Chinese Mathematicians Prove Poincare Conjecture

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.'"

Comments

Re:Dear God

[Anonymous Coward]

the new design is ok... I would have liked to see some changes around the colour schemes from certain topics etc... SECOND POST!!!!

It's all a conjecture

[The Bungi]

I looked at TFA, and I was kind of lost after reading this:

In its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher.

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

More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by H. Poincaré Eric Weisstein's World of Biography (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n==3.

More colloquially, it's homotopy-equivalent to the n-sphere! Of course!

Slow news day?

Re:It's all a conjecture

[joe 155]

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

[Umbral Blot]

You might think that this is useless to you. However simply memorize those quotes and you can be prepared for any situation. Boss unexpectedly wants a status report? Sure boss, currently my I'm developong a compact n-manifold that is homotopy-equivalent to the n-sphere if it is homeomorphic to the n-sphere. We'll be done in a couple of weeks. Wife bothering you to take out the trash? Sure honey right after I demonstrate that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher. Never be at a loss for words again!

Re:It's all a conjecture

[Anonymous Coward]

I don't understand the MathWorld article, therefore it's the article's fault for talking about complex mathematics that I don't understand.

Yeah, that makes a lot of sense.

Re:It's all a conjecture

[Barraketh]

Technically, it's more like proving P != NP, since that's the current accepted belief. Proving P=NP would be huge - this would give polynomial time algorithms for Travelling Salesman Problem, Boolean Satisiability Problem, and a slew of others (that all reduce to each other in polynomial time). Proving P != NP pretty much confirms what everyone believes to be true, similar to how the Poincaire conjecture was generally accepted to be true. Still, this is a major result, and clearly falls under the "News for nerds, stuff that matters" heading.

Re:It's all a conjecture

[Crayon Kid]

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

[Anonymous Coward]

Proving that P==NP wouldn't automatically give us polynomial time algorithms for any NP problem. The proof need not be constructive, and if it's not, it doesn't give algorithms. Granted, it seems easier to prove that P==NP by accidentally finding a polynomial time algorithm for an NP problem than otherwise, but don't assume that the prove would sove anything practical.

Re:It's all a conjecture

[giuntag]

Darn, I knew I should have never married a math teacher!

Re:It's all a conjecture

[S3D]

echnically, it's more like proving P != NP, since that's the current accepted belief. Proving P=NP would be huge - this would give polynomial time algorithms for Travelling Salesman Problem, Boolean Satisiability Problem, and a slew of others (that all reduce to each other in polynomial time).

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

Re:It's all a conjecture

[Anonymous Coward]

Thank you. This was a great reading after just waking up..

.. have a cup of coffee on me :)

Re:It's all a conjecture

[cp.tar]

My Boss replies with a frown and an "Are you bullshitting me?" to anything he can't understand.

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.

Re:It's all a conjecture

[whalewatcher]

Ha! I'm married to a mathematician and he won't use that old excuse again/i

Re:It's all a conjecture

[Anonymous Coward]

You live alone and are unemployed, right?

Re:It's all a conjecture

[Anonymous Coward]

Does anyone know how this might relate to Calabi-Yau (Kahler) manifolds and its possible affects on quantum field theory? I can't tell because I'm not a mathematician, but it seems this might help with the topology problems plaguing string theory.

Re:It's all a conjecture

[ZenSkin]

Though proving P != NP does not necessarily give any insight to heuristics for NP problems. The fact, in and of itself, has no value in engineering. But it would be a significant proof and highly newsworthy. Since both the P vs NP problem and the Poincare conjecture are pesky and hard problems that have received attention in the popular press, I would imagine, like you, it is worthy of mentioning on slashdot -- indeed if it wasn't mentioned here, slashdot would be suspect.

Re:It's all a conjecture

[IWannaBeAnAC]

That isn't quite true: you only need a polynomial time algorithm for a single NP-complete problem, and you can transform that into a polynomial time for all NP-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

[Anonymous Coward]

Only if you understand it. If you don't, or it isn't clearly explained -why- it's important, then it's just a bunch of garbage.

Re:It's all a conjecture

[Anonymous Coward]

Wrong again! If in fact P=NP, we already know polynomial time algorithms for NP complete problems. For instance (from http://en.wikipedia.org/wiki/Complexity_classes_P_ and_NP)

This is an Algorithm that accepts the NP-complete language SUBSET-SUM which is polynomial time iff P=NP.

Input: S = a finite set of integers
Output: "YES" if any subset of S adds up to 0. Otherwise, it runs forever with no output.

FOR N = 1...infinity
      FOR P = 1...N
              Run program number P for N steps with input S
              IF the program outputs a list of distinct integers
                    AND the integers are all in S
                    AND the integers sum to 0

              THEN
                    OUTPUT "YES" and HALT

Re:It's all a conjecture

[mOdQuArK!]

Dunno 'bout that, the "moron" politicians that I'm used to seeing practice reducing an incredibly complex situation down to a couple of sound bites, since that's the only footage that the media will report.

The "scary" moron politicians are the ones who believe that those situations really _are_ as simple as those sound bites.

The part about keeping a perfectly straight face while lying sounds perfectly accurate though.

Re:It's all a conjecture

[IWannaBeAnAC]

Hmm, I can hardly be wrong again, when this is my first post in this thread. I said:

True, that if there was a non-constructive proof that P==NP, it might not be obvious what the polynomial time algorithm actually is.

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

[grammar fascist]

Wife bothering you to take out the trash? Sure honey right after I demonstrate that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher. Never be at a loss for words again!

Alternatively, you could go to grad school. You'd have reservoirs upon reservoirs of useless phrases at your disposal.

"Sorry, honey, I'm currently working on a new correlation algorithm for stereo matching and subimage finding based on an inner-product L-k norm. I'll be done in a few minutes."

Re:It's all a conjecture

[steve_bryan]

No, in your effort to be flippant you've messed it up. The statement is that if the manifold is homotopy equivalent to a 3-sphere then it must also be homeomorphic to a 3-sphere. Note that the announcement was not about a proof for n-spheres but specifically for 3-spheres. The reason is that the corresponding proof for n > 3 has been known for decades. Smale proved it for n >= 5 and Freedman proved the result for n = 4.

Re:It's all a conjecture

[Anonymous Coward]

Did you copy that down, cause the statement there every "compact n-manifold is homotopy-equivalent to the n-sphere if it is homeomorphic to the n-sphere" is trivial. Its the opposite implication that is the conjecture. Anyway, this result is due to Perleman. This is just Chinese nationalism.

Re:It's all a conjecture

[Anonymous Coward]

> That such a polynomial time algorithm already exists, doesn't make my statement wrong. The algorithm is indeed non-obvious.

I will agree with you (myself?), in that I find the algorithm non-obvious. But can you imagine someone (in your world of acquaintances) who might find the algorithm obvious?

How about for "true" or "false," "I agree" or "I disagree." Likewise for "good" and "bad," "I like" and "I don't like."

thanks for reading

Re:It's all a conjecture

[IWannaBeAnAC]

I will agree with you (myself?), in that I find the algorithm non-obvious. But can you imagine someone (in your world of acquaintances) who might find the algorithm obvious?

Certainly. Practically everything is 'obvious', once you have studied it sufficiently. Even that algorithm looks quite straightforward, now: the trick is that the algorithm loops over all possible programs, and runs each program for a while to see if it actually solves the problem. But I must admit the proof is not obvious (yet!), I don't see why, even if P=NP, the program number that solves it is itself a polynomial of the input size (which is a distinct issue as to the number of steps of the program you need to run to get an answer). Is this property special to SUBSET_SUM ?

Anyway, as far as programs go, this is about as non-constructive as you can get with it still being a program ;-) Even if P=NP, this doesn't look like a good approach to get practical algorithms.

Re:It's all a conjecture

[Anonymous Coward]

The program number is a constant, it doesn't depend on the input at all. Of course the program number depends on the problem being solved, but if P=NP and the problem is in NP, then there is a constant-size program that solves the problem in polynomial time.

Re:It's all a conjecture

[IWannaBeAnAC]

Not in that algorithm; it loops over the program number P = 1..N. That is the whole point, if P=NP there is a program that solves it, but we don't know what that program is.

Re:It's all a conjecture

[bigg_nate]

Actually, if you can prove P=NP, you can construct a poly-time algorithm to solve any problem in NP -- see here for details. You're right that the proof wouldn't necessarily give a practical algorithm, but it would give an algorithm in P.

Re:It's all a conjecture

[bigg_nate]

every compact n-manifold is homotopy-equivalent to the n-sphere if it is homeomorphic to the n-sphere

I'm pretty sure this is backwards -- it should read "every compact n-manifold is homeomorphic to the n-sphere if it is homotopy-equivalent to the n-sphere." That's probably why you were confused ;)

Re:It's all a conjecture

[andy_shepard]

It only has to be polynomial time asymptotically, for sufficiently large N, so for sufficiently large N, N will be greater than the appropriate P, and the running time will be dominated by the time to run that particular program rather than the time to find it.

Re:It's all a conjecture

[Anonymous Coward]

how is it less grand? it is a much older problem with roots in a very old field of mathematics, and has inspired developments in nearly every other area of math and many fields of science.

Re:It's all a conjecture

[IWannaBeAnAC]

To clarify: to be a polynomial time algorithm, it must terminate at some N which is a polynomial of the input size (say, the total number of bits used to specify the input numbers S). The program that solves it is represented by some number P <= N, which implies P <= p(input_size), for some polynomial function p. The problem I have is that the number representing the turing machine P, is exponentially large in the length of the turing machine. Or conversely, as the input size gets large, the turning machine to solve the problem can increase in length only logarithmically. This is reasonable I guess (it may even be that the length of the program is bounded), but it still needs proof. And I don't see what is special about SUBSET_SUM here.

Re:It's all a conjecture

[andy_shepard]

There isn't anything special about subset sum. It doesn't matter how p varies as a function of the problem, because it's always the same problem. p is just a constant. The Turing machine needed is always exactly the same one regardless of the input size.

