Slashdot Mirror
The Poincaré Conjecture has Been Proved
Martin Dunwoody, a famous mathematician who works in the field of topology has a preprint that provides a proof of the Poincaré conjecture. This was one of the seven Clay Mathematics Institute millenium prize problems (reported on Slashdot here). The solution to each of the problems carries a monetary reward of 1 million dollars. However there are a number of conditions that still need to be met for the prize to be awarded in the case of the Poincaré conjecture.
If you follow the link to the description of the problem, it gets really wierd. Apparently this is one of those problems where you have to prove it for 1=7} but no one ever managed n=3 (which was the original, non-generalized conjecture anyways). Funny that this guy just had to fill in the last blank.
I think Mauve has the most RAM. --PHB (Dilbert Comic)
The Poincaré Conjecture proved, and microsoft ads on slashdot
"I think it would be a good idea" Gandhi, on Western Civilisation
You show great skill at cut & pasting from http://www.claymath.org/prizeproblems/poincare.htm : )
Just kidding. Go ahead, enjoy the cut & paste karma.
Wow. I wish I could highlight a section of your post to point out as being wrong as you did the the grandparent. Unforutnately, I can't. You're wrong all throughout.
First, how do you show something is proven? Well, you give a proof. How do I know the proof is correct? I work through all the steps... But what if I mess up and sneeze and my thinking gets confused and I accept something that isn't true? It could happen. Well, I'll just push it through a formal logic computer program that checks it.
But what if the computer has a glitch and a 0 or a 1 gets accepted. Or worse, I made the error while programming the formal logic system. Or more subtly, the compiler or hardware.
Basically, it's like this, proofs are as much a social event as a mathematical cedrtainty. Proofs are presented, and believed, and then refuted. Mathematical proof is a social process carried on by mathamaticians, and you can't forget that. I'm sure that I've proved things incorrectly before, and believed them. Just because nobody's found an error in a published and accepted proof doesn't mean one doesn't exist. If you think that humans can do ANYTHING with probability 1, you're sorely mistaken and are seeing the world in too convenient terms.
Sorry to burst your bubble, but there's a lot of thinking in this. Peer review does not imply flawlessness.
Which is exactly what they've done in the paper. They've depicted on how the mesh could possibly collapse.
:-(
They have depicted an 8-gon curve which satisfies the intersection properties, extrapolate using a 2 vertex model and use that to show the possible collapse. They've not depicted the collapse per-se in action tho.
That being said, Martin Dunwoody is a remarkable researcher and this work relies on important, ground-breaking work of Abby Thompson and Hyam Rubenstein, and this preprint sounds very promising!
It's psychosomatic. You need a lobotomy. I'll get a saw.
Surely it should read:
The conjecture that every *compact* simply connected 3-manifold is homeomorphic to the 3-sphere,
Normal euclidean space R^3 is simply connected,
and definitely NOT homeomorphic to to the
3-sphere !!
(That they are not homeomorphic can be proved by
comparing their homotopy or homology groups).
Liam.
http://www.math.princeton.edu/jfnj/texts_and_graph ics/erratum.txt
-Kevin
Maybe we should give these problems to the people at the next ACM International Programming Contest.
No, x/0 is undefined. However, you can do things like
because, when y approaches zero, x/y will obviously become larger. But that is not the same as0*infinity is undefined, however, continuing the example above, I could write:
i.e. in that example, "0*infinity" would be zero.
The problem with infinity is that you can't use it like a number, because it isn't one. Infinity literally means that there is an infinite number of things, e.g. the set of integers is infinite, meaning you can never list all integers because there is always a successor. You'll never "arrive at infinity" when listing integers. This means you can calculate with infinity only with equations that involve sequences and their limits. (Like the above-mentioned lim y->0 which means that y is a sequence of numbers approaching zero, and not y = 0. A suitable sequence might e.g. be y[n] = 1/n with n = 1, 2, .... Obviously, this sequence is approaching zero, but will never be equal to zero.)
Sig (appended to the end of comments I post, 54 chars)
I'm somewhat familiar with this proofs used in different dimension ranges. It's absolutely necessary to separate out the proof into separate cases because the topology changes wildly with dimension. Roughly speaking in dimensions 4 there is so much room that certain powerful general techniques become possible (essentially, half the dimension of the manifold is more than 2 dimensions away from the full dimension --- so submanifolds of half the dimension cannot be KNOTTED). In dimension 3 and 4 special techniques must be used (and they are different in each case). In dimension 4, a submanifold of half the dimension (i.e, 2) can be knotted in the full manifold, but one can analyze the types of knotting that occurs. Manifolds of dimension 3 need techniques UNIQUE to this dimension (incompressible surfaces, etc.). The case of dimension 3 has been the hardest.
