## Chinese Mathematicians Prove Poincare Conjecture 288 288

Joe Lau writes to mention a story running on the Xinhua News Agency site, reporting a proof for the Poincare Conjecture in an upcoming edition of the Asian Journal of Mathematics. From the article:

*"A Columbia professor Richard Hamilton and a Russian mathematician Grigori Perelman have laid foundation on the latest endeavors made by the two Chinese. Prof. Hamilton completed the majority of the program and the geometrization conjecture. Yang, member of the Chinese Academy of Sciences, said in an interview with Xinhua, 'All the American, Russian and Chinese mathematicians have made indispensable contribution to the complete proof.'"*
## It's all a conjecture (Score:5, Funny)

Homeomorphic. Thank god, they dumb it down a bit later:

More colloquially, it's

homotopy-equivalent to the n-sphere!Of course!Slow news day?

## Re:It's all a conjecture (Score:5, Informative)

Slow news day?this is actually quite a discovery; it's one of these things which has been hanging around for over a hundred years and it's good to finally have a proof... it's a little like proving P=NP... but a little less grand

## Re:It's all a conjecture (Score:5, Insightful)

## Re:It's all a conjecture (Score:4, Insightful)

## Re:It's all a conjecture (Score:4, Insightful)

singleNP-complete problem, and you can transform that into a polynomial time forallNP-complete problems.True, that if there was a non-constructive proof that P==NP, it might not be obvious what the polynomial time algorithm actually is. But since such a scenario would be probably the most astounding open problem in the history of mathematics, I don't think it would be an open problem for long ;)

## Re:It's all a conjecture (Score:2)

again, when this is my first post in this thread. I said:That such a polynomial time algorithm already exists, doesn't make my statement wrong. The algorithm is indeed non-obvious.

## Re:It's all a conjecture (Score:3, Funny)

Proving P=NP would cause doors to the Cthulhu dimention opened, as was shown by Charlse Stross [wikipedia.org] in The Atrocity Archives [wikipedia.org]

## Re:It's all a conjecture (Score:2, Interesting)

## Re:It's all a conjecture (Score:4, Funny)

## Re:It's all a conjecture (Score:2)

You might think that this is useless to you. However simply memorize those quotes and you can be prepared for any situation.My Boss replies with a frown and an "Are you bullshitting me?" to anything he can't understand.

The consequences of trying to pull a fast one like this on the missus are too gruesome to detail here.

## Re:It's all a conjecture (Score:2)

Well, at least

that's easy. You just say 'Of course not, Boss... how on Earth did you get that idea?' - with a perfectly straight face, of course.It's not that hard; look at all the moron politicians doing it on a daily basis, and professionally, too.

## A translation... (Score:5, Informative)

A 3-manifold is another four dimensional object - in fact, a class of objects. They are the analogies of surfaces in 3D space, only again we have it in 4D space. The 3-sphere, for example is an example of a 3-manifold. Simple, connected and closed are two topological properties describing what a surface is like. In layman's terms, simple connected and close means that the surface is well... just an obvious surface. The simple-connected-closed-3-manifold taken together essentially rule out the bizzare sorts of objects that mathematicians come up with. There won't be any 'holes' in the object, and there won't be any non-solid boundaries, the object can't go through itself, and you can't take two seperate objects and pretend the pair is a single one.

So what does the conjecture say? It says that if we have any 3-manifold satisfying certain properties, there is way of distorting it (that's basically what homeomorphism means. Like you take the object as a piece of putty and stretch and pull it, or fold it, or whatever without cutting or gluing bits together) to make it into a 3-sphere.

It's a sort of bubblegum theorem. You can chew up the manifold and blow it into a bubble. (Okay, it's not really like that, topologists.... But it's close enough)

## Re: They are the kind of people ... (Score:4, Funny)

knot.## Re:A translation... (Score:4, Funny)

## Re:A translation... (Score:5, Informative)

A closed surface is one that does not allow points to go off to infinity (technical term: compact) and has no boundary. So for instance all of three-space is a nice simply connected 3-dimensional manifold, but it is not closed because points can run off to infinity. It also doesn't have a boundary. How could something without a boundary keep points from moving to infinity? Well, consider the 2-sphere, torus, or 3-sphere. No points in these spaces can shoot to infinity, but yet they don't have a boundary (a boundary point is a point where the surface abruptly stops). Closed manifolds somehow have to fold back on themselves.