Re:It's all a conjecture

[cgibbard]

Two spaces X and Y are homeomorphic when there's a continuous function f: X -> Y which has a continuous inverse. Basically, spaces are homeomorphic when you can bend one into the other without gluing things together or cutting them apart.

It's saying that if you have a 3-manifold which is compact and simply connected, then it's possible to bend it nicely into the 3-sphere.

Saying that it's a 3-manifold means that at every point in the space, it sort of locally looks like the usual Euclidean 3-space. A reasonable analogy is the surface of the Earth is a sort of 2-manifold, since it locally looks like a plane -- enough so that it took us a long time to discover that it's actually connected up like a sphere.

With the 3-sphere, if you, as a 3-dimensional object, were floating inside it, it would be rather like being in ordinary 3-dimensional space that you're used to, but the space would be bent in such a way so if you travelled in any one direction for long enough, you'd end up back where you started. (Kind of like what happens in 2-dimensions while travelling on the surface of the Earth.) Rather than being on the surface of a 3-dimensional ball, you'd be *in* the surface of a 4-dimensional one.

Now, the words "compact" and "simply connected"...

"Compact", you basically might think of that geometrically as meaning that if you have an infinite sequence of points in the space, then some subsequence of those points will be converging on some point. It sort of means that the space must be "small" in some sense, because the points can't head off to infinity, and it also means that if a sequence of points are getting closer to each other, then they must be converging on some point. (The space includes its boundary.)

"Simply connected", basically means that if you're floating in this 3 dimensional space, and you have a stretchy loop of rope, you can move and stretch the rope without cutting it to be anywhere in the space. As an example of where you couldn't, if you were inside of a doughnut-shaped room, the rope might go around the "hole" in the middle, and you wouldn't be able to get it off without cutting it. (This can be expressed mathematically by talking about continuous functions from the circle into the space.) Another example of a not-simply connected space is just two ordinary rooms which are not connected to each other (like there's a wall between them). If the rope is in one room, you can't move it to the other. Of course, the individual rooms in this case might be simply connected, but considered together, they're not.

So, really loosely, the theorem is kind of telling you that if you're in a strange 3-dimensional space which seems to have a finite amount of room, and you can't seem to get a rope hooked on to anything, then you must be in the 3-sphere, there's no other strange way for a space to be connected up to itself.

It's interesting that the theorem has long been known to be true for 2 dimensional surfaces, and for 4 and higher dimensional manifolds for quite some time now, but the 3-dimensional case had eluded proof for quite a long time.

Re:It's all a conjecture

[cgibbard]

Re-reading the above, I realised that I should perhaps emphasize the point that to be a manifold, it has to look like 3-dimensional space at every point -- so there are no "walls" where the space just stops, or else at the wall, it would look like half of R^3 instead.

Re:It's all a conjecture

[Anonymous Coward]

Wow, if I use the Bork Bork Extention in firefox it actually makes more sense to me.

Re:It's all a conjecture

[Anonymous Coward]

...and the running time will be dominated by the time to run that particular program rather than the time to find it.

Re:It's all a conjecture

[Anonymous Coward]

(apologies for the earlier botched post - hit the submit button too soon)

...and the running time [of SUBSET-SUM] will be dominated by the time to run that particular program rather than the time to find it.

Let P be the first solution encountered, N the number of P, M the number of steps P needs to run to deliver the correct answer. The sum of the steps needed for SUBSET-SUM to run for k = 1 to N is at least N^2 (run each program k = 1 to N for N steps). At this point, P is found. However, the runtime M of P to produce the correct answer is not yet known (it could be N or less - we don't know). So we search up to around M^2 steps total to get the right number of run steps for P, or N^2 total if M < N (we have to find P first). The total runtime of SUBSET-SUM is at least max{N^2, M^2}.

The assertion that the runtime of P dominates means that M > max{N^2, M^2}, which is not true for M > 1. Choose S such that M or N is at least 2. Then this is impossible.

Re:Dear God

[Anonymous Coward]

The design looks fine but it's harder to read the article lead in on the front page. What's with the small text?

This is...

[mlow82]

This is one of the Millennium Prize problems! One down, seven more to go!

Re:This is...

[mlow82]

Six more to go actually...

Re:This is...

[smitingpurpleemu]

You mean one down, six more to go. There are 7 problems total.

Re:This is...

[nacturation]

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...

[mlow82]

There are actually 8 problems now. I added the Twin Prime Conjecture via one mighty edit to the Wikipedia article!

Just kidding, of course. ;)

Re:This is...

[o_miljac]

Everyone knows the final answer is 42, then why bother?

Re:This is...

[ozbird]

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...

[khallow]

Actually, it may still be unproven. High profile conjectures are notorious for generating false proofs.

Re:This is...

[Alsee]

maybe you know the mystery surrounding the elusive eighth question: "What is seven minus one?"

42

-

Re:This is...

[Anonymous Coward]

its already been proved that reimann hypothesis implies the twin prime conjecture, so it would be silly to have both as millenium prize problems....(yes i realize you were joking...)

Ok, in plain english

[AuMatar]

Can someone boil down what the Poincare Conjecture is for us? I've had up to linear algebra in college, but I don't understand what itsa saying.

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

Re:Ok, in plain english

[binarybum]

I think it basically has something to do with:

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

Re:Ok, in plain english

[Stalyn]

In topology spheres are identical to cubes and pyramids. However spheres are not identical to doughnuts. What PC says is that spheres are the only class of objects that are not doughnut-like (has holes). This seems trivial and obvious to most of us however to prove it is really hard. What it shows is that there is something fundamental and important about the sphere-like class of objects. It also says something important about space itself.

Re:Ok, in plain english

[Auxbuss]

From Millennium Prize Problems

In topology, a sphere with a two-dimensional surface is essentially characterized by the fact that it is simply connected. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question has been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds.

Re:Ok, in plain english

[mozu]

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

Re:Ok, in plain english

[david.given]

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

[Sqreater]

Yeah, it says space is continuous, that there is nothing outside of space. Big deal.

Re:Ok, in plain english

[uberchicken]

I'm very impressed with your explanation, however much you've simplified it. Thanks!

Re:Ok, in plain english

[ZenSkin]

Think of play-dough. If you make a ball of play-dough then stretch it or fold it or pinch it, it is still basically a ball of play-dough -- until you either pull it in two or poke a hole through it -- than it isn't like a ball of play-dough is a very fundemental way. Furthermore, this is true for every dimension. Easier said than proved.

Re:Ok, in plain english

[Darby]

A topologist is someone who doesn't know whether to dip their doughnut into their coffee mug, or vice versa...

I always preferred: "A topologist is someone who doesn't know the difference between their ass and a hole in the ground, but who does know the difference between their ass and two holes in the ground".

Re:Ok, in plain english

[Pendersempai]

Unfortunately this is wrong. A three-manifold is LIKE a sphere but in four dimensions. A sphere is a three-dimensional object with a two dimensional surface. A three-manifold is a four-dimensional object with a three dimensional surface. I suppose one way to visualize it is by taking our (three-dimensional) universe and imagining that if you traveled far enough in any direction you'd eventually end up back where you started, just like if an ant started walking in a straight line on the (two-dimensional) surface of a sphere he'd eventually end up back where he started.

The Poincare Conjecture says that every three-manifold that meets some conditions (no holes cut from its surface, it's all one object, etc.) can be smoothly distorted (through a process called homeomorphism) into any other three-manifold.

This is NOT true of two-manifolds: while you can smoothly distort (homeomorph) a sphere into a cube, for example, you cannot smoothly distort a sphere into a donut. This is because of the way we define a smooth distortion: at some point in the transformation you'd need to open up a hole in the sphere to make it into a donut, which disrupts the smoothness of the distortion. It's like if the cube were made of flexible rubber, you could bend it into a sphere, but you couldn't turn it into a donut without a pair of scissors and some glue. (This is all very hand-wavy, I know, but it's the best I can do without getting all technical.)

Re:Ok, in plain english

[Kehvarl]

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

That depends on the donut. A traditional toroidal donut is not simply connected, so in that case you're correct. However a jelly donut or one of the custard-filled subtypes is cylindrical rather than toroidal; this donut would then be a sphere, if you're a topologist.

Re:Ok, in plain english

[edew]

Let's look at 2-dimensional objects. There is a classification of 2-dim objects (done mostly by Euler in the late 1700s). Assume they're bounded (that is, doesn't go to infinity). There are the boundaried ones, like a sheet of paper: there's an edge to it. Then there are those without boundaries, like a sphere, or the surface of a donut, or the surface of a pretzel (three-holed). There are also orientable and non-orientable surfaces. A regular sphere is orientable: there's an outside and inside direction (if you're on the surface of the earth looking at the stars, no matter how your walk and travel, you can get back to where you were and look at the stars by looking in the same direction). A non-orientable surface would be a mobius strip. If you walk once around the strip, then looking at the stars in one direction will get to looking at a different direction. There is a non-orientable version of a sphere. The non-orientable version of a torus (the donut's surface) is called a Klein-bottle. In any case, the classification of 2-dim surfaces, orientable, compact, connected, no boundaries are basically the number of holes. Zero holes is the sphere. One hole is the torus, 2 holes is the two-torus (take two pairs of pants, sew the waists together. Then, sew the two leg holes of each pant together), 3-torus, 4-torus, etc. They're it, topologically. They're all equivalent (and equivalent here means homomorphic to each other; homeomorphic requires some additional geometric structure to stay the same). The genus of the surface is the number of holes. The main point for 2-dim is that any zero-genus surface is homomorphic to the sphere. The generalized conjecture is that any simply connected compact orientable n-dim manifold (fancy word for higher dimensional geometric object) is homomorphic to the n-sphere. Apparently, it wasn't very hard to prove it for very high dimensions. Supposedly, the constraints on the higher dimensions forced the case. The lower dimensions were rather difficult and the fifth and fourth dimension cases were solved only rather recently (in the late 80s?). The poincare conjecture was left to the 3-dimensional case, which apparently is now solved, if correct.

