[neverdem]

Xianfeng David Gu and Shing-Tung Yau
Even topologists don’t think this soap film can be made into a sphere.

Xianfeng David Gu and Shing-Tung Yau
To a topologist, a rabbit is the same as a sphere. Neither has a hole. Longitude and latitude lines on the rabbit allow mathematicians to map it onto different forms while preserving information.

Graphic: The Essential Grisha P.S. Enlarge the graphic to read it.

[HisKingdomWillAbolishSinDeath]

Nerd Nirvana.

[BreitbartSentMe]

bookmark for later

[RussP]

I'd be interested to know if this proof has any practical implications or uses.

[Mobile Vulgus]

Cool story.

I am excited that a new world has opened up for mathematicians. That means new worlds open for the rest of us down the line.

Not that I understand a THING about it, mind you!

[pcottraux]

Og bored!

[msnimje]

I am still working on this one...

[beaversmom]

Huh?

[ThePythonicCow]

The Poincare conjecture states (roughly) that 3-dimensional balls are, give or take some stretching, the only 3-dimensional object that has no holes. This conjecture has been generalized from 3 dimensions to N dimensions, but for the higher dimensions, it was already proved. Grigori Perelman proved it for the case of N == 3, which was Poincare's original conjecture.

[StJacques]

This is from: http://www.answers.com/topic/poincar-conjecture :

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology — what would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere.

Poincaré claimed in 1900 that homology, a tool he had devised and based on prior work of Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere. In a 1904 paper he described a counterexample, now called the Poincaré sphere, that had the same homology as a 3-sphere. Poincaré was able to show the Poincaré sphere had a fundamental group of order 120. Since the 3-sphere has trivial fundamental group, he concluded this was a different space. This was the first example of a homology sphere, and since then, many more have been constructed.

In this same paper, he wondered if a 3-manifold with the same homology as a 3-sphere but also trivial fundamental group had to be a 3-sphere. Poincaré's new condition, i.e. "trivial fundamental group", can be phrased as "every loop can be shrunk to a point".

The original phrasing was as follows:

Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

Poincaré never declared whether he believed this additional condition could distinguish the 3-sphere, but nonetheless, the statement that it does has come down in history as the Poincaré conjecture. Here is the standard form of the conjecture:

Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere. Loosely speaking, this means that if a given 3-manifold is "sufficiently like" a sphere (most importantly, that each loop in the manifold can be shrunk to a point), then it is really just a 3-sphere.

They never say too much about what Perelman actually did.

[HiTech RedNeck]

Rabbits have five holes (or more)

[snarks_when_bored]

Allyn Jackson, "Conjectures No More?: Consensus Forming on the Proofs of Poincaré and Geometrization Conjectures" (PDF)

[WestVirginiaRebel]

"I wish I'd read that book by that wheelchair guy!"

[lmr]

This is all fascinating. All Mathematics can be explained through nature in relation to pi. Duh...

[stands2reason]

I like pictorials.

Have you ever seen a one sided figure?

[Blue_Ridge_Mtn_Geek]

It should be noted that this proof may yet have a hole [no pun intended!], and if so it may take some time to discover it. There are precedents, e.g., Wiles agonizing extra year of work to plug the hole found after his initial announcement of a solution of the "Fermat Theorem" (for which he won a Clay Prize since that is one of the Millenium Seven):

http://en.wikipedia.org/wiki/Fermat's_last_theorem
(sorry, the syntax is correct, but the link generator must have a bug - - cut and paste the entire URL into your browser if you want to check out the reference)

"The story of the proof is almost as remarkable as the mystery of the theorem itself. Wiles spent seven years working out nearly all the details by himself and with utter secrecy (except for a final review stage for which he enlisted the help of his Princeton colleague, Nick Katz). When he announced his proof over the course of three lectures delivered at Isaac Newton Institute for Mathematical Sciences on June 21-23 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious error was discovered: it seemed to lead to the breakdown of this original proof. Wiles and Taylor then spent about a year trying to revive the proof. In September 1994, they were able to resurrect the proof with some different, discarded techniques that Wiles had used in his earlier attempts."

[Blue_Ridge_Mtn_Geek]

Sorry, one more reference I can't resist sharing, then to bed:

http://math.stanford.edu/~lekheng/flt/
Bluff your way in Fermat's Last Theorem

[camle]

"To a topologist, a rabbit is the same as a sphere. Neither has a hole."

wonder how a rabbit eats, etc. if it ain't got no hole?

[cricket]

There seems to be no concern espressed here; so am assuming 'someone' knows where Perelman is and that there is no reason for alarm at the 'lack of sightings' or the fact he does not respond to his e-mail. . .

. . .and for sure, the man knows his mushrooms. .

[Lil'freeper]

Pingy!

[Red Badger]

I think therefore I FReep.............