Not quite: in additions to stretching and pulling, a homeomorphism also allows cutting and gluing, as long as you first cut, then move and stretch, and thenglue together in exactly the same way that you cut earlier. So for example, take a little cylinder made from paper (without its top and bottom, just the side). Now cut it open along a straight line from top to bottom: if you unwrap it, you'll have a rectangle. Now create a double twist in that rectangle and glue it together along the same line again. The result is a terribly twisted "cylinder", and it is homeomorphic to the cylinder you started out with. (Had you made only a single twist rather than a double twist, then you wouldn't have glued points together that were earlier cut apart, and the result wouldn't have been homeomorphic to the cylinder--it would have been a Moebius strip.)## Re:A translation... (Score:5, Informative)

so you should be able to distort a shape in 4D to a 4D sphere! and it looks like it ought to apply to any number of dimensions as well - was proof required?There are some things which "seem" obvious to us which aren't necessarily so. In math classes that discuss Cantor's theorem [wikipedia.org], there are always a few holdouts that refuse to believe that one infinite set can be bigger than another infinite set. After all, they're both infinite. How could one be bigger than the other? And yet it's true, and Cantor demonstrated it in a way that's so cool that you can literally explain it on the back of a napkin.

Likewise, there are certain things that are accepted as a given, until someone discovers/proves something that causes the known world to fall around your ears, mathematically speaking. Kurt Godel pulled the rug out from a whole slew of logicians by demonstrating that not everything that's true can be proven [wikipedia.org]. Up until that time, the "completeness" of mathematics had been considered a given by some people.

So yeah - on a naive level, it may seem like "making things all bendy" is obvious, but that doesn't mean it wasn't in need of a proof.

## Re:A translation... (Score:2)

## Re:A translation... (Score:4, Funny)

## This is... (Score:5, Interesting)

## Re:This is... (Score:5, Funny)

One down, seven more to go!Given that there are seven questions total, maybe you know the mystery surrounding the elusive eighth question: "What is seven minus one?"

## Re:This is... (Score:4, Funny)

Just kidding, of course.

## Re:This is... (Score:3, Funny)

Given that there are seven questions total, maybe you know the mystery surrounding the elusive eighth question: "What is seven minus one?"Forty-two?

## Re:This is... (Score:2)

## Ok, in plain english (Score:3, Interesting)

Bonus points if you can explain some consequences of it being proven true.

## Re:Ok, in plain english (Score:5, Funny)

If poincare conjecture = proved , my homepage switches to harsh new look. QED.

## plain english? Maybe... (Score:2)

In a nutshell, and assuming I've understood it, if you just consider a normal sphere, then it has a 2D surface. That surface is "simply connected" which appears to mean that if you take any two points on the surface and join them, then you can (smoothly) transform that joining "curve" into any of the other possible joins between those chosen points. Basically, there are no holes.

If you then go

## Re:plain english? Maybe... (Score:2, Insightful)

## Re:Ok, in plain english (Score:5, Informative)

## Re:Ok, in plain english (Score:5, Funny)

In topology spheres are identical to cubes and pyramids.A topologist is someone who doesn't know whether to dip their doughnut into their coffee mug, or vice versa...

## Re:Ok, in plain english (Score:3, Insightful)

## Re:Ok, in plain english (Score:3, Informative)

## Re:Ok, in plain english (Score:4, Funny)

Seems like whats been proven is that a doughnut != sphere.

## more info (Score:3, Interesting)

## perelman rejected Clay's million (Score:2)

## Chinese == Good at Math (Score:3, Funny)

This is news?

(j/k... I am Chinese).

## Re:Chinese == Good at Math (Score:2, Funny)

*ducks*

## Chinese == Good at Math? Wrong! (Score:2, Insightful)

## Re:Chinese == Good at Math (Score:5, Interesting)

badat Math. In ancient times the Chinese innovated quite competitively but this hasn't been true for a long time. Since I just took issue with your equating mathematical ability with racial characteristics, you can probably guess that there's another reason, and as it happens, I am prepared to qualify my statements.The Chinese school system (and in ancient times, the scholar system, which stratified society into a "scholar class" and the "masses") is completely and utterly innovation stifling. It emphasises testing and memorization above all else, and curiosity and individuality are systematically beaten out of students. No snide comments about communism, please, it has nothing to do with that (any mathematician will tell you that the Soviet Union produced a metric tonne of talented mathematicians, my advisor was one). Chinese students memorize

