Poincaré Conjecture May Be Solved

Flamerule writes "The New York Times is now reporting that Dr. Grigori (Grisha) Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg, appears to have solved the famous Poincaré Conjecture, one of the Clay Institute's million-dollar Millennium Prize problems. I first noticed a short blurb about this at the MathWorld homepage last week, but Google searches have revealed almost nothing but the date and times of some of his lectures this month, including a packed session at MIT (photos), in which he reportedly presented material that proves the Conjecture. More specifically, the relevant material comes from a paper ("The entropy formula for the Ricci flow and its geometric applications") from last November, and a follow-up that was just released last month."

Comments

Explanation

[MaestroSartori]

Shamelessly stolen from here:

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?

Re:Explanation

[Gleef]

Some uses for topology:
http://www22.pair.com/csdc/car/carhomep.htm

Granted, none of this is stuff I would expect a gas station attendant to be playing with, but it's apparently increasingly important for researchers and engineers.

Google Partner Link

[Anonymous Coward]

For the lazy/paranoid.

What's that conjecture again?

[n3k5]

for the first time in ages, I'm looking forward to the discussion on this, in the hope that someone explains it in a manner I can understand

The explanation in the article is not too bad; the Wikipedia contains a better explanation:

[The Poincaré] conjecture is that every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere.

Loosely speaking, this means that every 3-dimensional object that has a set of sphere-like properties can be stretched or squeezed until it is a 3-sphere without breaking it. Note that a 3-sphere consists of all those points in 4-dimensional space R4 that have a distance of 1 from the origin.

Re:What's that conjecture again?

[Alsee]

It's so simple when you put it in plain english ...
[/sarcasm]

Ok, try this:

We long ago proved that an ordinary sphere is the only shape in 3 dimentions with no holes in it.

Note that the "shape" is "made of clay". You are allowed to stretch it, squish it, and bend it all you want. You aren't allowed to cut it or put a hole in it. And you can't "meld" parts togther.

A coffee cup is the same "shape" as a donut because you can smoothly "flow" the cup part into the handle and you get a donut.

What they just proved is that a 4-dimentional sphere is the only shape with no holes in it.

So what? Well if you have some wierd complex 4 dimentional "thing" and you know it doesn't have any holes in it then you now know it has to be equal to a sphere. It SEEMS obvious, but it was still important to prove. It is important for many other proofs.

Better?

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Re:What about the Dunwoody paper?

[Darnit]

Dunwoody

It seems as if he missed a step and couldn't figure it out.

Re:What about the Dunwoody paper?

[rasafras]

It doesn't appear that the paper will survive the two years...

From the site:
It is unclear as of this writing if Dunwoody's proof will last even a fraction of that duration.

In fact, it appears that the purported proof has already been found lacking, judging by the facts that (1) the abstract begins, "We give a prospective [italics added] proof of the Poincaré Conjecture" and (2) the revised April 11 version of the preprint contains a small but significant change in title from "A Proof of the Poincaré Conjecture" to "A Proof of the Poincaré Conjecture?" In particular, a critical step in the paper appears to remain unproven, and Dunwoody himself does not see how to fill in the missing proof.

Re:What about the Dunwoody paper?

[King Babar]

So, why the excitment about this later Perelman paper? Has the Dunwoody paper been debunked?

A gap or three in the proof were found within days, and a mathematician friend of mine reported that it didn't look like solutions to these problems were immediately forthcoming.

The excitement about this paper comes from the fact that the guy who did the work has come up with impressive results in the past, builds on important and cutting edge work, and seems to have a really thorough command of the potential difficulties. (In other words, when he is asked questions about the tricky points, he immediately responds with what look like strong and well-thought-out answers.) For that matter, his work claims to prove a more general conjecture of which Poincare is a special case, and so this work could have more general significance to many other problems, even if there turns out to be a glitch or two in this iteration of the proof.

It's a very hard problem, and this answer could be wrong, too. But there's a big difference between tossing a paper up on a preprint server and giving a lecture at MIT where nobody can (yet) touch you. :-)

Re:What is it ? (Translation to make it easier)

[MarvinMouse]

translation to make it easier.

basically all the poincare conjecture says is that if you have a 3 dimensional figure which is closed (therefore, it it bounded (doesn't go off to infinity in either direction), and doesn't have any "holes" in it (like a donut)) then you can take every point and map it to a point in an equivalent sphere without losing continuity (therefore, everypoint will have the same "neighbourhood" of points as it had in the initial shape.)

ie. You can map a cube into a sphere, or a dodecahedron, or a weird globlike thing that doesn't fold back on itself, or a whole piece of paper (without holes), or a pencil, or a lot of different figures.

As well, this conjecture also handles figures with holes in them (like donuts), and maps them all to simpler figures.

It's a very simple concept, but has been incredibly hard to prove, and what makes this conjecture even more frustrating is the fact that 1 and 2-dimensional forms of this conjecture were incredibly easy to prove, as well as 4 and up have been solved, and were reasonably easy as well. Yet for some reason the 3 dimensional version does not lend itself easily to a simple proof.

Everyone generally believes this is true, but no one has been able to prove or disprove it.

If proven, this is an important aspect of topology, because then we can map all n-dimensional figures to a simpler form (like a sphere) and know that the continuity and general structure of the figure will remain the same.

Actually, Perelman is claiming much more...

[Anonymous Coward]

Perelman isn't claiming just to have proved the Poincare Conjecture -- he's claiming to have proved the Thurston's Geometrization Conjecture, a much more general (and harder to explain) result. Basically, while the Poincare Conjecture just says things about 3-spheres (namely every "simply-connected" 3-manifold is a 3-sphere), the Geometrization Conjecture says that _any_ compact Riemannian 3-manifold is built in a specific way from a handful of basic building blocks (the important thing here is that you're not just considering the manifold structure, but the metric structure as well).

Anyway, if true, this is kind of like Wiles proof of Fermat's Last Theorem -- proving an old conjecture by proving a more general (and more modern) one (in Wiles case, it was proving part of Taniyama-Shimura).