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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.
You show great skill at cut & pasting from http://www.claymath.org/prizeproblems/poincare.htm : )
Just kidding. Go ahead, enjoy the cut & paste karma.
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.
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
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.