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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.'"
Homeomorphic. Thank god, they dumb it down a bit later:
More colloquially, it's homotopy-equivalent to the n-sphere! Of course!
Slow news day?
This is one of the Millennium Prize problems! One down, seven more to go!
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.
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More on the Poincare Conjecture: http://en.wikipedia.org/wiki/Poincar%C3%A9_conject ure
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Wow, Chinese people solved a math problem?
This is news?
(j/k... I am Chinese).
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.
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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.
-- If unsure, say "Why?"
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).
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?
There are so many Chinese, some of them are bound to be good at math.
If it doesn't contain holes like donut, it can be inflated until it's sphere.
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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.
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.
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)
Does He read /. ?
What are the useful applications of this? Can I get a quantum computer next week!?
I assert that there is a torrent of the proof somewhere on the net. Now can someone prove that, please?
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?
"isn't that like the Da Vinci Code???"
I think it makes a good thriller title... "The Poincare Conjecture"
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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.
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.
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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.