everything. Because I speak Chinese and love math, I have tutored quite a number of high school and university undergraduate students in math and the simple reason that they suck at it is they basically cannot wrap their head around proofs.Proofs are difficult for most people at first, but you have to understand that the way a typical mainland Chinese kid approaches math is by

memorizingevery formula in his math textbook and then trying as best he can to choose the one that "works" with the problem he is presented. He does not do this because he stupid: he does this because the Chinese standardized testing system reinforces the behaviour. The exam problems are expressly designed so that various formulas are the "keys" to the problem, that is, answering the (usually multiple choice) question correctly relies on your ability to quickly recall one formula (perhaps two) and plug the numbers in effectively. So many problems are presented and so little time is given that no time for derivation or logic is really provided. Because of this, essentially every Chinese kid can recite from memory a whole host of trigonometric identities without having the faintest idea why they work or how to derive them, even when the derivation is relatively simple.Because there's so much anti-Chinese sentiment in the west these days and on Slashdot in particular, I want to reiterate for a moment and say that this is not an inherent failure in the Chinese kids themselves -- they are not stupid -- but they are completely crippled by their education system. From day one they memorize everything. They memorize entire passages written in old Chinese and are asked to reproduced them from memory at exam time -- I've been told by several kids here in Beijing that writing even one character wrong is essentially equivalent to forfeiting the entire problem. These are not 3 line passages folks: we're talking two or three pages of

old Chinese.Imagine being told at 17 to memorize 3 pages of Beowulf. That's what we're talking about.The thing is (as any drama major will tell you) memorization, like all things, gets easier with practice. And from day one (when I first arrived in China I moonlighted as a Kindergarten teacher, so I have some first hand experience here) kids are memorizing stuff, from poems to proverbs to Chinese characters. It becomes easy for them, and over the years they depend on it more and more. The worst part is, high school and lower division level mathematics (if it can be called that) presents problems (like doing integrals or calculating derivatives) that lend themselves well to the "memorize a formula" method. And so Chinese kids tend to do exceptionally well in these courses, and then mistakenly assume they are good at math. This is in fact not

## Should share credit with Perelman (Score:5, Informative)

The proof is 300 pages but I would guess the majority of it is an overview of Perelman's extension of Hamiltion's Ricci Flow.

## Re:Should share credit with Perelman (Score:2, Insightful)

## Re:Should share credit with Perelman (Score:3, Informative)

I can't see why Perelman would share the credit. If his results are right, he proved it first.You haven't "proved" something until you have written it down in a form in which it convinces at least other specialists in your field. The fact that nobody knows for certain "if his results are right" is tantamount to the statement that he hasn't proven it yet.

So, I suggest a simple rule: whichever of the two proof attempts will be

verifiedfirst by at least a dozen other mathematicians or by a mechanical device,## Re:Should share credit with Perelman (Score:2, Insightful)

That assertion is simply untrue.

Suppose (and this a deliberately perverse example), Fermat had secretly developed all the machinery for Wiles' proof of his Last Theorem, and gone on to prove it. None of his contemporaries could possibly understand it. But the theorem

would'vebeen proved, even ifno-oneknew it.## Re:Should share credit with Perelman (Score:2)

Thanks for clearing that up!

-l

## Re:Should share credit with Perelman (Score:2)

somemathematicians does not mean it is less of a proof.## not necessarily (Score:5, Insightful)

communication, not merely about making true statements.Perelman apparently failed to do this: he may have produced a sequence of true statements that could somehow form a subsequence of a complete proof, but he has apparently not supplied enough detail to demonstrate his point to even specialists in his area. The fact that he may have done "the heavy lifting" or that he may have provided the key ideas doesn't change that.

I think it is valid to give all three mathematicians equal credit. And, strictly speaking, the people who actually have done the proof are the ones who "dotted the i's" because that's what ultimately constitutes a proof.

## Re:not necessarily (Score:2)

No. The one who proves get credit for the proof. Nobody else.Geez, talk about circular definitions.

His proof was basically non-readable and some Russians showed that his proof was indeed correct.By that reasoning, Poincare proved the Poincare conjecture, because he provided a single-step proof (the statement of the conjecture itself), and any subsequent work only filled in a bunch of missing steps.

