares conjecture

Whocarés Conjecture If we stretch a g-string around the surface of somebody's buttocks, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same g-string has somehow been stretched in the appropriate direction around someone's face, then there is no way of shrinking it to a point without breaking either the g-string or suffocating the person. We say the surface of the buttocks are "simply connected," but that the surface of the person's face is not. Whocares knew almost hundred years ago, knew that a well shaped pair of cheeks is essentially characterized by this property of simple connectivity, and asked the corresponding question for the rest fo the people still reading this, as to why they were doing so. This question turned out to be extraordinarily difficult, and slashdotters have been struggling with it ever since.

Riemann hypothesis reportadly also solved

According to the Guardian [guardian.co.uk] another clever Maths dude has proposed a solution to another of the 7 "million dollar" problems.

This particular problem has big implications for online cryptography as it deals with the distribution of prime numbers. Apparantly.

(I'm no mathematics person BTW.)

Re:Riemann hypothesis reportadly also solved

That's a great link, with a wonderful human-readable summary of the 7 problems.

For those too lazy to click:

Seven baffling pillars of wisdom

1 Birch and Swinnerton-Dyer conjecture Euclid geometry for the 21st century, involving things called abelian points and zeta functions and both finite and infinite answers to algebraic equations

2 Poincaré conjecture The surface of an apple is simply connected. But the surface of a doughnut is not. How do you start from the idea of simple connectivity and then characterise space in three dimensions?

3 Navier-Stokes equation The answers to wave and breeze turbulence lie somewhere in the solutions to these equations

4 P vs NP problem Some problems are just too big: you can quickly check if an answer is right, but it might take the lifetime of a universe to solve it from scratch. Can you prove which questions are truly hard, which not?

5 Riemann hypothesis Involving zeta functions, and an assertion that all "interesting" solutions to an equation lie on a straight line. It seems to be true for the first 1,500 million solutions, but does that mean it is true for them all?

6 Hodge conjecture At the frontier of algebra and geometry, involving the technical problems of building shapes by "gluing" geometric blocks together

7 Yang-Mills and Mass gap A problem that involves quantum mechanics and elementary particles. Physicists know it, computers have simulated it but nobody has found a theory to explain it

The Millenium Problems

Since a great deal of discussion and awe comes up anytime one of the millenium problems is mentioned (solved?) on Slashdot, I'd just like to say that any layman interested in learning more about the millenium problems should run to his/her library/bookstore and pick up The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time [amazon.com]. Although, perhaps, for the layman, the end may become a bit tricky (the problems are explained simply in order of increasing difficulty), it's a book worth sticking with, and ultimately worth a read.

- sm

Just like Linux configuration forums

But there's a snag. He has simply posted his results on the Internet and left his peers to work out for themselves whether he is right -- something they are still struggling to do.

Okay, so tell me how this is any different from every l33t user that tells me how to get my dual flat panel setup working under Xandros without editing the X files manually? Sounds like these kids just tried their hands at mathematics, too.

Mr. President...

...we must not have a poincare conjecture gap!

Hopefully he has better luck than de Branges

A few months ago Louis de Branges published his proof of the Riemann Hypothesis [purdue.edu] on the internet. This is also a Millennium problem. Apparently, no mathematician has read it [lrb.co.uk].

It is not that de Branges is unqualified: he settled Bieberbach's Conjecture [wolfram.com]. Interestingly, much of the validation of de Branges work on Bieberbach's Conjecture was done by a team at the Steklov Institute, referred to in the MathWorld link in the article.

Re:Hopefully he has better luck than de Branges

I go to Purdue, and de Branges is unable to explain himself at all. He has attempted to explain his process to other professors at a seminar here, and has only confused them. He also kicked first year grad students out of his seminar, stating they were to inexperienced. From these grad students, I have learned that he is pretty much and hotshot and an asshole. I'm thinking about going to his seminar on wednesday just to see how long it takes him to kick me out. (I'm a first year undergraduate). A note about his proof of the Bieberbach Conjecture. While de Branges did prove the conjecture, he overcomplicated it, as he does many things, and everybody and their thesis advisor has simplified his proof in some way. Mathworld really discredits his "proof" for one, it contains no proof, and his method was proven flawed by counterexample in 1998.

abstract- not complicated at all!

This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print; the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.

One thing he overlooked...

Solving a Millenium Problem carries a reward of $1M, but apparently Perelman isn't interested...

He does realize that's as good as *money*, right???

Re:One thing he overlooked...

