Elusive Proof, Elusive Prover: A New Mathematical Mystery

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

[neverdem]

Grisha Perelman, where are you?

Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.

After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.

Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.

As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.

“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”

In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.

Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”

But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.

Also left hanging...

[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?