In real life, there is some threshold above which a sequence of statements constitutes a proof and below which

## Re:Should share credit with Perelman (Score:2)

A collection of facts is no more a science than a heap of stones is a house.- Poincare

Somehow I think Poincare would appreciate the complete proof.

## Re:Should share credit with Perelman (Score:2)

## Don't jump the gun (Score:2)

## Re:Don't jump the gun (Score:2)

## Math isn't dead (Score:5, Interesting)

I was just having a conversation about this yesterday with my math teacher.

Lots of people think that high level math is just advanced adding and subtracting.

This is good stuff. Props to Zhu Xiping and Cao Huaidong- this shows people that a career in studying mathematics is actually an interesting and rewarding career.

## ... not yet. But it may die soon. (Score:4, Interesting)

## Re:... not yet. But it may die soon. (Score:2)

I don't really think that's true.

On our undergraduate history of science class, one of our texts was Kuhn's book. His arguement was that science doesn't work by continually adding more stuff on top of what is already there but works in paradigms.

So, for example quantum physics was a paradigm change. The photoelectric effect is explained by 7th graders but now higher level quantum mechanics cannot be. Every once is a while someone comes up with a total paradigm change on how problems are supposed to be a

## Re:... not yet. But it may die soon. (Score:2)

The actual proof could be just 1 line if we assumed the Taniyama-Shimura conjecture to be true.You have no idea what "proof" means if you think you can prove something by assuming that one of the premises is true when that premise is neither an axiom or a theorem that has been previously proved.

However, if you're right, I'd like 1 million dollars because I've just proved the Riemann hypothesis. My proof only requires that you assume the truth of the Spear Conjecture, which states that the Riemann Hypoth

## Re:... not yet. But it may die soon. (Score:3, Insightful)

And you apparently have no ability to read what the GP said. Specifically, he suggested that most of Wiles' effort was directed at proving the Taniyama-Shimura conjecture. From that point on, it was a simple step to prove Fermat's Last Theorem (for some extremely esoteric value of 'simple').

Note this line here:

Whether the grandparent poster's assertion about this method is accurate

## Re:... not yet. But it may die soon. (Score:2)

A good mathematical proof is as self-contained as possible; that's probably an utopia, but a proof of Fermat's last theorem which is just that line you mentioned does no good whatsoever to anyone reading it. If a proof is self-contained (or almost), someone who reads it and understands it can then safely say that he/she understands WHY the theorem is true. Otherwise, he may well be relying on someone else's mistakes (

adding more## Re:... not yet. But it may die soon. (Score:4, Informative)

However, today even verifying a proof is very hardWhile that's true of some proofs, it's certainly not true of all of them, or even most of them. Every year, hundreds of mathematics journals collectively publish thousands of new proofs. Some are more difficult to verify than others, but they are all verifiable (or falsifiable in the case of published errors).

the time may be near when no one on earth will be able to handle the complexity of this task anymoreI doubt we'll ever see that happen. Of course as a mathematical field matures, the number of accessible problems will approach zero and we're left with only the very difficult problems. However, new fields arise and give us a host of new problems to explore.

## Re:Math isn't dead (Score:2)

## Re:Math isn't dead (Score:2)

At this point though they're almost right

## Re:Math isn't dead (Score:2)

"Geometry. I was awesome at that, like side-angle-side and crap right?"

## What does it all mean? (Score:3, Funny)

## Here is a conjecture (Score:5, Funny)

## Great. Still waiting for peer review.. (Score:4, Insightful)

## Re:Great. Still waiting for peer review.. (Score:3, Informative)

## Joe Public goes (Score:5, Funny)

I think it makes a good thriller title... "The Poincare Conjecture"

## Re:Joe Public goes (Score:3, Interesting)

There's a whole slew of mathematical theorems, conjectures, hypotheses, et. al. that sound like Robert Ludlum novels [everything2.com]:

## wouldn't trust it yet (Score:2)

In the long run, mathematics really needs considerably more formality than it is using now, as well as mechanical support for the bookkeeping necessary for long and involved proofs. Actually, the tools already exist, it's just that wor

## Re:wouldn't trust it yet (Score:4, Interesting)

Unfortunately, automated proof tools are not sophisticated enough to handle the kind of maths seen in solving the Really Big Problems. Not yet, anyway.

## Re:wouldn't trust it yet (Score:2)

## Re:The proof is due to Perleman (Score:4, Informative)

## Re:The proof is due to Perleman (Score:2)

Still, I'd like to give Cao and Zhu the benefit of the doubt, for now.