He was working on "A special theory on winning a million dollars with math". Being a real mathematician since he has proven to himself he can get the reward, he is satisfied.

Just like the joke about the mathematician who woke up and discovered a fire in his room. After working out exactly how much water to use and what direction to throw it, he said "There is a solution" and went back to sleep (without putting out the fire - that's a job for the physics/engineering folks).

Wake me up when it's peer reviewed and accepted

I'm tired of seeing these 'please make me famous even though I didn't really prove it' threads. The little boy has cried wolf too many times. We don't care unless it's really solved.

Editors, I'm talking to you.

Racist title

I can't believe slashdot would run a story with that title. "Perelman May Have Solved Poincare Conjecture" would have been much more dignified. You would never see "Muppet May Have Solved Poincare Conjecture" would you? Please, Perelman is a mathematician first, Russian second.

Paincare conjecture

Wow! Someone finally solved the paincare conjecture... wait, didn't morphine do that? and the Christian Scientists [wikipedia.org]?

A Christian Scientist from Theale
Said, "Though I know that pain isn't real,
When I sit on a pin
And it punctures my skin
I dislike what I fancy I feel".

Oh! It's poincare... forget it...

tr/Russian/Grigori Perelman/ ..?

Why doesn't this article's title read:

"Grigori Perelman May Have Solved Poincare Conjecture"

I've noticed that these kinds of announcements often make a point of appending a nationality to the name of the person involved in the discovery. Surely this proof builds on mathematical knowledge from around the world. Or was Grigori Perelman standing solely on the shoulders of "fellow Russian" mathematicians? I highly doubt it...

The "reclusive nut's" picture

http://www.wordiq.com/definition/Image:Perelman.jp g

The guy is a bit scary..

Interesting View

This is all very interesting and I like the way Perelman has gone about working out this whole genius and fame, and money. I wonder what if movie stars ever found out or the RIAA or the music industry, they might license him. Interestingly there was also a breakthrough in the Riemann Hypothesis, I wonder if anyone has ever heard of Louis de Branges de Bourcia at Purdue and his paper on the Riemann Hypothesis [purdue.edu]. The person who posted the news article did not tell use what Poincaré Conjecture is? Well this is slashdot not, mathdot :) { Just Kidding Dawgs, aite } . Anyway Perelman has a very ascetic way about him, maybe he sees beyond the materialsitic, and media oriented consuermism. Anyway interesting it is to see someone who sees beyond himself. Just because google news bot picked this up don't make it that great of a post. It was known for the last 6 months that Perelman and colleagues had been working on this. PS ::- buying != happiness Saw this at NYC Penn Station {not a good sign}

Perelman and the prize

Perelman was unemployed for 10 years while he worked on the problem. His last job was in the States in the early 90s, where he saved enough money to live in Russia for the whole time he worked.

So think about his perspective: he's a complete loner who was ignored by the mathematical community for 10 years! Now that he's going to be a "certified" genius (with the $1M prize) why exactly should he care.

Also, it's worth pointing out that like Wiles (who solved the Fermat Conjecture), Perelman's work develops a theory that has the Poincare conjecture as a corollary which is interesting but not of central importance.

Re:Perelman and the prize

Perelman was unemployed for 10 years while he worked on the problem. His last job was in the States in the early 90s, where he saved enough money to live in Russia for the whole time he worked.

What I find particularly interesting is that this guy was able to devote 10 years of his life to solving a problem so complex that there was no intermediate output. The same happened to Wiles, who took 7 years to get hold properly of the Fermat theorem.

Obviously, in both cases it would have been impossible to reach such great results if the authors had had to keep a steady pace of lesser publications. But this is the rule in the academic world: "publish or perish". You must prove yourself "productive" year by year, otherwise you're out.

I've always thought that applying industrial methods of prouctivity measurement to research is utter madness (I am an academic myself). IMO, Perelman's and Wiles' cases show it clearly.

Poincare Conjecture link sucks!

Can someone explain this better? That link to the conjecture is plain awful.

Here are my questions (in parens):

If we stretch a rubber band around the surface of an apple, then we can shrink (huh? what do they mean by "shrink" here?) it (rubber band or apple?) down to a point by moving (huh? again, what does "move" mean here) it (rubber band or apple) slowly, without tearing it and without allowing it to leave the surface (okay, must mean rubber band doesn't leave the surface of the apple then?). On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction (what direction would that be?) around a doughnut, then there is no way of shrinking it (rubber band again I assume) to a point without breaking either the rubber band or the doughnut. (why? the writer made a big leap here, but it's not obvious)

Re:Poincare Conjecture link sucks!