Re:Ok, in plain english

[Stalyn]

A 3-manifold is a 3 dimensional object. A 3-sphere is a 3-manifold in a 4-space. A "typical" sphere is a 2-manifold in a 3-space.

The generalized PC says every simply-connected closed n-manifold is homeomorphic to a n-sphere. The unproven case and the original conjecture was n=3. Actually n=2 had been proven a long time ago.

Re:Ok, in plain english

[coopex]

Ruining the joke, if you think about it, humans are equivalent to a torus, so the hole would have to have a singular exit for the topologist to be unable to differentiate the two.

Re:Ok, in plain english

[Darby]

Ruining the joke, if you think about it, humans are equivalent to a torus, so the hole would have to have a singular exit for the topologist to be unable to differentiate the two.

Taking the joke way too far (and using a bunch of irrelevant big words to try and sound cool and failing ;-), what if you embedded just the ass in a locally convex topological vector space? Then there wouldn't be another exit.

Re:Ok, in plain english

[coopex]

Well played, well played indeed.

Re:Dear God

[Anonymous Coward]

What's with the small text?

Microfont is de rigueur amongst amateur web-artistes.

more info

[airbie]

More on the Poincare Conjecture: http://en.wikipedia.org/wiki/Poincar%C3%A9_conject ure

Re:more info

[MattWhitworth]

Looking at that article, you can see Grigori Perelman solved the Conjecture way back in 2003, but hasn't yet submitted it to a journal. The general expectation is that he will claim the Clay Maths prize however, because the peer reviewing on arXiv has recieved a lot of review already.

How many Millenium Problems have we solved now? If we end up solving two, it's been a pretty efficient decade already.

Re:more info

[smitingpurpleemu]

I prefer the Mathworld web page to Wikipedia. The information is more reliable, and if you're unclear on the definitions, the terms are all hyperlinked so you can see what stuff means. http://mathworld.wolfram.com/PoincareConjecture.ht ml

Chinese == Good at Math

[hyeh]

Wow, Chinese people solved a math problem?

This is news?

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

Re:Chinese == Good at Math

[dartarrow]

NO the news is that the lest of the wold undestan wen thee Chinese peopre EXPRAIN the sorusion to the plobrem.

*ducks*

Re:Chinese == Good at Math

[Anonymous Coward]

Did you solve it? Then quit boasting. There are a billion Chinese, it's not exactly an exclusive club.

Re:Chinese == Good at Math

[Anonymous Coward]

Yes,

and old Soviet == Good At Sports. And Chess

Somehow it seems that it is easier to get people to push "themselves" in states with totalitarian regimes. Strange that.

I'll be impressed if you manage to keep that up when you're a democracy.

Re:Chinese == Good at Math

[Anonymous Coward]

Chinkalicious.

Re:Chinese == Good at Math

[Anonymous Coward]

I get the impression this would be considered more of an assist rather than an actual basket (to use a sports metaphor). Not that it matters what nationality the people are who solve very important mathematical problems such as this one. But, of course, appropriate credit is due the individuals commensurate with their individual contributions. Because there is a large money award involved, I hope its eventual disbursement will provide a clear showing of who did what.

Re:Chinese == Good at Math

[Anonymous Coward]

Toucheee ...

Were you fired because your job was outsourced. It is ok, drive your SUV and eat McDonalds.

Re:Chinese == Good at Math

[808140]

As someone who has lived in China for a long time and was formerly a mathematician, I think that your statement is sort of ridiculous. For one thing, as others have pointed out, saying "some race is good at math" as if being good at math were something in your blood is silly. Having said that, the Chinese (as in, those from China) are, unfortunately, overwelmingly bad at 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 memorizing every 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

Re:Chinese == Good at Math

[Pinback]

If it follows the recent pattern, there will be a follow up story about this pair of Chinese mathematicians hiring other Chinese mathematicians to pretend the proof is real.

Re:Chinese == Good at Math

[Tesla Tank]

Well written, and long :) , post. Unfortunatley, it's buried too deep in the comments for it to be modded up.

I was part of the Chinese system for 5 years before moving to the Western system. I hated all the memorization. That's a major reason why I despise biology, too much memorization. Anyway, I don't actually have a point. Just thought I'd say that your post was an interesting read for me.

Re:Chinese == Good at Math

[deodiaus2]

An excellent article summarizing the history and current situation of the educational system in China. I liked its length because the author took the time to explain his thoughts and justification. This differs from the "USA Today" mentality where most "information" is presented with bold and misleading statements and no explanations whose intention is to brainwash to populace into a way of thinking. BTW, I love in the USA. Elsewhere I'd have to serve in the military. Here, the Red Necks, Hispanics and Blacks sign up and die in my place serving the country! How can you not love a country like this?

Should share credit with Perelman

[Stalyn]

I thought the general consesus was that Perelman had proved Thurston's geometrization conjecture. If this proof by Zhu Xiping and Cao Huaidong is correct it must be a rephrasing of Perelman's work. Perelman is credited with making the major theoretical advances in order for any such proof. Basically he did most of the heavy lifting while these Chinese mathematicians basically dotted the i's and crossed the t's.

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

[gowen]

I can't see why Perelman would share the credit. If his results are right, he proved it first. A second proof is impressive (moreso if it contains anything particularly new), but until shown otherwise, Perelman was the first, so he gets all the marbles.

Re:Should share credit with Perelman

[Anonymous Coward]

Perelman wasn't able to prove it. The two Chinese professors did. Though he did provide some groundwork which they acknowledged.

The dotting the i's and crossing the t's are very important in mathematics.

Re:Should share credit with Perelman

[Stalyn]

Many believe he did prove it. It was just his work was so difficult to fully understand. These Chinese mathematicians it appears just gave a more detailed account and thus easier to see how the proof "falls out".

Re:Should share credit with Perelman

[jalet]

Exactly what I thought too !

Re:Should share credit with Perelman

[m874t232]

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 verified first by at least a dozen other mathematicians or by a mechanical device, its author(s) should get recognized as the people to prove the conjecture.

Re:Should share credit with Perelman

[gowen]

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.

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've been proved, even if no-one knew it.

Re:Should share credit with Perelman

[igny]

The fact that a proof is beyond comprehension of some mathematicians does not mean it is less of a proof.

Re:Should share credit with Perelman

[Luyseyal]

So, it is true that the falling tree makes a sound even when no one is listening!

Thanks for clearing that up!
-l

Re:Should share credit with Perelman

[Anonymous Coward]

Perelman did prove it. He posted his papers on the arXiv. The problem is, the proof deals with some very obscure areas of mathematics. Few people understand it, so there aren't that many people in the world who can verify it. The consensus in the mathematical community, after looking at his work for awhile, is that his proof is correct.

Re:Should share credit with Perelman

[xrz1138]

While I cannot speak authoritatively, and am not familiar with Perelman's work, I thought that the work
of Richard Hamilton was also fundamental to this new work. He researched Ricci Flow
(http://en.wikipedia.org/wiki/Ricci_flow)

I want to add that he is a cool guy too. I recall him working out an analysis, finding that it was
failing, and then turning to the class to say "Hmmm..... it worked in the car". I also heard him give
a talk a few years later at UCLA that had those people cheering. ....a weird sight really, to see such
folk all giddy. ;0)

Re:Should share credit with Perelman

[Intron]

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

[Anonymous Coward]

Dotting the i's and crossing the t's is an essential part of completing a proof.

I once referreed a journal article that was correct except that the author had used a "+" instead of (the correct) "-" in one inequality. Unfortunately, that error invalidated the entire proof (and I could see no "patch").

The author had clearly spent a lot of effort checking his/her work, but this one error slipped by. Without the i's dotted and the t's crossed, there is no way to know that there are not such errors in Perelman's work.

Re:Should share credit with Perelman

[Anonymous Coward]

Perelman's "proof" is apparently beyond the comprehension of all mathematicians since nobody has claimed to have been able to verify it and since Perelman hasn't supplied more details to clear it up.

The proof is due to Perleman

[Anonymous Coward]

Although the ideas used are extensions of that of Hamilton. I don't know why this is being attributed to "Chinese mathematicians".

Re:The proof is due to Perleman

[mlow82]

From the Wikipedia article:

In June 2006, the Asian Journal of Mathematics published a paper by Cao Huaidong of Lehigh University in Pennsylvania and Zhu Xiping of Zhongshan University in China, which has filled in the details of Perelman's work, thus "putting the finishing touches to the complete proof of the Poincare Conjecture", according to the Fields medalist Shing-Tung Yau.

Huaidong and Xiping "filled in the details", meaning that some important details must have been missing from Perelman's work which they were able to provide.

Re:The proof is due to Perleman

[Propaganda13]

My understanding is that Perleman and Hamilton did the groundwork, but never created a proof. It's like coming up with an idea for building a flying car, but never building one. When someone takes your ideas and expands upon them and creates a flying car, they get the credit.

Re:The proof is due to Perleman

[Haeleth]

It's like coming up with an idea for building a flying car, but never building one. When someone takes your ideas and expands upon them and creates a flying car, they get the credit.

And you sue them for patent infringement. After waiting for their flying car to become a commercial success, of course.

Re:The proof is due to Perleman

[cnettel]

Hey, wait, you just stole my joke!

Re:The proof is due to Perleman

[smitingpurpleemu]

You're right... but this is mathematics, not engineering. The idea is of central importance in a proof, and the person that comes up with it deserves a lot of credit, unlike in other realms of invention. Perelman deserves the credit he gets, and the Chinese and American mathematicians who filled out the proof also deserve credit.

Re:The proof is due to Perleman

[FhnuZoag]

Erm, a wikipedia quote hardly proves anything, especially given that it has only recently added.

