Perelman, Poincare Conjecture solved now?

[Clausius2]

Perelman, Poincare Conjecture solved now??

Seems that this guy has solved the Poincaré Conjecture:

http://en.wikipedia.org/wiki/Grigori_Perelman

He is supposed to get the Fields Medal in Madrid this year, in the next international congress of mathematics. But it is likely that he won't show up.

Is anybody able to give a plain explanation of the conjecture to an engineer?

Thanks in advance.

 

[matt grime]

Yes, many exist out there. The New York times carried one explanation this week, and the clay mathematics institute has explanations of all the millenium prizes aimed at the 'general public' on its website. These are both easily accessible, though I believe the NY Times wants registration before you can read some things, or at least they used to; they may have stopped this silly practice by now. (Incidentally, the news that he did this came to light 2 years ago. If you look at the Clay prize material you will see why this is important, if you do not know already.) In anycase, he is a lot of a recluse and certainly won't collect any of the prizes in person, and has shown little interest in the prizes fullstop, which I think is a refreshing change.

 

[Clausius2]

Thanx matt, I've looked at Clay's webpage. I have found an edible explanation. From the point of view of a mathematician, what is the utility of the discovery? We better don't talk from the point of view of an engineer...

 

[matt grime]

I don't feel the particular need to justify any mathematical research on grounds of utility to the real world, or even what an engineer thinks the real world is. In fact the geometrization conjecture, which Perelman is supposed to have proven probably has implications to the 'engineer's real world', just as a lot of things in 'pure' mathematics do. Surely you know that there is no nowhere vanishing vector field on the sphere, but can you prove it? I know it follows from the vanishing of certain cohomology groups, for instance, but I suspect you'd treat those ideas similarly as being things not of utility to an engineer.

Let me explain a bit more lest you think I am accusing you of dismissing mathematics as pushing around symbols meaninglessly; I am not.

The thing is I think you are overlooking the fact that most of the underpinnings of engineering are solid abstract mathematics, and such a result as this might well have implications in such areas as those which wish to model things using 3 dimensional manifolds.

Now, as for the mathematics of it: the result is old, and has challenged some of the finest minds in mathematics. That they have not found a result and someone else has means that this person has probably just invented a very powerful new technique (or realized how to use an old one in a different situation). It is not the direct implication per se that is important in mathematics: the Clay prizes were not chosen because the solution of them would revolutionize mathematics in and of itself. After all we are free to assume that they are true and make deductions from this position. They were chosen because it was deemed that any solution to them would require ideas that would revolutionize parts of mathematics. As an example one of the biggest 'myths' is that if the Riemann Hypothesis is true internet banking (or cryptography in general) is doomed. That is patent nonsense. Here, I assume it is true, now, does that do anything? No. I can assume all I want about the distribution of the primes but it won't do much. However, a proof of RH might (should in the opinion of the Clay Institute) contain some ideas of fundamental importance to mathematics.

 

[Clausius2]

Hey matt.

Turns out that you are talking to an unusual engineer, whose main tool in his thesis is applied mathematics to fluid mechanics, such as asymptotic analysis. I don't diminish this stuff, and I was not actually being ironic when I asked the question. For me I think the solution itself is not extraordinary useful, but I envisage that while working out the solution thousands of mathematicians have looked carefully at the structure of threedimensional manifolds and have brought light to that part of the science.

 

[mathwonk]

one basic problem in mathematics is classification: i.e. how do you recognize and describe something you have encountered. the poincare conjecture tells you how to recognize a sphere, but one dimension up from the usual sphere, i.e. a CONNECTED space X is topologically the same as the 3-sphere defined by x^2 + y^2 + z^2 + w^2 = 1, if and only if every point of X has a nbhd that is topologically the same as a ball (x^2 + y^2 + z^2 < 1), and IF EVERY loop in the space X bounds a disc in the space X. i.e. every simple closed curve in X can be shrunk to a point with passing out of the space X.this took 100 years to resolve.

 

[asar]

Is Spacetime Quaternionic?

one basic problem in mathematics is classification: i.e. how do you recognize and describe something you have encountered. the poincare conjecture tells you how to recognize a sphere, but one dimension up from the usual sphere, i.e. a CONNECTED space X is topologically the same as the 3-sphere defined by x^2 + y^2 + z^2 + w^2 = 1, if and only if every point of X has a nbhd that is topologically the same as a ball (x^2 + y^2 + z^2 < 1), and IF EVERY loop in the space X bounds a disc in the space X. i.e. every simple closed curve in X can be shrunk to a point with passing out of the space X.


this took 100 years to resolve.

Thanks for a clear and concise explanation. Looks like a X is a unit quaternion! I have maintained that our Cosmos is quaternionic, that is a point in spacetime is p=ct + ix + jy + kz = ct + r, where c is the speed of light; 'ct' the scalar dimension and 'r= ix + jy + kz' represent the three vector dimensions.

p^2 is also a quaternion:
p^2 = ((ct)^2 - r^2) + 2(ct)r = (ct)^2(1-(v/c)^2 + 2v/c).

The norm is a 3-sphere.
pp* = p*p = (ct)^2 + x^2 + y^2 + z^2