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: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

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: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: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: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?

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...

[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...

[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.)