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

Re:The proof is due to Perleman

[Anonymous Coward]

Well, its more like IBM / Exxon getting patents (Nanotechnology/Solar cells), but never really coming up with working products. During the internet (gold rush) boom, I heard many an IBM managers saying how it is more profitable to be making internet technology tools (like WebSphere) rather than actually finding end products (gold).
Exxon holds a great many patents on solar cell technology, yet it has not really developed solar cells. The patents are there to siphon off money from future endevours [hedging a bet] and to ensure a demand for oil.
The funny response to this was in "The Hitch-Hikers Guide to the Galaxy", where the philosophers were complaining that there was no money to be made in philosophy. A comment was made to philosophers that they should patent/trademark modes of thinking. Hence, you could enforce consessions from people when they made arguements which encroached on your mode.

WDWC query

[Toba82]

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?

Re:WDWC query

[Umbral Blot]

You might think that this is useless to you. However simply memorize a few quotes from the article and you can be prepared for any situation. Boss unexpectedly wants a status report? Sure boss, currently my I'm developong a compact n-manifold that is homotopy-equivalent to the n-sphere if it is homeomorphic to the n-sphere. We'll be done in a couple of weeks. Wife bothering you to take out the trash? Sure honey right after I demonstrate that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher. Never be at a loss for words again!

Re:WDWC query

[jopet]

Why do you care about the arts, a clean apartment, love? Well, judging from your question, you probably don't but a lot of people do.
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

[Anonymous Coward]

Well done, you posted the same comment twice with the obvious spelling mistake intact.

Re:WDWC query

[Umbral Blot]

Thank you I try hard to please. What did you expect me to do, post a reference to my earlier comment?

Re:WDWC query

[Lord Crc]

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 figured out a way to use finite fields in a way that enables you to still play a scratched cd, 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

[xquark]

not only scratches, but even holes, rs codes can correct holes on cds with diameters upto ~5mm!

Re:WDWC query

[m874t232]

For the people doing it, it's fun. It's no different from playing games, playing the piano, or posting on /.

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.

Re:WDWC query

[ZenSkin]

Why read Shakespeare? Why listen to music composed by some dead guy? Why go to see art hanging on a wall? The proof of the Poincare Conjecture offers new insights into the human understanding of the advanced mathematics of topology and geometry. To ask, "what do we care?" is the archetypal philistine question. If you don't care, than you're not interested in knowledge ... that's fine. But those of us who have studied these problems see a thing of beauty. The question is, why don't you see a thing of beauty? Because you do not understand it or because your eyes are shut?

Re:WDWC query

[grungy hamster]

Wow, read the subtitle for the site.

"News for nerds, stuff that matters."

It probably should say "to nerds" afterwards, but it's implied nontheless.

You sir should probably not be reading slashdot if you don't appreciate the significance of such a discovery.

Re:WDWC query

[Iron Condor]

Making pigs fly?

It is well known in aerospace engineering that

1) given enough thrust, pigs fly just fine.

2) However this is not neccessarily a good idea.

Re:WDWC query

[exp(pi*sqrt(163))]

People studying topology found they needed some pretty abstract tools. Some people came along and found they could divorce these tools even from topology making the esoteric branch of mathematics popularly known, even by mathematicians, as "abstract nonsense", though properly known as "category theory". And yet today people use category theory to understand datatypes in computer languages leading to ways to write code that put more burden on the compiler to find errors before your code starts running leading to happier end users.

This particular topology problem may or may not have a practical application. Topology itself has a few applications. And the mathematical tools developed by topologists will probably make their way into household items before too long.

Don't jump the gun

[twistedcubic]

This has not shown up in the mainstream Western press, which is very curious. A more believable article would be a report that Perelman's proof works.

Re:Don't jump the gun

[twistedcubic]

Oh, I was just kidding. I read the wrong article.

Re:Don't jump the gun

[lxt518052]

I'm sure it would appear on the Western press quicker if it's titled "Chinese mathematicians stole the proof of Poincare Conjecture from Perelman and Hamilton." And the slashdot comments on the article would probably double.

Two very good reasons

[Silver Sloth]

1. Firstly you never know when a mathematical oddity will turn out, years later, to be an essential part of something else. Both the sqare root of minus one and matraces had no posible application when they were firts investigated. Now both are essential tools for engineers.
2. Secondly for the same reason that we flew to the moon, because if we lose our inate curiosity then we lose our humanity. There's far more to being alive than materialism

Re:Two very good reasons

[Toba82]

I'm not saying it doesn't matter, I'm asking what made it front page news.

Re:Two very good reasons

[Anonymous Coward]

How, exactly (or roughly) does the square root of minus one tie into engineering?

I ask because this startled me quite a lot. I've only done the very basic classes that covered that area, but it seemed quite frankly like mathematical dickwaving (nothing wrong with that, it just contrasts what you're saying), and not very.. practical, for lack of better word.

I'm quite curious. Short explanation? Links? Google suggestions?

Re:Two very good reasons

[Silver Sloth]

I don't normally reply to ACs but in this case...

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 I said, I studied engineering thirty five years ago and I'm very rusty, I'm sure that other /.ers could provide plenty of examples

Re:Two very good reasons

[grolschie]

Both the sqare root of minus one and matraces had no posible application when they were firts investigated. Now both are essential tools for engineers.

I guess this is similar to the case of spelling (or proofreading). :-)

Re:Two very good reasons

[pedantic bore]

Two words: Fourier Analysis.

Re:Two very good reasons

[Mathiasdm]

I was thinking the same thing.

Maybe that's just because I have an exam on that stuff... In two days.
Damn you, Dirichlet problem... DAMN YOU!

Re:Two very good reasons

[mprinkey]

The key word for the airfoil problem is "conformal mapping." It is a technique used to map 2D space into the complex domain and in the process manipulated its shape. So what was a sphere or straight line segment is now an airfoil. It is used to make the solution of "potential flow" possible, so called because the velocity field of the flow is generated by the gradient of a single scalar potential.

Re:Two very good reasons

[Silver Sloth]

That's the bunny! Thanks.

Re:Two very good reasons

[Threni]

> I'm not saying it doesn't matter, I'm asking what made it front page news.

No, you effectively asked why it mattered - what it's consequences were. That's what the answer you got explained.

In this post you're asking why it made front page news, and the answer, as you can see from the number and quality of responses, is that it interests a lot of people who come to Slashdot to read tech/maths/physics related articles.

Re:Two very good reasons

[Anonymous Coward]

Journalism is the first draft of history. If it's not front-page news today, then when should it be news?

Re:Two very good reasons

[rbarreira]

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

[dollargonzo]

all the other responses are correct, but they do not get at why i was necessary in the first place (IIRC). in general, in order to solve any real quadratic equation, i.e an equation of the form ax^2 + bx + c = 0 you can find all the real solutions, if they exist, entirely without using i, i.e. entirely using real numbers. there are some imaginary solutions, but this is certainly not a reason to acknowledge i as necessary. however, if you try to solve cubic equations, there exist equations with real coefficients, i.e. where a,b,c,d in ax^3 + bx^2 + cx + d = 0 are all non-imaginary but the solution to the equation requires steps that involve i. this shows that i is indeed necessary. hope that helps.

Re:Two very good reasons

[NeoSkandranon]

Good luck dude, I had that class (Signals and System I, in my case) last year, it was a beast.

Re:Two very good reasons

[NeoSkandranon]

in Electrical and Computer engineering, dealing with AC electricity in terms of a sine wave involves some imaginary tinkering when you're determining how bits of a circuit effect the wave Also as another poster mentioned, in signal analysis when you get into Fourier and Laplace transforms.

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

Re:Two very good reasons

[Duhavid]

What makes it front page news is that we only have a front page.

Re:Two very good reasons

[chthonicdaemon]

The entire field of Linear Systems (start at http://en.wikipedia.org/wiki/System_analysis and click out from there) is greatly simplified by converting systems of linear differential equations to Laplace domain ( http://en.wikipedia.org/wiki/Laplace_transform ) and getting a frequency response by substituting s = iw where i = sqrt(-1) and w is a frequency. Due to the Euler identity ( http://en.wikipedia.org/wiki/Euler's_identity ), we can then transform this all back to time. The simplification is that we manipulate differential equations as algebraic equations in Laplace space.

I do this stuff all day and I can not imagine a world where sqrt(-1) is just undefined. Of course, the Laplace domain is not really restricted to linear systems, but it makes things easier when they are.

Now, this could all seem like "mathematical dickwaving" to you if you have not used it before, but it seems pretty damn useful to me that I can now determine optimal controller settings mathematically instead of having to guess values and have my plant blow up in the process. So there you have my little piece of the usefulness of imaginary numbers. (pure) Mathematics are not really involved in solving practical problems, but engineers can use all this rigorous theory to solve real problems. It is possible to prove that we don't need more than simple operators and a few digits to solve all problems -- having powerful mathematical tools just makes the journey shorter and more enjoyable.

Re:Two very good reasons

[Tesla Tank]

It's quite sad that I took the exam just a month ago, and now I'm having difficulty recalling what I learned in Signals & Systems.

Re:Two very good reasons

[NeoSkandranon]

I had it almost a year ago (Spring 05) and I can't remember jack squat from it.

Math isn't dead

[colin353]

This is another reason why math isn't dead. The world's problems aren't solved, and they aren't impossible, either.

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.

Re:Math isn't dead

[mlow82]

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

I know exactly what you mean. Whenever I mention that I'm a math major, most people instantly think that I like numbers and do a lot of arithmetic in my spare time, as if I cuddle in bed with my scientific calculator. On the contrary, most of the proofs I read/write do not mention numbers at all. It's much more abstract than that and unfortunately I don't know if the general public will ever understand that.

Re:Math isn't dead

[raddan]

Tell them that you're a philosopher. It's closer to what you actually do than what a bean counter does.

Re:Math isn't dead

[NeoSkandranon]

Funny, people thikn the same thing when I tell them I'm in engineering(student) ;)

At this point though they're almost right :\

Re:Math isn't dead

[fishbowl]

> I don't know if the general public will ever understand that.

They might not. They don't seem to understand that a person who is good a Math might be bad at Arithmetic, either.
The big problem here, is that it may be possible that people are not being led to Mathematics *because* they do not show an aptitude for Arithmetic.

I think sets and logic should be taught, starting at kindergarten, as a fully distinct subject from "Math".

