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A (Correct) Poincare Proof!?
aphyscher writes "About a year ago, there was an
announcement that M.J. Dunwoody had proved the (in)famous
Poincare conjecture.
His paper turned out to have a slight problem, and so it remained unsolved...
until perhaps now!
Sergey Nikitin has posted a preprint of what may perhaps be an actual proof."
Here is an example with all sorts of definitions you can read.
t ml
http://mathworld.wolfram.com/PoincareConjecture.h
Thanks,
--
Matt
Here is step 2! You win 1 million dollars for the correct proof from claymath!
h tm
http://www.claymath.org/prizeproblems/poincare.
A 2 dimensional sphere is one where the sphere is "locally" the same as a flat 2-dimensional plane. That's what we call a sphere. A 3-dimensional sphere is really a four-dimensional object, because it consists of those points in four dimensional euclidean space that are equidistant from the center of the 3 dimensional sphere. So, a 3-dimensional sphere looks "locally" like flat 3-dimensional space. Its hard to visualize.
All is Number -Pythagoras.
For super-geeks, here is is a more thorough discussion of the Poincaré Conjecture.
t ml
http://mathworld.wolfram.com/PoincareConjecture.h
Probably the simplest layman's explanation I can think of: if something feels like it is the 3-sphere, then it is the 3-sphere.
By "feels like" I mean that it has certain properties which strongly suggest that it is the real thing.
There's a $1,000,000 award for one thing...
This problem is priced at $1 million if solved.
3.243F6A8885A308D313
Simpy put, the poincare conjecture implies that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. (From Wolfram's MathWorld) Actually I don't know what that means, but having read and studied a bit about math, I can offer some explanation on the importance of such a proof. When a proof attempts to show that two algebraic structures are equal, as does this conjecture, it allows mathematicians the freedom to look at a problem in two ways instead of one. At last, a compact n-manifold problem can be safely regarded as an n-sphere problem and all the rules regarding n-spheres can be applied to certian n-manifolds. On another topic, these long-standing, but near-universally-believed-to-be-true conjectures are often assumed to be true in order to prove other theorems. i.e. a ground-breaking new primality testing algorithm ASSUMES the truth of the unproven Reimann Hypothesis. So, future encryption keys may rely on unstable hypotheses for their unbreakability.
This has a lot of implications for anything in 4d space.
Basicly before the proof you couldn't be sure what limits existed, now an extra limit has been placed on 4d environments.
The proof may also point the way to other proofs
thank God the internet isn't a human right.
I don't know which is worse; a problem like the Poincare problem, which has been definitively solved for 1-manifolds, 2-manifolds, and n-manifolds where n > 3, leaving only one little hole; or something like Femat's Last Theorem, which was solved for everything up to n equals a million billion and most numbers beyond that, before someone finally come up with a definitive proof.
The Poincaré Conjecture is widely considered the most important unsolved problem in topology. It was first formulated by Henri Poincaré in 1904. In 2000 the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution.
The conjecture is that every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere. (Loosely speaking, 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).
Analogues of the Poincaré Conjecture in dimensions other than 3 can also be formulated. The difficulty of low-dimensional topology is highlighted by the fact these analogues have now all been proven, while the original 3-dimensional version of Poincaré's conjecture remains unsolved. Its solution is central to the problem of classifying 3-manifolds.
On April 7, 2002 there were reports that the Poincaré conjecture might have been solved by Martin Dunwoody; on April 12 Dunwoody acknowledged a gap in the proof and was attempting to fix it. In October of that year, Sergei Nikitin of Arizona State University announced that he had proved the result.
(Note to reader: I'll ignore the obvious troll potential in that statement and go for the semi-serious approach that tapers out at the end) IIRC, he noticed the displacement of a fluid when a body is submerged in it. This lead to displacement of a goldsmith's head since it provided him with a method to test the density (and hence deduce the proportions of the different metals) of a newly manufactured golden crown for the King (whose name I have conveniently forgotten, let's hope no one knows who George Bush was two thousand years from now, but everyone has heard of Stephen Hawking).
Little known conjecture: If Alexander Graham Bell had been alive at the time, Archie would have forgotten the whole thing when he had to climb out of the bath to answer the phone. Let's decapitate telemarketers!
Money for nothing, pix for free
No, the proper name for a donut shape is 'torus'. An annulus is the figure bounded by and containing the area between two concentric circles.
We used to treat 'mathematics' as a plural, like you still sometimes hear data treated ("the data were tested..."). When lazy schoolboys and Cambridge ugrads abbreviated it to maths, they kept their plural, as changing it to a singular (at that time) would have felt odd. Now, we treat it as singular, but continue to call it maths. Obviously. One sheep, two sheeps, fish.