It's very easy. A rubber band around a sphere can slide along the surface so that the circle it forms becomes smaller and smaller, until it converges into a point. But if a rubber band is wrapped around a torus (doughnut) like a link in a chain (so that it goes through the hole in the doughnut), you can't slide it along the surface to make it any smaller than the cross-section of the torus nor can you detach it without cutting the band or the pastry.

The Poincare Conjecture involves hypothetical 4-dimensional shapes with the same properties, and isn't very easy.

Any relation to Yu Perelman

Does anyone remember the book "Mathematics can be fun" maybe published some 40 years ago
which made learning mathematics as a kid absolutely wonderful ? Wonder if Grigori Perelman
is of any relation to the author of that book Yu Perelman ?

So he can solve it...

I want to meet the guy (or gal) who came up with a question worth $1 million!!!!

Fritz
__________

Turning down the money is nothing special

I don't think that there's anything inherently honorable or dishonorable about taking the money. If he wants to take the money and blow it on hookers and Ferraris, that's just as honorable as getting satisfaction because your brain gives you some sweet endorphins because you think you've made the honorable statement that "I'm not about the money, I'm about the math".

Time

The main problem with all of these solutions especially in math is that time is the largest factor in determining if the solution is correct. Give you 2 years and its marginally okay. Give you 40 and its accepted as a standard etc...

Russian? Brit?

http://www.newscientist.com/news/news.jsp?id=ns999 92143

So did the British man or the Russian solve it? April 02 newscientist has the same basic story with the names changed.

Russian may have proved Poincare Conjecture

In Soviet Russia...

they prove conjectures.

If he doesn't care about the money

... then maybe he should consider donating it to the town of Beslan. I'm sure they could use the help.

Good!

Now that's what I call altruism.

Good article here

For an accessible math article on this, try http://mathworld.wolfram.com/news/2003-04-15/poinc are/

isn't interested?

what a twat.

i mean really, if you don't want the money, then take it and give it to charity .

In Russia

Poincare Conjectures solve YOU. SCNR

Two yars to prove this.

According to this [wikipedia.org] article he published the proof in 2002 and as of August 2004 was still being checked.

Seems to be not that easy stuff, if it takes a world full of Mathematicians two years to check a single proof...

This proof has been around for a while

http://www.findarticles.com/p/articles/mi_m1200/is _17_163/ai_101339488
The paper link is inside.

My Solution to Number 5

I've solved it:

5 Riemann hypothesis Involving zeta functions, and an assertion that all "interesting" solutions to an equation lie on a straight line. It seems to be true for the first 1,500 million solutions, but does that mean it is true for them all?

Answer: NO it doesn't mean it's true for all of them. You would have to prove that.

Where do I get my money?

Old news?

Extracted from: http://www.nacho.unicauca.edu.co/Maticias/0309ConP oi/0309ConPoi.htm

Translated for better reading (I use to speak spanish):

Robinson, S., Russian reports he has solved a celebrated math problem, New York Times, 15 April 2003, F3

In November, 2002 appeared a rumor on the Internet saying that the mathematician Russian Grigory Perelman, from the Institute Steklov of the Russian Academy of Sciences in Saint Petersburg, had published in arXiv a preprint presenting a proof of Poincaré's Conjecture.

Trivial to prove / the description must be wrong.

Uhm maybe that link describing the Ponticare conjecture described it incompletely, because the question as described is trivial to prove. I can see it geometrically.

Cut a 4 Sphere with a plane right down the center.

The cross section is a 3 sphere. Consider that section to be the section wrapped with your 3 sphere "rubber band".

Now move a short distance perpenducular to the this slice and take another slice. It will be a smaller sphere. You've just slide your "rubber band" down the apple a bit.

If you keep doing this the 3 sphere slices get small and smaller, converging to a point.

Viola, it's simply connected.

sheesh

Somebody needs to slap this guy and scream "TAKE THE MILLION BUCKS!!!!"

I cannot stand people who rub a lamp, get a genie, and then can't think of anything they want.*

It's like buying the last orange cream soda in the Gobi Desert Gift Shop, deciding you don't want it, and pouring it out.

If you think the money can be put to better uses, well then DO THAT.

* I'm not implying that this was luck. This behavior is worse than when it's luck and you're unprepared.

Book title?

At first glance I thought that a Russian had finally managed to read one of those holiday books and make sense of it... you know, those with titles like "The Bourne Identity", "The Omega Sanction"...

Millennium, not millenium

Just remember, it's from the Latin "annus", not "anus".