Re:Math isn't dead

[Sage Gaspar]

"Man, you're in algebra? I remember that way back in high school. I was really good at algebra. Can't believe you're still doing that though." Yeah, I spent the last couple years researching the parabola. I might move onto the cubic next.

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

Re:Math isn't dead

[pooly7]

Hurry then ! There's only 6 carrers^Wproblems to go with the million $ reward !

Is there a math geek in the house?

[ocelotbob]

Unfortunately, my higher level math skills are a bit rusty, but would it be safe to say that this conjecture proves that if it looks like a sphere, acts, like a sphere, then it's probably a) a sphere, and b) can be broken down into a circular-shaped plane. Or am I missing something here?

Re:Is there a math geek in the house?

[Wooster_UK]

I'm not too sure what you mean by "broken down into a circular-shaped plane", and I'd much sooner you lost the word "probably". I'll explain the conjecture by means of the two-dimensional version. Before I get there, I've got to explain what I mean by a "sphere", because the mathematical definition is quite specific. A "sphere" is the skin of a ball, okay, so it's all the points lying at a distance r, say, from the origin. Having been so specific about all that, I'm now going to be dreadfully, appallingly loose in the rest of my language. Here we go.

Now, suppose you've got a surface, let's call it S, which is bounded (so it's finite in any direction), closed (so it's not got an edge), and simply-connected (so it's got no holes). Then by twisting, stretching, moving and generally deforming S in any way you like, but without taking scissors to it, you can turn it into a sphere. That's the Generalised Poincaré Conjecture, reduced to 2 dimensions, and it was proved, oh, ages ago. To understand the higher dimension versions, just imagine doing that for an n-sphere, which is the set of all points lying at a distance r from the origin in n-dimensional space.

Double the post, double the karma?

[Anonymous Coward]

You might think that this is useless to you. However simply memorize a few quotes from the article and you can be prepared for any situation. Boss unexpectedly wants a status report?

Wow! This is impressive. Post the same comment twice and get modded up for it twice! At first I thought it might be a bug with the new look slashdot but as the posts have different timestamps I guess you just put in a second time. It was funny the first time but....

Re:Dear God

[Teddy Beartuzzi]

Yup. Reading the first page of comments on the top story, and my eyes are already killing me. Just way too much brightness there. Checked the preferences page, no option for a different css style. :(

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

plain english? Maybe...

[N Monkey]

I didn't know what it was either but Wikipedia does have some simple descriptions which I'll try to summarise.

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 up to the next dimension (a 4D sphere?), which I guess means the "surface" is 3D, does the "simply connected" property still hold?

It's been a long time since I did maths at Uni so take with a grain of salt!

Re:plain english? Maybe...

[The Mathinator]

Not quite. The fact that the n-sphere is simply connected is pretty easy to prove. Poincare asked whether every closed simply connected 3-manifold is a 3-sphere. A surface is a 2-manifold. The sphere, plane, Mobius strip, Klein bottle, and so on are all 2-manifolds. A 3-manifold is just a natural extension of that idea, except instead of a surface, you have a 3-dimensional object. They're a bit hard to visualize, since most of them don't "fit into" our notion of space, in the same way that a sphere doesn't fit into a plane. Anyway, Poincare's original question In English: if you have a 3-manifold with no holes and no border, is it necessarily the 3-sphere? Translating the more general version into English is a bit more difficult, and I'll leave it to those who actually have experience with the problem. I just read the Wikipedia article. Just a bit more information from there that might be interesting: The problem is actually easier for higher dimensions. It was first shown for dimensions 7 and above, and then worked down to the lower dimensions.

And where was the DMCA ?

[tokyoahead]

Oh my gawd! How the chinese and russians violated the DMCA by understanding what the US-Scientists did!
How dare they build upon their knowledge! Lets sue them.
I am glad there are some copyright laws to protect others knowledge.

Chinese == Good at Math? Wrong!

[Anonymous Coward]

There are so many Chinese, some of them are bound to be good at math.

Math geeks are in their own L-Space

[Yetihehe]

If it doesn't contain holes like donut, it can be inflated until it's sphere.

Re:Dear God

[Anonymous Coward]

Horrible new design kids.

The amusing thing about this is that the parent post was moderated Redundant.

... not yet. But it may die soon.

[Anonymous Coward]

The problem is rather that the complexity of current math problems has approached the limit of what humans are able to handle. Any 8th grader can verify Pythagoras, but verifying a proof like the one at hand can only be done by a handful of the world's best mathematicians and may take weeks to complete (remember what happened when Wiles proved Fermat's Last Theorem). A proof is meant to demonstrate that a given conjecture is true by splitting it up into many small steps which are considered self-evident. However, today even verifying a proof is very hard and the time may be near when no one on earth will be able to handle the complexity of this task anymore, so that even if a proof is given it may be impossible to say with certainty whether it is valid. Computers may help here, but other problems arise in that context.

Re:... not yet. But it may die soon.

[colin353]

But the point I'm trying to make is that despite common perception that current problems are "too hard", people are still solving age-old problems today. Sure, it may be only a handful of mathematicians that can do this on Earth, but it means that problems remain to be solved and the solutions are possible to find. And as time goes on, more solutions, computer-based or otherwise, will appear. That'll be interesting to see.

Re:... not yet. But it may die soon.

[mochan_s]

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 approached and it makes a lot of previous hard problems easy.

Actually, Wiles proof of FLT is a simplification of the Taniyama-Shimura conjecture which he proved I believe. The actual proof could be just 1 line if we assumed the Taniyama-Shimura conjecture to be true. Most of Wiles effort was spent on proving the TSC I believe.

Re:... not yet. But it may die soon.

[ClamIAm]

Well, you definitely understand one of the limits that we have in science. If we keep going along on the same track, little additions and proofs to the current system get more and more difficult to do as we go along.

However, most branches of science do not go along the same track forever. If I remember correctly, physicists in the 18th or 19th century were telling people to not go into physics, because all that needed to be done was some cleaning up of the things they already knew, and they would be done. Of course, we know how that turned out.

The reality here is that every once in a while, someone comes along with a theory or something that is radically different from the accepted system. Things change. New things become "possible". Look at the development of electronics, for another example.

Re:... not yet. But it may die soon.

[geoffspear]

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 Hypothesis is true. QED.

Re:... not yet. But it may die soon.

[rbarreira]

Your post was quite good, but the last paragraph is contradictory with the rest.

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 stuff on top of what is already there, as you said).

The really good advances come if what you said on the first part of your post happens - someone coming up with a new fundamentally different way of seeing concepts and problems, which makes it easier for someone to fully understand them.

Re:... not yet. But it may die soon.

[Anonymous Coward]

This is exactly how new proofs are formed. The poster should not have say "assume" the conjecture was proven but
ONCE the TS conjecture was proven the FLT could be proven (and validated by other mathematicians) with a small amount of additional effort.

Millennium problems tend to become Millennium problems because there is no short jump from existing proofs.
Their solution draws on a wide range of existing proofs that have to woven together into a self consistent whole, which become the new proof.

I believe in the case of FLT the forward development of TS conjecture had nothing to do with FLT until a mathematician (I don't remember who) proposed the short jump from TSC to proof of FLT. After that proof of FLT became the same as proof of TSC.

One day the Riemann hypothesis my be proven the same way.

Re:... not yet. But it may die soon.

[Ithika]

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:

Actually, Wiles proof of FLT is a simplification of the Taniyama-Shimura conjecture which he proved I believe.

Whether the grandparent poster's assertion about this method is accurate or not is neither here nor there. You managed to quote him grossly out of context and completely twisted the original message.

Re:... not yet. But it may die soon.

[RackinFrackin]

However, today even verifying a proof is very hard

While 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 anymore

I 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:... not yet. But it may die soon.

[MattWhitworth]

I'm not sure if that sentiment is true. If many people work on the same problem - each having a different line of attack, they'll slowly work their way towards a proof. For example, the Riemann Hypothesis. The knowledge around the problem has definitely enlarged over the past century, and we can see possible line of attacks such as field theory (IANAM). On your last point, I do think computers will play a large part in proofs in the future - but we're miles away from giving it a statement and asking it to prove it's correctness :)

Re:... not yet. But it may die soon.

[fishbowl]

> Of course, we know how that turned out.

Yes! Now they are telling people not to go into physics because politics has made physics nearly impossible to pursue.

Re:... not yet. But it may die soon.

[fbjon]

But how do you know? Is the field of mathematics finite or infinite? This problem needs a solid proof, methinks..

Re:... not yet. But it may die soon.

[njh]

I think Goedel proved this in 1939.

Re:... not yet. But it may die soon.

[fbjon]

So in essence, not only is the list of axioms that cannot be proved infinite, the list of axioms that can be proved is also infinite?

Re:... not yet. But it may die soon.

[njh]

I don't think you can prove axioms?

Re:Dear God

[bmo]

The design is also fubar. It looks like Poo in the left hand corner in Konqueror/Safari.

It looks fine in FF, but I hardly ever fire up FF now that Konq works.

--
BMO

A translation...

[FhnuZoag]

First, think in Four Dimensions. Not in terms of time, or something, but as a fourth spacial dimension - like in terms of up down, left right, in out, and foo bar. A 3 sphere is a sphere in that sort of space. For example, in three dimensions, a 2-sphere is just a normal sphere - a group of points that are all the same distance from a certain centre point. A 3-sphere in 4 dimensions is just the set of points in four dimensional space that are the same 'distance' from a point in 4D. (We define distance using the pythagorus formula sqrt(x^2 + y^2 + z^2 + k^2).)

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:A translation...

[Handyman]

Hmmm, this sounds kind of related to things like planar graphs (i.e., graphs that you can paint on a flat surface without edges crossing eachother). I think the idea was that the a graph is planar if and only if it does not contain a subgraph that is homeomorphic to either the fully-connected five-node graph or the fully-connected 3/3-bipartite, 6-node graph. Whoa!

Re:A translation...

[Anonymous Coward]

Let's say your makes a little sense. Why is the result important? Just because the question has been around a long time?

