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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.'"
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
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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.
The best education consists in immunizing people against systematic attempts at education. - Paul Feyerabend
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.
The best education consists in immunizing people against systematic attempts at education. - Paul Feyerabend
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?
Marc
If it doesn't contain holes like donut, it can be inflated until it's sphere.
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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)
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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.
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.
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.
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.