Why aren't proofs verifiable via software?

One thing I don't get is why isn't there some software out there to verify the proofs? I mean math follows rules and these rules should be convertable into a piece of software, shouldn't they? So why do I always read that somebody might have proofed this and that, yet nobody has yet verified it and often there are even just a few people with enough knowledge to verify the proof at all so it takes quite some time until a proof get verified.

I am not talking about having a computer generate the proof itself, which can be difficult of course, I am just talking about verifing a given proof.

Millenium Problems ?

Solving a Millenium Problem carries a reward of $1M

Is one of them spelling millennium correctly ?

some terminology

I'll try and explain what the Conjecture is, because it's not entirely obvious. First of all, I need to explain what the 3-sphere is.

The n-sphere (which mathematicians generally denote by S^n) can be thought of as `all points in (n+1)-dimensional space which are at unit distance from the origin'. So S^2 is the surface of a solid 3-dimensional ball. This sometimes surprises people, who expect this to be S^3 but the key observation here is that the 2 refers to the intrinsic dimension of the object, rather than the extrinsic dimension of any space you might happen to put (`embed') the object in. The fact that we often think of the 2-sphere as being embedded in 3-dimensional space doesn't change the fact that it's inherently a 2-dimensional object. An ant wandering around on it still only has two degrees of freedom.

The 3-sphere (S^3) locally looks like ordinary, flat, Euclidean 3-space, but on a larger scale it kind of doubles back on itself - if you keep walking (or floating) in a `straight line' (well, actually the 3-dimensional analogue of a `great circle', but never mind) in any direction, then you'll eventually get back to where you started.

The Poincaré Conjecture says

Any homotopy 3-sphere is homeomorphic to the 3-sphere

This, by itself, isn't particularly enlightening to the non-topologist, but what it actually boils down to is:

Any closed, compact, simply-connected 3-manifold is homeomorphic to the 3-sphere

What does this mean?

Well, an `n-manifold' is a space which locally looks like ordinary, flat, Euclidean n-dimensional space. So a 3-manifold is a space (like S^3) which locally looks like ordinary 3-space (but which might twist back on itself in a peculiar way on a larger scale).
`Closed' means that the 3-manifold doesn't have a boundary - no matter how far you walk, you're not going to run into a brick wall, or fall off the end. `Compact' is a bit more technical, but in this context essentially means you don't get odd shooting-off-to-infinity stuff you have to deal with.

And `simply-connected' means that the first homotopy group (the `fundamental group' of the space) is trivial. What that means is that any closed loop (of string, if you like), in the manifold, can be continuously shrunk down to a point. Here `continuous' means that you're not allowed to cut or glue the string while you're doing it.

To use a 2-dimensional analogy, the 2-sphere (the surface of the 3-dimensional ball, remember, or alternatively a British doughnut) is simply-connected, because given any closed loop in the surface, you can shrink it down to a point without it getting snagged on anything. Whereas the 2-torus (the surface of an American doughnut) isn't, because you can't shrink all closed loops down to a point - one which goes all the way round the central hole, for example, can't be shrunk.

Finally, `homeomorphic' is basically a technical word for `topologically equivalent' - we allow continuous deformations (stretching, twisting, etc, but not cutting or pasting), rotations, reflections, or any combination of these.

So, the (classical) Poincaré Conjecture is essentially a technical way of saying ``If it looks like a 3-sphere then, basically, it is''. (For certain definitions of `is', and `looks like'.)

The analogous conjecture in n-dimensional space is known to be true for n=1 (trivial), 2 (pretty simple), and 5 and above (the 5-dimensional case was proved by Zeeman, who is my PhD grandsupervisor - my supervisor was one of his students). The 4-dimensional case is weird, and there are three different forms to consider - the `piecewise linear' and `topological' cases have been proved, but the `smooth' case is still unproven.

As I understand it, what Perelman claims to have done is prove Thurston's Geometrisation Conjecture, which implies the Poincaré Conjecture as a special case - rather lik

consequences?

Does this have any real world consequences? Like I know if P=NP and P ain't that bad, there could be dogs and cats living together, mass hysteria, giant marshmallow men in new york, etc... What about this?

Ironic

That an article reporting on a mathematical problem being solved
equates 1 million dollars = 560 Million pounds!

http://story.news.yahoo.com/news?tmpl=story&cid= 85 7&ncid=757&e=10&u=/nm/20040906/od_uk_nm/oukoe_scie nce_maths

Riemann hypothesis

this may also be of interest, it appears that
another one of the so called "millennium problems",
may have just been solved, that is Riemann
Hypothesis:-
http://www.vnunet.com/news/ 1157891
we are all lucky to live in such exciting times.

