Slashdot Mirror
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."
Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?
Game dev and music blog
For the lazy/paranoid.
For those who do not know about the Poincare Conjecture, copied from http://www.claymath.org/Millennium_Prize_Problems/ Poincare_Conjecture/
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.
but what do i know, i'm just a model.
Dunwoody
It seems as if he missed a step and couldn't figure it out.
Easy, i shall explain
The conjecture that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincaré (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n = 3.
The Poincaré conjecture has proved a thorny problem ever since it was first proposed, and its study has led not only to many false proofs, but also to a deepening in the understanding of the topology of manifolds (Milnor). One of the first incorrect proofs was due to Poincaré himself (1953, p. 370), stated four years prior to formulation of his conjecture, and to which Poincaré subsequently found a counterexample. In 1934, Whitehead (1962, pp. 21-50) proposed another theorem which proved to be incorrect, then discovered a counterexample (the Whitehead link) to his own theorem.
The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical, n = 3 (the original conjecture) remains open, n = 4 was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), n = 5 by Zeeman (1961), n = 6 by Stallings (1962), and by Smale in 1961. Smale subsequently extended his proof to include
you see ?, its all quite clear if you think about it
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.
webpage
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. :-)
Babar
Yah. Looks like NYT got wise to us. Replacing 'www' with 'archive' no longer works. Just redirects to the main page.
So here is the Google/NYT partner link
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.
~ kjrose
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).
Because if you are standing on a "good-old friendly" sphere then it looks 2-dimensional. This is describing a local property of the object, in other words, if you stand at any point on the sphere what is in your immediate viscinioty looks like a 2-dimentional disc. Simmilarly a circle is called a 1-sphere, because if you pick any point on it it what is around you looks like a 1-dimensional line segment(you could also call it a 1-dimensional disc). You can then carry this up to higher dimentions, so that a 27-dimentional sphere is a sphere that looks like a 26-dimentional disc locally. This way of thinking doesn't really break down until you get to an infinite dimentional sphere.