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...

[ThePythonicCow]

They never say too much about what Perelman actually did.

More accurately, from Wolfram Mathworld comes this statement of what he actually did, which is establish Thurston's Geometrization Conjecture's, a generalization of Poincare's Conjecture.

Thurston's Geometrization Conjecture

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds.

Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.

The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."

Thurston's conjecture is that, after you split a three-manifold into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:

1. Euclidean geometry,

2. Hyperbolic geometry,

3. Spherical geometry,

4. The geometry of ,

5. The geometry of ,

6. The geometry of the universal cover of the Lie group ,

7. Nil geometry, or

8. Sol geometry.

Here, is the 2-sphere (in a topologist's sense) and is the hyperbolic plane. If Thurston's conjecture is true, the truth of the Poincaré conjecture immediately follows. Thurston shared the 1982 Fields Medal for work done in proving that the conjecture held in a subset of these cases.

Six of these geometries are now well understood, and there has been a great deal of progress with hyperbolic geometry (the geometry of constant negative scalar curvature). However, the geometry of constant positive curvature is still poorly understood, and in this geometry, the Thurston elliptization conjecture extends the Poincaré conjecture (Milnor).

Results due to Perelman (2002, 2003) appear to establish the geometrization conjecture, and thus also the Poincaré conjecture. Unlike a number of previous manuscripts attempting to prove the Poincaré conjecture, mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any mistakes (Robinson 2003).

[ThePythonicCow]

A 3-sphere is a solid ball - or warped equivalent. In 4-dimensional space, the set of points one unit from the origin is one example of a 3-sphere, just as in our ordinary 3-dimensional space, the set of points one unit from some point is an example of a 2-sphere that is more commonly called a circle.

[bruinbirdman]

I have detected a flaw in Dr. Perelman's work. You have 100 years to prove me wrong.

yitbos

[ThePythonicCow]

Rabbits (and humans) don't have five holes. If you model the alimentary canal, from mouth to anus, they have one hole. Indentations don't count as holes. If you model the sinus cavities and nasal passages, and their connection to the alimentary canal, then they get holier - I guess you might be able to get to five holes that way. I'm no otorhinolaryngologist, so couldn't say for sure.

[snarks_when_bored]

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

[Vanders9]

ooh...philosophy!

If we are to believe Einstein, then energy and matter are interchangeable. This however, does not account for the transphysical and supernatural, so I would concur with the opinion that matter is not everything.

The point is moot however, as my use of the term "matters" is in its alternate meaning of "having significance", rather than that of "corporeal existance". I suspect that you were aware of that point prior to your reply, but chose to ignore it for jocular reasons.

:)

[Vanders9]

Do they?

Cats only have two holes in their skin. I cant believe rabbits are more endowed.

[Vanders9]

...now you're being silly...

[HiTech RedNeck]

Don't forget the tear ducts. And if you want to get really technical, the porous skin.

[bruinbirdman]

"A 3-sphere is a solid ball - or warped equivalent. In 4-dimensional space, the set of points one unit from the origin is one example of a 3-sphere, just as in our ordinary 3-dimensional space, the set of points one unit from some point is an example of a 2-sphere that is more commonly called a circle."

Give that man $1,000,000; if you can find him.

yitbos

[Vanders9]

Its their EYES!!


ooh...what you were all thinking!

[rawcatslyentist]

I'm havin a BALL !!!

[Blue_Ridge_Mtn_Geek]

"Rabbits (and humans) don't have five holes. If you model the alimentary canal, from mouth to anus, they have one hole. Indentations don't count as holes. If you model the sinus cavities and nasal passages, and their connection to the alimentary canal, then they get holier - I guess you might be able to get to five holes that way. I'm no otorhinolaryngologist, so couldn't say for sure. "
- - - - - - - - - - - - -

That's true, there are only four holes (mouth, anus, and the Eustachian tubes (containing our hearing apparatus to the hole connecting the mouth and anus). The "hole" in mammals from which urine emits, and which serves the reproductive function does not connect to these other passageways,and does not exit at some other point so it is only an indentation, not a true hole. Thus the topology of the mammalian body surface (ignoring, as you say, the possibility of multiple true holes in the sinus cavity) is topologically equivalent to a donut with four holes.

Is there an anatomy specialist in the house who can comment on the topological character of the sinuses, and the lungs which could also have large numbers of true holes (in the topological sense)?

[ThePythonicCow]

the Eustachian tubes

I don't think the Eustachian tubes go all the way through, unless you have a burst ear drum. So they are long skinny indentations, not holes.

[Blue_Ridge_Mtn_Geek]

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

Poincaré Conjecture

= = = = = = = = = = = = = = = = = = = = = = = = = =
Yes, one of the implications for the author, if this proof holds up and is published in a vetted math journal, is a Clay prize of $1 million. The Poincare Conjecture is one of their "Millenium Problems". For the other six, with formal statements of all seven, see the Clay Institute site (Google it!).

[ThePythonicCow]

mouth, anus

And the mouth and anus are not two holes, but two ends of one hole, the alimentary canal.

[WestVirginiaRebel]

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

[Blue_Ridge_Mtn_Geek]

the Eustachian tubes

I don't think the Eustachian tubes go all the way through, unless you have a burst ear drum. So they are long skinny indentations, not holes.

= = = = = = = = = = = = = = = = = = = = = = = = = =
Perhaps so. I thought the ear drums were suspended on bones, but didn't necessarily cover the entire passage (but I'm no expert on this, just recalling some high school anatomy studies from 40+ years ago). Is there another here familiar with mammalian anatomy who can help us out?

[ThePythonicCow]

The ear drum covers the entire passage -- that's why your ears pop when your airline flight comes in for a landing. The air in the outer ear cannot quickly equalize pressure with that in the middle ear. The ear drum separates the outer and middle ear canals.

[Blue_Ridge_Mtn_Geek]

"It takes a thousand pages of math to do that?"

= = = = = = = = = = = = = = = = = = = = = = = = ==

That's one of the fascinating things about some of the "really hard" math questions - - -fairly easy to state, in a way that non-specialists can understand, but brutally difficult to answer rigorously, requiring the development of whole new mathematical apparatuses and methods to solve.
Two recent (late 20th century) examples are the solutions of "Fermat's Last Theorem" and the "Four Color Problem" (Google for details).

Invariably the spinoff results from these new techniques and insights are far more fruitful than simply the resolution of the question that is the object of the investigtion. That's why military establishments and modern technological enterprises commit a certain part of their budgets to funding "pure math research". See

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html

for two papers, one by a physicist and the other by an applied mathematician, both called "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" for further discussion of this point.