Re:A translation...

[chicken_moo]

Thank you sir, for explaining what this all means in plain english. Not all of us were math majors, but we do still like to understand what's goin on in the world. Cheers!

Re:A translation...

[Wierdy1024]

Seems obvious to me:

you can distort a square into a circle in 2D (by squishing the corners in)
you can distort a cube into a sphere in 3D (by rounding the points)
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?

Re:A translation...

[sammy baby]

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, 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. 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...

[Anonymous Coward]

It proves that topologically there is no difference between you yesterday, and you now with your head up your ass.

Re:A translation...

[Intron]

What the Poincare conjecture shows is that you can distort any bounded (closed, simply-connected) 3D space into the surface of a 4D sphere.

Re:A translation...

[RackinFrackin]

Kuratowski's Theorem!

Re:A translation...

[mike2R]

Why is the result important?

Because it's the truth.

Re:A translation...

[Jo Owen]

If I had any mod points, they would all be yours.

Re:A translation...

[OldManAndTheC]

You lost me after 'First,'

Re:A translation...

[mOdQuArK!]

Haven't you heard? In today's fast-moving & hip society, truth is _so_ passe...

Re:A translation...

[Anonymous Coward]

This is going to be huge in the glassblowing world ...

Re:A translation...

[msuarezalvarez]

Why is the result important?

The result is important because it provides a vital piece of information needed when you are trying to understand certain objects which we mathematicians call manifolds. It is quite hard to explain why one wants to do this, though.

Put briefly, the truthness of the conjecture gives us a rather simple method to recognize a (3-dimensional) sphere when you see one. That may sound like little, but, actually, it's turned out to be a very difficult problem.

Re:A translation...

[AxelBoldt]

This is a really nice description of the theorem. I have just two small additions:

simple connected and close means that the surface is well... just an obvious surface

Simply connected means "no holes that you could capture with a loop". For instance, an ordinary sphere (what mathematicians call a 2-sphere, the surface of a ball) is simply connected: if you have any closed loop on the sphere, you can shrink it to a single point without leaving the sphere. The same is true for the 3-sphere. With a torus (surface of a donut) you can't always do that: there are certain loops that you can never shrink to a point without leaving the donut's surface. So the torus is not simply connected.

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.

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

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 then glue 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...

[msuarezalvarez]

Hm. I forgot a big thing...

But probably the most important thing about what's been done is that it is a step in the direction of what's called Thurston's Geometrization Programme, which is a project which would provide an incredible amount of information on manifolds---this is not my subject, so I cannot really explain muche here.

As it usually happens, it is not the actual result that bears the most importance, but the methods developed to solve it, which will be applied to solve further problems.

Re:A translation...

[Stalyn]

A 3-manifold is a 3D object.

Re:A translation...

[The Bungi]

Thanks for this. I was obviously being facetious but it's good to get an understandable explanation of an interesting topic.

Re:A translation...

[Anonymous Coward]

Your third inference does not follow from the former statements. Can you disort or stretch a figure eight into a circle in 2D? It is a "shape", after all. Even in 2D, not all "shapes" can be distorted into a circle without gluing or tearing.

Re:Dear God

[Anonymous Coward]

There must be something wrong with your installation. I also use Konqueror and the new design looks uber cool.

With a bit of imagination....

[FhnuZoag]

Well, General Relativity works in 4D, where this result applies, and lots of things in general relativity are basically 3-manifolds. So, if warp drives are invented at any point in the future, the proof of this conjecture will reassure us that Picard can get from Rigel to Farpoint station without being spewed out as salami.

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

Re:Dear God

[grolschie]

Does He read /. ?

What does it all mean?

[distantbody]

What are the useful applications of this? Can I get a quantum computer next week!?

Re:What does it all mean?

[Anonymous Coward]

Hell, no. You'll get your quantum computer last week.

Re:What does it all mean?

[Anonymous Coward]

What are the useful applications of this? Can I get a quantum computer next week!?

Yes, just as soon as you build it. Now, go to your garage and get to work

Here is a conjecture

[2Bits]

I assert that there is a torrent of the proof somewhere on the net. Now can someone prove that, please?

Grade A Kokgobbler

[Anonymous Coward]

Thank you for posting that exacty same comment again, you fat lazy american P o S.

You are a grade A kokgobbler.

Gobble! Gobble! Gobble!

Re:Dear God

[iBod]

Typical Slashot arrogance.

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

[cnettel]

The contest results story is kind of relevant, and there were certainly a lot of comments made there. I might agree that it would make sense for them to post a poll where we could vote on the results, and also leave relevant comments.

No sharing credit.

[Anonymous Coward]

As Perlemans proof implies the Poincare conjecture, he should get all deserved credit and no one else.
If we don't stop this bullshit right now, people will stop handing out preprints because some gonzos "fill in the gaps" and "completing the proofs" and steal their work.
I mean look at the journal. If these guys had anything important to say about the Poincare conjecture, they wouldn't publish to a very different journal.

Re:Dear God

[linvir]

There have been at least two front page stories about the redesign contest, and Rob ran a long series of comments-enabled journal entries about designs he liked. There has been a constant stream of tonnes of feedback since this thing started.

Great. Still waiting for peer review..

[Anonymous Coward]

I'll accept the proof when it's been properly reviewed by peers. Just publishing a proof in a journal doesn't equate to a correct proof, now does it?

Re:Great. Still waiting for peer review..

[Metasquares]

The Clay prize isn't given out until 2 (IIRC) years after publication, so there will be plenty of time for it to be reviewed.

Re:Great. Still waiting for peer review..

[Anonymous Coward]

Journal articles are peer-reviewed before they are published. Of course, mistakes do happen, and more reviews of the work will help to confirm it, but it would be misleading to claim that this proof has not been checked. It *should* already have been.

Re:Great. Still waiting for peer review..

[zen-theorist]

I think the Asian Journal of Mathematics is a peer-reviewed journal, since it has an editorial board.

However, with such profoundly deep theorems, it might be the case that only a handful of people in the world understand what's going on, and it is up to them to come to the right consensus on its correctness.

Re:Great. Still waiting for peer review..

[Horde]

"Just publishing a proof in a journal doesn't equate to a correct proof, now does it?"

Neither does peer review

Re:Great. Still waiting for peer review..

[edew]

The Poincare Conjecture, along with several other notorious famous conjectures (Riemann Hypothesis, for example), have the nasty problem of making people who try proving it to use it as an assumption somewhere within the proof. Basically, the PC is logically equivalent to several other statements. Somewhere along the proof, in many prior failed cases, the author uses one of the other statements and then somehow claims, voila! PC is proved. Then, someone shows where the mistake by showing where the equivalent statement is used as an assumption.

Other conjectures, like the twin prime and such (most are number theory based) are so enigmatic that no one has even a slightest inkling on how to proceed.

Joe Public goes

[advocate_one]

"isn't that like the Da Vinci Code???"

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

Re:Joe Public goes

[dido]

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

1. The Riemann Hypothesis
2. The Eisenstein Criterion
3. The Fredholm Alternative
4. The Poincare Conjecture
5. The Fourier Transform

Re:Joe Public goes

[Gryle]

Tom Hanks as a mulleted mathematician...*shudder*

Re:Joe Public goes

[mako1138]

Heh, now I've got some ideas for this year's CampusMovieFest.

Re:Joe Public goes

[Anonymous Coward]

Sounds like Ludlum (The Parsifal Mosaic, The Chancellor Manuscript). Clearly, he's touched on art and literature, already. Why not a post-humus math adventure?

wouldn't trust it yet

[m874t232]

Serious errors in mathematical papers are so common that I wouldn't put any trust in this until the proof has been around for a decade or two; even if Cao and Zhu did everything correctly, there's a good chance that something they relied on turns out not to be true after all.

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 working mathematicians usually don't use them.

Re:wouldn't trust it yet

[Yrd]

This is something I'm peripherally involved in - automated proof tools are becoming more capable all the time, and I was at a keynote address by Tom Hales (University of Pittsburgh) who has been using such tools to formalise one of the proofs he's known for. There's some resistance (a lot, perhaps) to using such things in the mathematical community, but as a mathematician who's decided to use them rather than a computer scientist who's trying to prove that they're useful, he's hoping to change some minds and it's also nice for those of us in AR research to hear that there are mathematicians out there using them!

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

[cool_arrow]

forgive my ignorance, but didn't a guy name Hilbert propose, long ago, to automate proof checking. Didn't Godel prove it couldn't be done?

Re:wouldn't trust it yet

[rbarreira]

I think he just proved that NOT ALL the statements in a mathematical system are provable. He didn't prove that it is impossible to prove ALL the statements.

Re:wouldn't trust it yet

[caramelcarrot]

No. In fact, part of the proof of the incompleteness theorem involved the fact that it is #trivial# to check a proof, as all you have to do is to work through each line of the proof and confirm that it logically follows by the given axioms. Of course, it may not be so easy to actually state everything properly as axioms and may inflate what seems like a short mathematical proof a lot, but it's what is needed.

Re:wouldn't trust it yet

[ObsessiveMathsFreak]

In the long run, mathematics really needs considerably more formality than it is using now....

No, no and no. What mathematics needs is to escape from the rigid and inflexible formality that has been holding it back for the last 80 years.

Frankly, many mathematical topics have become so obfusacted that their transition from theory to application has been made nigh impossible. This is accomplished through the procedure of precise, immaculate, terse and above all convoluted definitions and theorems. Oh, and no pictures. I've seen books on topology completely devoid of diagrams. Thank you Bourbaki.

Simple mathematical concepts have become clouded by a fog of unnessesary cruft. The beauty and power of mathematics is being lost to serile exactness. It's holding progress back, and it needs to be stopped.

not necessarily

[m874t232]

The purpose of a proof is to communicate a sequence of statements such that each and every individual step is easily derivable from axioms or well-known theorems. Let me emphasize this again: a proof is about 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

[Anonymous Coward]

No. The one who proves get credit for the proof. Nobody else.
The situation might be similar to DeBranges proof of the Bieberbach conjecture.
His proof was basically non-readable and some Russians showed that his proof was indeed correct.
However, they wouldn't dream of taking the credit for the proof.

