Poincare Conjecture Proof Completed

Flamerule writes "A New York Times article has finally provided an update on the status of Grigori Perelman's 2003 rough proof of the Poincaré Conjecture. 3 years ago, Perelman published several papers online explaining his idea for proving the conjecture, but after giving lectures at MIT and several other schools (covered on Slashdot) he returned to Russia, where he's remained silent since. Now, mathematicians in the US and elsewhere have finally finished going over his work and have produced several papers, totaling 1000 pages, that give step-by-step, complete proofs of the conjecture. In addition to winning some or all of the $1,000,000 Millennium Prize, Perelman now seems to be the favorite to receive a Fields Medal at the International Mathematics Union meeting next week, but it's not clear that he'll even show up!"

Comments

A rabbit is a donut, not a sphere.

[Vellmont]

What kind of strange rabbits have these topologists seen? The rabbits I've seen have a hole from end to end through them called the digestive tract.

The tone of the summary is typical

[blueZ3]

The incredulity that this mathematician might have been more interested in the challenge of the work than fame and fortune in the Western world practically oozes from each sentence.

I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake? Perhaps he's gone on to other challenges, or he's wrapped up in some research that has his complete attention. Heck, perhaps he just enjoys math for its own sake and doesn't want to deal with all the side-effects of notoriety.

Re:The tone of the summary is typical

[smallpaul]

I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake?

It isn't a shock that he did it for its own sake at all. Look at the thousands of open source programmers. The shock is that he's been given a million dollars and seem uninterested. Linus Torvalds does Linux for its own sake but if someone gave him a million dollars, he'd take it. Even someone who is not materialistic might think: "hmmm. A million dollars might help many Russian orphans or deliver AIDS drugs to Africans or ..." It is strange for a single person to be neither greedy, nor ambitious nor altruistic ... merely obsessed.

Yes, that's strange. It's rare and therefore strange.

Re:The tone of the summary is typical

[aiken_d]

Oddly enough, people tend to form their expectations based on past experiences. Is it so unreasonable for the tone of the article to be incredulous when the situation is unprecedented?

Where you see value judgments and a jaded reporter, I see a pretty reasonable surprise. I don't see anything in the article where the reporter suggests that Perelman "should" do anything other than what he is. Surprise, and remarking on an unusual behavior, is *not* approbation.

-b

Re:The tone of the summary is typical

[eddy]

I think that never is this more amply examplified than when the people who manage 'rights holders' "explain" how, if it weren't for copyright, there would exist no art.

Re:The tone of the summary is typical

[Thisfox]

Sadly, yes, doing something for it's own sake rather than for monetary gain is frowned apon, and sometimes viewed with fear and confusion, not that I'm saying this review goes THAT far (if you don't believe me, try smiling at someone while in a subway one of these days: the person will generally check that you haven't got someone stealing their wallet while they are distracted. Or busk without a hat out: no one realises that an orchestral musician might just enjoy playing music in the sun in winter, and they search madly for a way to throw a coin into my closed music case). Perhaps he sees the money as a complication rather than a useful item: instead of assuming he could donate it, there would be all the trouble of getting the money into his country, bank balances, taxes, and more questions and papers to fill out to get it donated, and all the rest of it. All of which is time he could have been spending on solving another interesting question, or gathering mushrooms, or whatever. Coming into a fortune is not always fortunate.

Recognition = Worry

[BoRegardless]

"Perelman now seems to be the favorite to receive a Fields Medal at the International Mathematics Union meeting next week, but it's not clear that he'll even show up!"

The curse of the gifted is that niggling worry in the back of the mind that if one accepts praise, one may lose his focus, drive or muse, if you will.

Re:TFA is well worth reading

[OldManAndTheC]

Side note: the Millenium Prize is a cool million. Which is $24 million less than Adam Sandler makes per movie.

Hurray for the free market! The true value for a personal accomplishment has once again been properly determined and awarded!

name change?

[bark]

Now that the conjecture is proved, do they change the name to "theory"? Or does the name stay put because that's what everyone knows and refers to it as?

Re:I remain skeptical

[spuzzzzzzz]

First of all, I highly doubt that all of those 1000 pages are devoted to solving the Poincare Conjecture. Perelman, if I remember correctly, studies Ricci curvature flows which is a large area of mathematics in its own right. In the course of his research, he discovered some things that led to this proof of the Poincare Conjecture. I would expect that the 1000 pages referred to by this article deal with many different consequences of Perelman's work. Mathematicians like to do things in full generality, so they would have studied broader consequences instead of focussing for so long on only one result.

Secondly, I would invite you to write down a complete proof of some well-known mathematical fact, the Stone-Weierstrass theorem say. You must prove this from first principles, starting with axiomatic set theory. I would be very surprised if you even managed to finish and even more surprised if the proof came in at under 1000 pages. This highlights what was mentioned by a sibling of mine: mathematics is divided into small steps and you would never dream of trying to prove something all at once.

Thirdly, this is the first ever proof of the Poincare conjecture. It is quite common in mathematics that a nicer proof of a known fact will be found.

On the contrary...

[moly]

A Scottish physicist two centuries ago sees a strange bump-like waveform in a canal. It persists for over three miles, moving at nearly constant speed along the canal trench. He writes a paper, calling it a soliton wave and two Dutch mathematicians find a nonlinear partial differential equation that describes its motion. The equation, the Korteweg-De Vries Equation, proves fiendishly hard to solve. Finally, the crew working on the hydrogen bomb, finish the job early, so Ulam decides to use ENIAC to help him solve the Korteweg-De Vries Equation. He attains the first analytic solutions, and the study of soliton waves begins in earnest.

How does this earn a quid? Well, solitons model the way that blips of light move down a fiber-optic cable. The military decides that DARPA-net could run on fiber-optic cables, and uses them in building the early internet. Cellular telephone companies begin using fiber-optic cables to pack 100,000 phone conversations into a single pipe in such a way that they all get separated on the other end of the pipe-- one of the great engineering marvels of our time. We owe the modern internet, cell phones, anything that uses fiber-optics, to the solution of the Korteweg-De Vries equation. There was a similar burst of technology earlier in the last century when some closed-form solutions of the Schrödinger Equation were found.

Truth is, when we solve a major math problem like the Poincaré conjecture, billions of dollars of revenue are generated by new technologies that spring into being because of the new scientific understanding that the solution affords us. A thousand Adam Sandlers will not generate the amount of capital that the solution of the Poincaré conjecture will generate, especially considering that Perelman has shown the world that the Millenium Prize Problems are actually solvable.

Re:Square Pegs in Round Holes

[ThePhilips]

Nothing extraordinary really. In USSR, mathematics (as well physics) was just one of the top prioritized subjects. As one of my german friends compared me and his son, we soviet pupils have had about twice more mathematics during our school times.

Mathematics is not about numbers and problems - it teaches brain to think. Nothing more.

Re:High Mips, Low I/O

[Ruie]

Nothing freaky about it. IO is often the bottleneck, minimizing it is just good common sense.

Next time you are in a meeting think about this..