[FUCK BETA]
this is not quite right. when people say "three sphere," they mean the same object that you call a sphere in four dimensional space. however, it doesn't necessarily need to be embedded in any four (or higher) dimensional space. the important part is that viewed in isolation, it's got three dimensions of its own.
the "sphere" you know and love, we call the "two sphere."
- pal
Is it trying to prove that there can exist a 4-dimensional object that has all points equidistant from a single point in space-time or something?
Assume that you have a sculpture made of Play-Doh® modeling compound, without any holes in it. If the Poincaré conjecture is true, then you can reshape the sculpture into a ball without breaking or joining anything.
Will I retire or break 10K?
In the mathematical sense, the dimensionality of an object refers to how many dimensions the object itself has, not the dimensionality of the ambient space. A 2-sphere, denoted S2, lives in R3. However, if you examine the surface itself, it is a 2 dimensional surface (as in, a basis for the points on the 2 sphere has only 2 elements). When doing math, dimensionality is a property of the object, not of the ambient space. When you think about this, it makes sense, since there are plenty of examples of 2 dimensional objects which cannot be found in less than 4 examples, the most common of which is the klein-bottle (think of a torus, except it intersects itself if represented in 3 dimensions, since it's made out of a mobius strip instead of an anulus).
There's no sig like SIGSEG
What he's saying is, the...er...well, he means that the, uh...
Look at it this way:
Suppose the universe doesn't have any "edges" -- you can keep on going forever in a straight line without "falling off the edge of the world". Suppose further than there aren't any "wormholes" -- that given two paths between a pair of points, you can continuously deform one into the other. Finally, suppose that the universe is finite in volume.
Now, the first and third conditions above imply that the universe "folds in on itself". Add in the "no wormholes" condition, and Poincare's conjecture/theorem, and you find that there is only one possible way that it can fold in on itself -- as a hypersphere.
At least, that's the best explanation I can provide without any formal background in topology or astrophysics.
Tarsnap: Online backups for the truly paranoid
This is partially correct and partially misleading. The part that is misleading is to think of a n-sphere as necessarily being embedded in some other space. I think that is where UberQwerty gets confused.
A topological space is a set of points with the notion of a neighborhood of each point. If every point in the space has a neighborhood that looks like (homeomorphic) familiar 3-space then it is a 3-manifold. Similarly for n-manifolds. (Example: the ordinary hollow sphere is a 2 manifold because little sections of it look like a plane.)
Our ordinary experience (excluding relativity, string theory, etc.) says we live in a universe that is a 3-manifold.
Poincare says that a 3-manifold that is simply connected (e.g., able to draw a curve between any two points without going out of the space) and closed (any sequence of points that tend to a limit have that limit in the manifold) is actually topologically eqivalent to the set of points in 4-space equidistant from a given point.
So thinking of the apparent 3-dimensional universe, it doesn't have "holes" or weird twists like you can do in 2 dimensions on a Mobius band.
For example. You can draw a smooth shape (no sharp corners, no intersections) on a piece of paper. Assume you can come up with some equation that defines that shape. Then there exists not more than 2 equations that will transform (map) your original equation into a circle. Now the problem is prooving this in 3 dimensions. You start with a blob (actually just the surface of the blob), the blob has no holes and no sharp angles and doesn't intersect with itself (just your standard blob). There exists not more than 3 equations that will map your original blob defining equation into a sphere.
I'm no mathematition, but thats how I read the description. I think where this would be useful is in manipulating very complex shapes. You start with a shape, find the transformation equations. Manipulate a shpere (easy), then apply the inverse maping back to the original, and you get the result.
Like I said, I may be way off on this, but I did pass high school.
Thanks for modding redundant even though mine was posted 9 minutes before the last link. :(
Thanks,
--
Matt
Here is the importance of this conjecture. It's really about a 3-dimensional subset of 4-dimensional space, but think of the usual 2- in 3- situation if it helps.
Basically, if somebody gives you a twisted and knotted object, you would like to be able to say whether its really just a twisted and knotted sack (the sphere). It could in principle have any weird basic shape, you can't identify it when it's all twisted up. Showing the object is just a sphere, would require you to try to unknot it and smooth it out and say: look, I told you it was really just an ordinary sack.
In pathological cases it can be really hard to figure out how to undo all the knotting and twisting possible, and the case of dimension 3- in 4- was the only one still unkown.
