Elusive Proof, Elusive Prover: A New Mathematical Mystery

New York Times ^ | August 15, 2006 | DENNIS OVERBYE

[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?

[Vanders9]

EVERYTHING matters...

[wyattearp]

But is matter truly everything?

[StJacques]

ping

[garyhope]

I'm still working on how they got all those tomatoes in that little bitty can.

[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)

[HiTech RedNeck]

It takes a thousand pages of math to do that?

[neverdem]

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

Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.”

--snip--

In the early 1980’s Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein’s general theory of relativity and string theory, to investigate the shapes of spaces.

I don't think it should be discounted.

[ThePythonicCow]

Ah - from Clay Mathematics Institute comes this more accurate statement of this conjecture:

Poincaré Conjecture

---

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

[Greystoke]

What's a 3-Sphere?

[rawcatslyentist]

>"EVERYTHING matters..."

Even antimatter?

What about Cotton Matthers?