He also has some other books on more advanced topics in algebraic topology, in various stages of completion, but I haven't read those yet.
4 (Insightful)!? Almost every statement in this post is incorrect.
The description of simply connected is a description of connectedness. Simply connected means your space doesn't have holes in it, in addition to being connected. This is required, since there are obviously 2-D surfaces (think of donuts) that are connected, yet not homeomorphic to a 2-sphere.
A manifold is a space that is locally homeomorphic to Euclidean space. i.e. if you take a very small piece of the space near a point, it looks like a small piece of R^n. A figure 8 curve is an example of a 1-dimensional space that is not a manifold.
Homeomorphic means that there exists a bicontinuous (continuous in both directions) one-one correspondence between the spaces.
Compactness has precisely nothing to do with surface areas and volumes. If an objects surface area is as small as it can get wrt its volume, it's a sphere, and this has been known for a long time. Secondly, circles are 1-D, not 2-D.
Intuitively the notion of compactness corresponds to being `finite'. In R^n, a set is compact if it is closed (i.e. contains its boundary) and bounded (doesn't stretch off to infinity). The general definition of compactness is more hairy: one way of stating it is that every infinite sequence in the set has a convergent subsequence (note that the limit also has to be in the set).
What the Poincare conjecture states, roughly, is that any closed bounded d-dimensional object in R^n that doesn't have any holes in it (this makes it homotopy equivalent to a d-sphere) is actually homeomorphic to a d-sphere. (Note: it's non-trivial to prove that a compact d-dimensional manifold can actually be embedded in R^n for some n).
Be careful how you phrase that last sentence - your carefree use of the word "obvious" in reference to math calls to mind an old joke:
Two mathematicians were talking one day about some recent work they'd done. One described a proof to the other but quickly glossed over a complicated step. The second one said, "Wait a minute - you didn't prove your last assertion." The reply: "It's obvious."
So the second mathematician wordlessly took a piece of chalk, went to the nearby blackboard, and began to fill it with long statements full of obscure symbols. Nearly half an hour later, he stopped writing, turned around, and said, "You're right. It is obvious."
As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.
-- Albert Einstein
Really, we do have proofs in physics(for example) that are just as provable as those in mathematics. You just have to understand that proofs of any kind are made based on certain assumtions (axioms + rules of logic).
For instance, the quantum no-cloning theorom states that you cannot exactly duplicate an unknown quantum mechanical state. This is an absolutely proven theorom -- one of the axioms of which is the Schrodinger equation. If we ever find that quantum mechanics is not the correct description for our universe, the no-cloning theorom will still be entirely valid within the constructs of QM, as well as the regime of the universe under which QM is applicable.
Likewise, Euclid said the sum of the angles of a triangle is Pi, but this is only true for trinagles in spaces that have a certain structure, which is why we call it Euclidian. It turns out that in general, space is non-Euclidian, though unless you are near a black hole or a neutron star, the difference is hardly noticable.
Computer scientists have "proven" using very general methods, that there are no algorithms for computing certain things that are faster than a given bound -- There is no way to search an unordered list in faster than O(N) time, no way to sort arbitrary numbers in less than O(N*Log(N)) time, etc. However, this is based on a Turing machine model of computation, and the laws of quantum mechanics as we understand them allow computers intrinsically more powerful than a turing machine. We still don't understand much about what these quantum computers can and can't do better than a classical computer, but we do know that they can search unordered lists faster than any classical computer, though I think it has been shown that they cannot sort lists faster than a classical computer.
This reminds me of another anecdote - which I believe is true. I don't recall who it is about, though. The story is that at a seminar, a respected mathematician was giving a proof when someone questioned one step. The speaker said, "it is clear," and moved on. A bit later, he turned back to the questioner and said "it can be shown," then continued once more with the talk. A few minutes later, he paused, thought for a few seconds, turned to the questioner, and said "It is well-known." Moving on with the argument, a few minutes later he paused again, turned once more to the questioner, and said: "It is wrong."
It's always easy to take things for granted that look obvious; to some extent one always has to do this. The trick is knowing when you can do it and be right.
Matt Reece
Main Entry: prove
Pronunciation: 'prüv
Function: verb
Inflected Form(s): proved; proved or proven
You can say it either way. It's standard usage. Idiot.