Bah!

Call me arrogant, call me a troll ..

If he truly had a brain in his head, he would accept the million and donate it to some organization/charity that needs it!

wow

ok (trying) to read and understand the millenium problems makes me feel dumb. but that aside what practical uses do any of these solutions or problems have?

Re:Duplicate?

RTFA. He published another paper on it recently.

Re:Duplicate?

Think nothing is impossible? Try slamming a revolving door.

Place a 2 by 4 on the floor in the door.
Slam the revolving door.

Another impossible problem solved.

Re:If he doesn't want the money

They should give it to me so I can buy a 200,000,000 page Slashdot subscription.

Re:Duplicate?

According to Wikipedia [wikipedia.org], his proof of this surfaced around 2002 and he was lecturing on it in 2003. I guess it's not new news per se, but a Millennium prize problem is a big deal no matter how you look at it.

Re:Problems with the Millenium Problems

You're an idiot. The Poincare Conjecture has direct application to streching rubber bands around apples.

I'm joking, but you're still an idiot.

Re:Well, at least the Russkies are good at math...

Who would have thought that those that worship islam would kill kids by shooting them in the back and laugh at them as the tortured them.

After all islam is the Religion of Peace (TM)

Re:Wake me...

Wake me when someone verifies his work. I can claim to solve anything, but it doesn't mean much unless the community says I'm right. Right off the bat it seems fishy: no journal submission, just a web post? No referee? And he's not answering questions about his work? He's either a genius or a nutcase, possibly both.

The claim has been around for a while. From the referenced MathWorld article:

Almost exactly a year later, Perelman's results appear to be much more robust. While it will be months before mathematicians can digest and verify the details of the proof, mathematicians familiar with Perelman's work describe it as well thought out and expect that it will prove difficult to locate any significant mistakes.

That was in April 2003. It's now over a year later again and it hasn't been disproven.

Re:Problems with the Millenium Problems

If I'm not mistaken, the N-body problem was shown to have no solution in elementary functions like 100 years ago, or something like that. Possibly by someone really famous (as far as mathematicians go). He won a prize from a Danish king by doing so, as the king had made a contest to see who could solve it.

There are a number of reasons why these problems should have prize money attached to them without direct practical applications that are curreently known. First, their results are important from a purely mathematical standpoint. Second, the techniques that must be invented to solve these problems are important in their own right. Third, the mathematical problems that we can't find uses for now could very easily have applications in 100 years. Number theory is being applied to computers, group theory has practical uses now, and I'm sure that many other brahcnes of math have found applications after long periods of having little to no practical value.

What the fuck is up with the world and its instant gratification, "but what does it get me NOW" attitude these days? Oh, if only the 80s had never come....

Re:The Poincare Conjecture?

You're thinking of the Bourne Again Christian Conjecture.

Re:Confused

You are forgetting that this is about an Apple, so obviously there's a Reality Distortion Field at play somewhere. Anything is possible. Including 2D fruit, 4D donuts, and G5 sunflowers. Oh wait, the last one isn't.

I wonder if this Russian fella used RDF as a factor in his equations?!

Re:Confused

A better analogy would be to continuously move a circle on the surface until it becomes a point. In the case of a donut, you could draw the circle through the middle hole and around again, so you can't "shrink it to a point" my continuously moving it anywhere; it goes around the donut anywhere you put it. With a sphere, though, you can continuously move the circle to a "pole," where it becomes a point. This property is called simple connectivity.

It's pretty easy to see that all simply connected 2-manifolds (in 3 dimensions, at least) are homeomorphic to the shell of a sphere, i.e. they may be stretched and contorted to look like it. The question answered here is whether the same is true in the next dimension.

Re:Confused

Mods:

You suck, this is not Flamebait. Someone mod me as flamebait, and mod this something Positive.. beacause I guarantee 95+% of slashdot does not understand the description.. As someone else earlier said, the page linked makes a HUGE leap, and doesn't take anyone with it, in the explanation of the conjecture.

Re:Problems with the Millenium Problems

I understand that it's supposed to generate "good mathematics" that will supposedly help us solve practical problems eventually .. but why not offer the reward for actual practical mathematics

Because practical mathematics already has a monetary reward?

Re:PoincarÉ

"Is it that difficult to respect international spellings?"

Yes.

Re:Problems with the Millenium Problems

You're right because you can't see anything practical coming from it? *That* makes you right? Someone needs to take some logic courses.