## Re:The proof is due to Perleman (Score:2)

## Two very good reasons (Score:5, Interesting)

## Re:Two very good reasons (Score:2)

## Re:Two very good reasons (Score:2, Interesting)

If you think in cartesian terms the the unreal axis is at 90 degrees to the real axis. This maps very neatly to where entities work at 90 degrees to each other, as in electric currents running through a wire in a magnetic field. Additionally, although I can't remember the details as it was thirty five years ago when I studied this, the shape of an aerofoil can be described using

i, and if you do so the maths relating to the airflow becomes much simpler.As

## Re:Two very good reasons (Score:3, Informative)

## Re:Two very good reasons (Score:3, Informative)

## Re:Two very good reasons (Score:2)

## Re:Two very good reasons (Score:2)

## Re:Two very good reasons (Score:2)

THere may be other things I've forgotten, but those are the major applications I believe.

## Re:Two very good reasons (Score:2)

## Re:Two very good reasons (Score:2)

I'm asking what made it front page news.Because it's a well known mathematical problem, it's even one of the seven Millennium Prize Problems!

## Re:Two very good reasons (Score:2)

## Re:WDWC query (Score:3, Insightful)

Not everything worthwile doing needs to result in amazing products.

Apart from this, mathematical insights, sometimes of the more dry and abstract sort *have* already resulted in amazing products (take public key encryption, the application of insights gained from number theory).

## Re:WDWC query (Score:5, Informative)

This leads us to the answer to another pressing problem in mathematics - Why Do We Care?

Really, what does this have to do with how we deal with reality? Will we be buying amazing products that are based on this? Breaking encryption? Making pigs fly?

In the the 18th and 19th century, the foundations were laid for something called finite fields, which had little to no impact on reality then. Fast forward to 1960, when a couple of guys [psu.edu] figured out a way to use finite fields in a way that enables you to still play a scratched cd [wikipedia.org], or ensuring your raid-5 is working properly when a disk fails.

So do you still think the mathematicians back in the 18th and 19th century should have done something else, something with direct applications in their time?

## Re:WDWC query (Score:2)

## With a bit of imagination.... (Score:2)

More generalised versions might also result, which will help us along with string theory and move us closer to cool stuff like this. [everything2.com]

## Re:WDWC query (Score:2)

For humanity, understanding low-dimensional spaces better is important: we apparently live in a 3+1D world, and there may well be fundamental mathematical reasons for that.

## Because: (Score:2)

This leads us to the answer to another pressing problem in mathematics - Why Do We Care?Often, mathematical advances have no use in the time when they are discovered but later prove to be valuable, either inside mathematics or not.

For example, who could foresee that non-euclidean geometries would be used by Einstein in his theories? Einstein's theories are quite useful today (GPS comes to mind). QED

## Re:WDWC query (Score:2)

## Re:Dear God (Score:3, Funny)

## Re:Dear God (Score:2, Troll)

They impose a new design with IMO rather poor usability. They don't provide an article where users can comment.

Comments in other threads are then deemed 'offtopic'.

Nice!

## Re:Dear God (Score:2)

## Re:Dear God (Score:2, Interesting)

Hopefully enough folks complain to get the runner up and a few others added. It's easy to provide the choice.

## Math geeks are in their own L-Space (Score:2, Informative)

## Re:Is there a math geek in the house? (Score:3, Insightful)

r, say, from the origin. Having been so specific about all that, I'm now going to be dreadfully, appalling## Re:name change - Mod parent UP (Score:2)

## Re:What a bunch of crap (Score:2)

I must disagree. I could claim to have solved some very hard problem, and present some gobledy-gook nobody else understood as my "proof". But nobody would give me credit for solving the problem, even if my conclusion eventually turned out to be right. If you're going to get credit for proving something, your proof is going to have to pass some threshold of understandability by others.

I've not read Perelm

## Re:The Cube (Score:2)

I've been staring at a cube for a half-hour now and I can't for the life of me find a hole. Can somebody please explain why my cube has no holes? Does your cube have a hole? Someone please show me the hole!!Wait, that came out all wrong.Actually, if you are talking about a regular box (with 2d surface), it does have a 3-dimensional hole in it.

Look up homotopy and mappings of 3-dim sphere into 2-dim one for more details (sorry, there is no way to post a formula on Slashdot..)

## Re:The Cube (Score:2)