Claiming the proof in such a case is just academic theft.

Re:not necessarily

[Anonymous Coward]

correct, and considering there are maybe 1000 people on the planet competant of reviewing perelman's papers, it's pretty fucking likely he knew it was proven years ago via the thurston's conjecture which is a superset of poincare, and just didn't give a shit and moved back into the woods of russia.

Re:not necessarily

[m874t232]

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 it doesn't. There is no a priori reason to draw it such that Perelman's sequence of statements constitutes a "proof".

Ultimately, mathematics and science is not about people figuring things out for themselves, it's about communicating results. If you don't communicate, you shouldn't get credit.

Claiming the proof in such a case is just academic theft.

Claiming credit for a result without delivering a clear, readable, and understandable explanation of your results is academic theft as well.

Re:not necessarily

[Anonymous Coward]

suggestion, why dont you cry about it?

you'll feel better and then us adults can go back to the math.

Re:not necessarily

[notnAP]

a proof is about communication, not merely about making true statements.

Am I the only one who thought Bush and WMD when he read this?
[ducks and prepares for the -1 offtopic/flamebait/troll mods]

Re:not necessarily

[Stalyn]

Actually Perelman did prove it. The thing is he doesn't really care about the Millennium Prize. So he wasn't too concerned about those who were unable to see the proof fall out. These Chinese mathematicians just did a more lucid proof. Without Perelman's work there would no "other" proof. He solved the major problems and completed Hamilton's work on Ricci flow. In a historic sense Perelman will be the major figure while Huaidong and Xiping will be mere footnotes.

Re:not necessarily

[mrgeometry]

Troll.

Re:not necessarily

[m874t232]

Ah, yes, the refuge of people who just can't put a clear argument together: calling people they disagree with "trolls".

I have a more apt moniker for you: "incompetent". Unfortunately, your kind of incompetence is pretty common in the sciences and mathematics, which is why the published literature is so rife with poorly written papers and incorrect results. No doubt, you have contributed your share.

Fortunately, this kind of mess will be cleaned up and people like you will be forced to change your ways or leave the field; mathematics and science can't progress otherwise.

Re:not necessarily

[mrgeometry]

LOL. Thin-skinned troll.

Re:not necessarily

[m874t232]

"Thin skinned"? No, merely frustrated at the flood of shitty papers that come across my desk every day: sloppy write-ups of half-baked ideas that even the authors didn't completely understand and where they felt it beneath their dignity to work out the details.

Whether people like you want it or not, this will change over the next few decades. We now have both the technology and the economic incentive to fix it.

Doesn't look too new...

[Anonymous Coward]

Seeing that the Millennium Prize Committee has been checking Perelman's proof of the Poincaré Conjecture for at least the last 2 years, maybe even people who don't have a background in homotopy theory can suspect that Perelman has something that can be considered an "attempted proof" (drop 'attempted' when mistakes are ruled out). And he had it first.
So this great Chinese proof (as reported by Xinhua...) is probably a re-write of the original one; the filled-in details are presumably of the kind "carrying out some straightforward calculations that distract from the main idea and therefore weren't explicitly detailed in the original paper, although the author obviously did them when coming up with the result". Being a mathematician, I have seen more than one paper that has appeared a second time in a Chinese Journal, with the names on it changed, and this kind of details "provided". Well, their government seems to need something to boost patriotism... :-/

don't make me laugh

[Anonymous Coward]

"The findings would help scientists to further understand three-manifolds geometrization and heavily influence the development of physics and engineering," said Yau

yeah right. As if there already isn't plenty of research based on the assumption that the proof exists.

Re:don't make me laugh

[efagerho]

Usually the proof itself contains more information than the actual statement of the theorem, so you usually don't get much by just assuming stuff.

Re:Dear God

[supasam]

It looks fine in my safari.

BUT

[Frightening]

If you stretch the putty far enough, it becomes thin, then you fold it over and voila! You have the same effect as cutting and glueing.

Killed a 300 page proof in 2 minutes. I rule.

Because:

[rbarreira]

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:Because:

[emplynx]

Though empirical data would have allowed for the necessary corrections to GPS that Special Relativity provides mathematically.
None-the-less, existing mathematics has contributed on multiple occasions to future scientific discovery (in contrast to mathematics which is created to fit the discovery).

The Cube

[Frightening]

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.

Re:The Cube

[0ptix]

AFAIK topologically speaking your cube IS (at least homomorphic to) a sphere => it has no hole.

Re:The Cube

[Ruie]

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

[Bourbon Man]

*My* cube has a hole in it; it's where the the power, monitor, mouse, etc cords come thru the desktop.

Re: They are the kind of people ...

[140Mandak262Jamuna]

... who would pour their coffee into their doughnuts and dunk their cups in the soggy mess and look surprised. They are topologists. They cant tell a cup from a doughnut. When they need a ball in a hurry they will break off the handle of their coffee cup and try to bounce it on the court. These topologists are the most confused/confusing mathematicians around. ;-) They could make their math easier for us lesser mortals to understand. But they would rather knot.

What really happened

[szembek]

I bet they just sanded the Motorola logo off of another proof and passed it by as their own.

sic?

[lousyd]

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.'"

"contribution"? Not "contributions"? sic?

Re:sic?

[fishbowl]

Even the most English-literate Chinese make this kind of error.
When you can write an article in some dialect of Chinese without
the slightest error, start throwing stones.

Re:sic?

[lxt518052]

Fortunately, when it comes to writing, all the hundreds of dialects look basically no difference to each other. Hopefully it will make mastering the Chinese lanuage less a daunting task. (I know, I know, there're two writing systems, Simplified and Traditional. But if one knows how to write and read in Simplified Chinese, there's little difficulty to understand the Traditional version.)

Of course, lots of native speaker of Chinese make mistakes in grammar too. So it is a bit too harsh to require the GP like that. However, don't native English speakers do the same?

Re:sic?

[fishbowl]

I have a small number of friends who were born in China. I help them with their English when I can. I explain idiomatic expressions, teach them slang, and I proofread their papers. One of my friends learned English in Georgia. So her English has a Deep South accent, with a lot of expressions that are distinctly Southern, together with a lot of expressions that are distinctly from what I would call an African American dialect. I don't want to come across as racist here, and I am certainly not a linguistics researcher, just trying to describe a phenomenon.
She learned English in a public school in Georgia, where a certain dialect prevails, and most of her peers were black teenagers in a segregated society. I find the results rather interesting...

Re:sic?

[lxt518052]

Couldn't agree more. Diversity of a language is normally a result of its popularity. Internationally, we have Indian English, Malasian English, Japanese English and Chinglish^_^, to name but a few. Even here in England, the Brits alone have numerous accents. My girlfriend, who's also a Chinese, once told me the reaction of her English colleague when she used the idiom "raining cats and dogs", which she learned back in China. He was very surprised and told my GF, "that's a decades old saying, nobody still uses it nowadays."

Guess that is another example. Language is indeed a living thing.

The English language has only been gaining its popularity in the past few hundred years since Chaucer and Shakespear. The Chinese, however, has been spreading over the continent and its surrounding isles since the Qin dynasty, 221B.C. The aformentioned universal writing system has been proven instrumental to keep the such a complex language relatively stable.

Re:Dear God

[Anonymous Coward]

How can you complain? The plans have been available in the local planning office for the last nine months.

name change

[big dumb dog]

Once others verify these results, does it become Poincare Theorem or does Lau add his name to it?

Not that I will ever need to know, but how does the process of going from conjecture to theorem work?

Re:name change

[flooey]

Once others verify these results, does it become Poincare Theorem or does Lau add his name to it?

Not that I will ever need to know, but how does the process of going from conjecture to theorem work?

It may or may not still be called the Poincare Conjecture. It basically depends on what it's cited as in future work. There isn't a standards body somewhere that decides on the names of mathematical concepts, they just pick up whatever name they pick up.

From the PDF, it's easy to see

[int19h]

"
(2.15) S (g) = | min(sg , 0)|3/2 dVg.
Then it is easy to see that
(2.16) |(M )|3/2 = inf S ^ 3/2; (g).
"

Of course, we all knew that.

Truthiness

[hao2lian]

To me, a whole lot of time could've been better used if the mathematicians realized that conjectures are true if they felt that it was true in their gut. Sure the Chinese mathematicians could've looked Perelman's proof up and add details, but they could've just looked it up in their gut first.

Re:Truthiness

[monteneg]

Is this a joke? Engineering shows us that if we feel something is "true in the gut" then it may work most of the time, but when it gets taken to the most extreme case it can end up with a building or bridge falling down (Tacoma Narrows Bridge for instance). In math we like to take something that is "true in the gut" and see just how extreme we can take it and still have it be true, just like architects want to stretch construction principles as far as they can. Once we find the must fundamental principle involved in the proof then it often gives insight into what other things may be true or false, again just like Tacoma Narrows taught the engineerings about the limits of their techniques at the time.

What a bunch of crap

[Nimduin]

Just how in the hell is it correct to attribute this to "Chinese mathematicians?" Everything indicates that the proof was successfully completed by Perelman. Whether or not Perelman's proof is difficult to understand doesn't make a flipping bit of difference -- this was a millenium problem for a reason! It's therefore reasonable to expect that the proof might necessarily be tricky to comprehend.

What's more, it's math. No wishy washy bullshit involved (cue "obviously you've never studied postgraduate mathematics.") If Perelman is correct, he and only he deserves credit. Whether or not he was clear does not farking matter.

Re:What a bunch of crap

[2short]

"Whether or not Perelman's proof is difficult to understand doesn't make a flipping bit of difference"

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 Perelmans work, and I'm not enough of a topologist to judge it if I did. But it is my understanding that quite a few people who have and are were not entirely convinced. Cue the traditional denigrating of the abilities of ones doubters.

