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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."
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
:)
Rational thought is the only true freedom
Only two years more of eating noodles before he's rich!
The link to mathworld.wolfram.com from the post says:
So, why the excitment about this later Perelman paper? Has the Dunwoody paper been debunked?
You win again, gravity!
The subject of 3 dimensional objects with holes is quite fascinating... wouldn't it be awesome if it was discovered that toroids are actually some extradimensional manifestation... Or even that Toroids have special properties allowing FTL travel...
Food not Bombs is a nice platitude but it breaks down when you notice that the Bombees are usually well fed
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.
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
"Though his early work has earned him a reputation as a brilliant mathematician, Dr. Perelman spent the last eight years sequestered in Russia, not publishing."
"However, according to the rules of the Clay Institute, the paper must survive two years of academic scrutiny before the prize can be collected."
So, all told, Perelman is going to wait a total of 10 years from the time he started to work on the solution to the Conjecture, to the time where the scientific community lets him know if his answer is correct. Wow.
Complex mathematics? Looks like its time for Matt Damon and Pretty-Boy Affleck to write Good Will Hunting II.
http://www.theinformationminister.com/press.php?ID =612212491
we got this ages ago. i swear
Silly people... this is TOPOLOGY! It's not meant for people to USE it! It's just for mathematicians to RUMINATE UPON!
Has Fermat's Last Theorem actually been used in practical applications? I don't think so...
If everyone thought like you we'd still be living in caves.
Just because practical applications aren't totally obvious for a layman (or even a matematician) doesn't mean this will never be of practical use.
Even if no practical applications are ever found, this proof (if it survives peer review) may well pave the way for something else that is immensly useful.
There's just no way to tell right now.
"First lesson," Jon said. "Stick them with the pointy end."
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
Mathematical rigor demands we see the proof first. One decade ago Wiles thought he solved Fermat's Last Theorem but a mistake was found and worked again for several months before ultimately solving it. Faulty proofs are made all the time. Until it undergoes peer review I will be very skeptical.
It appears most people are spelling incorrectly! Including the sites included in the post!
..... its "mathemagician"
:)
It is not "mathematician"
Please make the appropriate corrections.
[I can picture a world without war, without hate. I can picture us attacking that world, because they'd never expect it]
Me. Hammer. Pliers. Every available 3-manifold. Can I have my $1 million please?
(This of course assumes that 3-manifolds are malleable.)
Note to M1-ers: a curt but otherwise insightful message is not "Flamebait" or "Troll".
I thought that this Wolfram guy was the smartest man in the universe and had all the answers. Now some brie-muncher comes along and proves something in math that Wolfram couldn't? This can only be due to one of three reasons:
1. When Wolfram and Hart were all killed by the Beast, Wolfram was in the house.
2. Wolfram is human and isn't as smart as the papers say.
3. He stepped up to MCHawking and is now hanging from a tree with a sign pinned to him that reads: WHACK EMCEE.
If Slashdot were chemistry it would look like this:Cadaverine
I swear that looks like perl.
By reading this comment, you immediately waive any and all rights regarding it.
Math is one of those disciplines where you just can *not* skim the problem and expect to understand it... you have to load into memory every word that is in the text (like 'manifold' etc), and create a working instance of that object in your brain...
It's basically like launching a heavy app like Photoshop.
So yeah, to answer you: even when I was right in the middle of studying this stuff, there were moments when I would think I was stupid too... but if you concentrate *and* you know what they're talking about, it makes sense.
Conclusion: it's knowledge, not intelligence.
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).
> Now, can someone tell me what practical applications
;-)
> there might be of this? Or is it strictly an abstract concept?
Speaking as a layman, the practical application of these sorts of proofs is that you can use them to prove equivalent, more practical questions.
One of the references in another comment explained that this conjecture has been proved for all other dimensions, and this 3-sphere seems to be a special case, as far as proof is concerned.
If the Poincare' conjecture were proved, then the general case could be solved. After that, "simply" proving that another hard problem is equivalent to the Poincare' conjecture is enough to prove that other problem.
Now, I've heard the problem described with a lasso instead of with a rubber band. I can imagine times when I'd really like to know when my lasso is going to close around something or if it's just going to slip off
Are you on a computer right now? Ever heard of a guy called George Boole? Does a "boolean" sound familiar? Well you see, this guy called George Boole he hated mathematicians so much he decided to invent this thing called Boolean Logic. You know the, 1 & 1 == 1, 1 || 0 == 0 stuff? As it turns out it was totally useless and that's what he intended, to invent something mathematically correct that is totally useless. So thanks to George Boole for accidentally inventing the foundation of computer architecture, logic gates and boolean logic - and he has something to do with you being on the computer right now. Indeed he is pissed off as he intended it to be useless.
Give maths time and it will applicable to your everyday life. What has been going on for the past 3,000 years?
Analytic & algebraic topology of locally Euclidean meterization of infinitely differentiable Riemmanian manifold
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.
100 years ago a proof of the difficulty of factoring large numbers might only have been interesting to mathematicians. Now that we use encryption based on the difficulty of factoring products of large primes, it's very important.
Galois fields are used for checksum algorithms, something I'm sure Galois never thought of.
Fourier transforms are used for image compression (JPEG).
Who knows what Poincaré's topology might be used for in the future?