So what you would like to do is not have to provide instructions for untangling the object, but rather just put it into a CAT scan, map its shape and perform some kind of calculation to verify it must be a sphere.
This is the homotopic equivalence of the conjecture. You can calculate the homotopy groups of the sphere. You can also calculate the homotopy groups of the weird twisted object. If they are equivalent, you don't have to go to the effort of unknotting the shape. Before you didn't know for sure it must just be a sack, but now you do!
Actually, that's assuming the proof holds up. Don't rush to judge these things so fast.
If you imagine that the Earth was a perfect sphere (it's not, but just for the sake of argument let's say it is) and that the equator was the rubber band. See how it slices the Earth into two bits?
Start sliding the equator up towards the geographical north pole.
Keep sliding. See how the total length of the "equator" has shrunk? See how there is one slice of the Earth that's bigger than the other? Imagine taking the top off of a boiled egg, if that helps... Slide some more.
Stop right there! You're just about to reach the north pole. Push it perfectly onto the north pole...
See? It is still on the Earth, but the "slice" of the Earth formed by the "equator" here is so thin that the "equator" now has zero length, and the second slice has no volume.
This, of course, requires a degree of perfection mere humans could neverachieve. I'm talking perfect perfection here. Not one merest of iota away from where it should be. Hey, it is theoretical...
Move the "equator" back anywhere near where it should be...
and it gets a non-zero length again. Push it even slightly further than the north pole...
And it is no longer on the Earth.
Congratulations on pinging the "equator" at the Sun, by the way. You've just annoyed every geographer on the planet. It looks like you've hurt the Sun as well... oh dear, it's going supernova! We're all going to die!
The preprint is posted on the arXiv.org web site, which is exactly that, a place to put preprints. Preprints that appear there have not been subject to peer review, so at this point, this is an annoucement of a result, which is very different than a number of mathematicians with the appropriate background agreeing that this is a proof.
The abstract from the arXiv is:
This paper proves that any simply connected closed three dimensional stellar manifold is stellar equivalent to the three dimensional sphere.
and the intro of the paper says that "Since every 3-dimensional manifold can be triangulated and any two stellar equivalent manifolds are PL homeomorphic, our result does imply the famous Poincare conjecture."
There is a nice front end to the math part of the arXiv from UC-Davis at this link
It's psychosomatic. You need a lobotomy. I'll get a saw.
If I recall correctly, assuming that the Reimann Hypothesis is true results in an algorithm that runs *faster* (has a lower polynomial running time), but they also provided an algorithm that is polynomial time *without* assuming Reimann hypothesis is true.
In other words, primality testing is in P -- unconditionally.
A.
Piece by piece:
By which he means: there is one equivalence set of loops through the manifold. Every possible loop (A path that returns to its beginning point. Duh.) in the manifold belongs to one set of loops that are pretty much the same - you could push any one of them around and get any other one. A sphere has this quality - any loop you draw on the surface of a 2-sphere (the one that exists in three dimensions), but a torus doesn't - there are the loops that are equivalent to the loop around the outside of the torus, and the ones that run through the hole.
What he's asking is, is it possible there could be an object in 4-dimensions, that has some kind of tangle in it such that you can't make it into a hypersphere, but isn't so mangled that there are fundamentally different ways to "draw lines" on it. The suggestion is fairly reasonable, I think.
For instance, any loop drawn on a plane is homeomorphic to a circle. You're options in connected finite 1-d manifolds amount to line segments, or cirles. But when you upgrade to 2-d, suddenly you've got holes. You can't have holes in 1-d and still be connected, but in 2-d you get donuts and dresses and honeycombs and whatnot. But the thing about a hole is that it means that your fundamental group have more than one set in it, and without a hole, you wind up being a sphere.
So why shouldn't there be another characteristic of 3-d objects, one which allows for more than one kind of simply connected, non-contractable manifold?
IP is just rude.
Is there any torture so subl
It's also interesting to note from his CV, that he's only an Associate Professor. ASU might want to make sure this guy's on a tenure track (if he wasn't before, I'm sure he is now.)
Also from his CV: "1996-t/n Linux Consultant in Arizona"
09 F9 11 02 9D 74 E3 5B D8 41 56 C5 63 56 88 C0
this is the property of a non-euclidean riemann geometry. suppose that you had a front yard, and you wanted to put a fence around it, to show it was yours. the yard is 2D, so the bigger the yard, the bigger the fence. however, since the flat surface of the earth curves and folds on itself as a sphere, you can own a yard the size of the earth and NOT need a fence, since there are no edges. the same thing applies here.
BSD is for people who love UNIX. Linux is for those who hate Microsoft.