Maybe the chinese mathematicians have added something real to the proof. Maybe they've filled in stuff that just seemed obvious to Perelman. I don't know. Frankly, with my limited topology knowledge, the whole theorem seemed obviously true to me, but that doesn't mean I could prove it.

Priority disputes are annoying. I've no idea who deserves how much credit in this case, but I do know that whether one is clear absolutely does farking matter. Theorems are true or not on their own; the point of mathematics is to add them to human knowledge, and that requires clarity.

Re:What a bunch of crap

[Anonymous Coward]

Frankly, with my limited topology knowledge, the whole theorem seemed obviously true to me, but that doesn't mean I could prove it.

Given the amount of effort put into it by people with far greater knowledge than yourself, I'd say it's all but certain that there's no chance in hell you could prove it.

Re:What a bunch of crap

[2short]

Um, yes, I agree. What's that got to do with anything?

perelman rejected Clay's million

[weierstrass]

My understanding was that Perelman basically proved the Conjecture to the satisfaction of other mathematicians, but didn't try to get his work published in a peer-reviewed journal, expressly because he was not interested in the Clay Foundation's money, and wanted to show that.

Re:name change - Mod parent UP

[lord_rob the only on]

Interesting question I was about to ask. Given that Pointcarre's conjecture has been proven, it's no more a conjecture, but indeed a theorem. The most important thing is not *how* it has been proven (don't get me wrong, of course the way it has been proven is important, if it wasn't anybody could prove anything ...) but that is HAS been proven. So now any mathematician can use this theorem to prove further results.

the missing parts of the proof were ..

[zen-theorist]

.. QED

Yes, this can be applied to something now.

[Anonymous Coward]

Black Holes are considered to be a 1-sphere in our Space. But the proof can be applied to this and will change the perception of it to a 3-sphere - which makes sense of it all. Since the proof is that there are no holes in a 3-sphere, this means that wormholes do not exist in R4 and gravity is pulled equidistant to this anomaly (barring normal gravinometric pressures from other sources). matter is compacted and warped but does not leak out into another plane of existence or reality because there is no "hole".

So that also means framing is the R3 manifestation of R4 disturbance waves.

Black holes suck but they do not swallow.

Re:Yes, this can be applied to something now.

[Anonymous Coward]

Black holes suck but they don't swallow.

Don't you mean, "they don't spew"?

Re:Yes, this can be applied to something now.

[chawly]

Nope ! Everybody (you are the only exception) knows that they (Black Holes) are caused by God when he tries division by zero. Happens to everybody once in a while. It's called a programming error. No need to be embarrassed.

homeomorphic???

[Anonymous Coward]

turns out homeomorphic means "can be deformed into each other by a continuous, invertible mapping", and not at all what i thought it meant... i was pretty close, only instead of "mapping" mine said "pumping"

Math for CS Majors

[Dareth]

Calculus is okay, but set thoery and basic logic is the math that CS majors should be required to master in college. Dealing with databases and groups of employees with similar and different permission schemes will require some understanding of sets. Nothing like a good ole Venn diagram to help put things into perspective.

Re:Math for CS Majors

[fishbowl]

"Calculus is okay, but set thoery and basic logic is the math that CS majors should be required to master in college. "

There's surely not a CS major out there that doesn't have at least 2 semesters of discrete math, requires at least single variable integration, and at least some elective that gets deep into algorithms, which means the typical CS major gets a LOT more experience writing proofs than people in other sciences, aside from math (which is a science.)

Re:Math for CS Majors

[Anonymous Coward]

>There's surely not a CS major out there that doesn't have at least 2 semesters of discrete math

I was going to debate with you but, when I realized that my rebuttal was going to require some sort of example permutation or statistical analysis, all I could think of was... "You bastard!" :P

Your radical ideas have already occurred to others

[Anonymous Coward]

Theorems whose names sound like Robert Ludlum titles

The Answer is...

[Anonymous Coward]

42

Zombies

[DaddyDonMynack]

The real issue is how the use or abuse of this new knowledge puts us at risk of a zombie outbreak/invasion. C'mon scientists, the world needs to know!!!

What's up?

[lxt518052]

Be it Chinese mathematicians, Indian mathematicians, Russian, American, or French, who did the final work deserve the credit. All men/women are equal in the face of science. Perelman and Hamilton's work deserves credit too. Xinhua has acknowledged that in the article. So there's nobody stealing other's credit.

Why is there always people nitpicking the Chinese?

Re:What's up?

[Anonymous Coward]

I scanned all the comments on the Poincare Conjecture and I feel I have to add to this a little bit. I am a professional mathematician and I work in the area of geometry and topology; I have known the Poincare Conjecture and Perelman's work for a long time. The ONLY person who deserves credit is Perelman. Someone talked about building a flying car so let me illustrate by way of an example: someone said (W. Thurston) that if we build Battlestar Galactica (the Geometrization Conjecture), then we can also build spaceships (the Poincare Conjecture). Hamilton drew a schematic picture of a spaceship but didn't know how to design the propulsion system. Then for nearly 15 years after Hamilton no one had a clue what to do next. Out of the blue Perelman published three absolutely stunning papers in which he designed Baattlestar Galactica along with careful and detailed diagrams of EVERYTHING. Since the proof is incredibly sophisticated (those of us in the know are in awe of his staggering intellectual genius) Perelman's diagrams are missing some details like how many screws in this part of the hull or how wide to build the door to the captain's room etc. So several mathematicians have spent years unravleing his intricate and extraordinary work. Now Zhu and Cao claim they have done it which in my professional opinion is downright disgusting. They may have helped fill in lots of details but please believe me when I say that Perelman's work is more than foundational -- it is THE proof, the key idea. My request is to not mention Zhu or Cao anywhere along side Perelman or Poincare. By the way, I am happy to explain the actual details of Perelman's proof if you are so inclined. The entire proof can also be found on Los Alamos e-Print ArXiv.

sorry, but that's just wrong

[m874t232]

Sorry, but that's just wrong. Correct mathematical proofs aren't something informal or vague, they are a well-defined reduction of theorems to axioms.

Mathematical proofs as written up in the literature are usually merely an informal notation, but this informality is only permitted because there is an implicit assumption that a competent reader can fill in all the missing steps. If a published mathematical proof is of the form where competent readers can't fill in the missing steps, then the author hasn't published a proof at all.

So, either you need to publish a complete, formally verifiable proof, or you publish a proof where other competent mathematicians can fill in the missing pieces themselves. If you do neither, you haven't proven anything.

Re:sorry, but that's just wrong

[gowen]

If a published mathematical proof is of the form where competent readers can't fill in the missing steps, then the author hasn't published a proof at all.

Who ever said there were missing steps?

So, either you need to publish a complete, formally verifiable proof

A proof may be verifiable even if there are no people alive capable of verifying it.

Re:Dear God

[elvum]

Typical Slashdot reader arrogance... ;-)

Who are the morons again?

[Eternal Vigilance]

Umm, the politicians are taking it to the bank and we're all getting fucked in the ass.

Who are the morons again? :-(

Oh, just to make it relevant to TFA, in social interactions I highly prefer that my ass remain homeomorphic to the 3-sphere.

You should watch more TV ...

[Anonymous Coward]

... it was a joke alluding to Colbert's performance at the White House Press dinner.

Yau (and his students) always pull this crap

[Anonymous Coward]

When Givental proved (part of ) Mirror symmetry, Yau finds some little thing in his proof, rewrites it and then spends the next 3 years running Givental down in talks and speaking about how he proved Mirror symmetry. I cannot believe he is going to do this again. There a LOTS of people (eg John Morgan - did he prove the Poincare conjecture too now??) writing notes on Perleman's proof. No one has found any serious mistakes as far as I know. If for example this lot had found an Error, Perleman had failed to correct it for several years, and then they had corrected it then maybe you mention their name in the proof. But that isn't the case is it...

The worst about this is that Perleman won't even defend himself. The guy is like some sort of Math monk. Yau is a fields medalist, but a cancer. And worse he seems to be training his students that this is the way things are done.

-- and no I can't sign this. Because Yau and his fricking math mafia will end your career.

Re:Yau (and his students) always pull this crap

[Anonymous Coward]

They now claim that Perleman contribute 25%, the chinese 35% and Hamilton 40% in Chinese media. And H. get all his ideas originally from Yau. I bet two years later, it will be Yau the major contributor for this according to himself.

Re:Yau (and his students) always pull this crap

[chawly]

Let's hear it for Poincaré - he died in 1912, and can't defend himself. I hope he pisses on good old Yau from way, way up there. And copiously.

Re: They are the kind of people ...

[thechao]

Any topologist worth his deformable-doughnut knows it has one-handle, whereas a cup has no handles!

Re:Ok, in plainer english

[Anonymous Coward]

By a surface a mathematician means the surface of a ball (e.g. the surface of the earth), the surface of a donut, the surface of pretzel, etc... Roughly speaking two surfaces are equivalent topologically if one can be deformed into the other without ripping or tearing. So the surface of a ball and the surface of a donut are not equivalent; however the surface of a donut is equivalent to the surface of a coffee mug with a handle and the surface of a sphere is equivalent to the surface of a coffee mug without a handle. Of course there is still something to distinguish between the members of a given class of topologically equivalent surfaces - the surface of a coffee mug with a handle and the surface of a donut are clearly different! What distinguishes them is their geometry. Here it gets more difficult to quickly describe how one distinguishes them - and how one decides which topologically equivalent surfaces should be regarded as geometrically distinct (in fact there are various notions used depending on what properties of the surface one wants to study). The geometric classification of surfaces is called the uniformization theorem.

The Poincare conjecture is a special case of a much more general conjecture called the Thurston Geometrization Conjecture which is essentially the analogue of the uniformization theorem for three-dimensional surfaces, which unfortunately cannot be visualized in any naive way. (to understand these things in any serious way one would need to study for many years, maybe 5 years, maybe a decade, depending on one's preparation going in). From the perspective of a differential geometer, the enunciation of the Thurston conjecture is one of the sublime intellectual accomplishments of all time, and its proof would be one of the crownig achievements of geometry.