## Poincare Conjecture Proof Completed 222

Posted
by
samzenpus

from the show-your-work dept.

from the show-your-work dept.

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!"*
## Square Pegs in Round Holes (Score:2, Funny)

## Re:Square Pegs in Round Holes (Score:2)

freaky misfit mathematical geniusesMake for good stories too... Wait, I saw this coming! There's also something about a black box, a blind guy, passports, and a guy who looks like he spent a little

toomuch time in prison.Hmm, lost it now.

## Re:Square Pegs in Round Holes (Score:2)

## Re:Square Pegs in Round Holes (Score:2)

## Re:Square Pegs in Round Holes (Score:5, Insightful)

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

## Okay, so what you're saying is... (Score:5, Funny)

In Soviet Russia, mathematics teaches you.

## Re:Grigori Perelman, please give us a sign! (Score:5, Funny)

human rightsanddemocracy. Come toAmerica!"The reason they can't find him in Russia is because he's already living in Sweden.

## Re:Grigori Perelman, please give us a sign! (Score:3, Funny)

## Re:Grigori Perelman, please give us a sign! (Score:2, Insightful)

## Re:Grigori Perelman, please give us a sign! (Score:2)

## Re:Grigori Perelman, please give us a sign! (Score:4, Funny)

## Re:Grigori Perelman, please give us a sign! (Score:3, Informative)

We should be quite concerned about Grigori Perelman since he returned to Russia.Nice bit of jingoistic xenophobia there, but that's about all that's nice about your post.

Gang Tian, who has co-wrote a guide to Perelman's proof, said in 2004: "He certainly has no interest in material things. If he gets the Fields Medal, there is the issue of whether or not he will accept it." He also refused a prize from the European Mathematical Society many years before that.

He is not being threatened, he is simply a pe

## Re:Grigori Perelman, please give us a sign! (Score:2)

## Re:Grigori Perelman, please give us a sign! (Score:2, Troll)

30% Flamebait

30% Troll

30% Interesting

There's nothing remotely Flamebait or Troll in that message. TrollMods spew cosmoturf to suppress discussion of growing Russian backsliding towards tyranny. Slashdot's mod system really is disgusting sometimes when it's abused by political operatives.

## Too Many Pages (Score:3, Funny)

## Re:Too Many Pages (Score:3, Funny)

Trust me, 99.9999% of the folks will never follow the link if your short blather is at all close to an accurite summary.

## Re:Too Many Pages (Score:3, Interesting)

## A rabbit is a donut, not a sphere. (Score:4, Insightful)

## Re:A rabbit is a donut, not a sphere. (Score:3, Funny)

What kind of strange rabbits have these topologists seen?Chocolate ones

## Re:A rabbit is a donut, not a sphere. (Score:2)

Reputedly, there exists a book with a picture of Borromean humans.

## a million, a thousand, roundness (Score:4, Funny)

## Re:a million, a thousand, roundness (Score:3, Funny)

## Re:a million, a thousand, roundness (Score:2)

7097556CL3 = CF93

## who cares Fields medal? (Score:2)

## Re:who cares Fields medal? (Score:2)

Is Madam Curie well known cause she won the Nobel?

Is Neils Bohr well known cause he won the Nobel?

Is Dirac well known cause he won the Nobel?

Is Watson/Crick famous cause they won the Nobel?

The point I am trying to make is that GP is 100% right.

Nobel/Fields doesnt come anywhere near, if I were to prove the Poincare Conjunture.

Maybe I can put it across in another way.

Is Ramanujan any less well known since he did not win a Fields?

Is Mahatma Gandhi any less a person since

## nytimes is more realistic (Score:2, Informative)

http://news.xinhuanet.com/english/2006-06/04/cont

## Re:nytimes is more realistic (Score:2)

Did these two guys have ANYTHING to do with solving the proof?

## How does this relate to string theory? (Score:2)

## Re:How does this relate to string theory? (Score:2)

## Re:How does this relate to string theory? (Score:5, Informative)

According to string physicist Lubos Motl [blogspot.com] the proof indeed important to string theory. The proof based on the flow on the manifold (surface), analogous to heat dissipation - Ricci flow [wikipedia.org]. This flow deform metrics (distance between points of the surface). But this process also describe renormalization [wikipedia.org] of worldsheet - how the physics of the worldsheet [wikipedia.org] (surface which string drawing, moving in space and time) change with changing of the observation scale. That is how phisics of string change then the scale of calculation changed.

## Re:How does this relate to string theory? (Score:3, Funny)

... ANAM (I'm not a matematician) ...IANAA (I am not an acronymist)

## Re:How does this relate to string theory? (Score:5, Interesting)

The Ricci Flow [wikipedia.org] was defined by Richard Hamilton in 1981 as a step towards classifying topological compact 3-manifolds. Classifying 3-manifolds would certainly decide The Poincare Conjecture, as it states that all

simply connectedcompact 3-manifolds are homeomorphic to the sphere. This is an important special case: most proofs of the classification of compact 2-manifolds start out by proving the an analogous statement for the 2-sphere. The Ricci Flow is a differential equation which defines how the shape of a manifold changes in time: given an arbitrary manifold M(0), you can apply the differential equation to it to get manifolds M(t) for (some) positive t, which gradually change shape. However, the Ricci Flow is not volume preserving, so you "renormalize" so that M(t) has constant volume.The Ricci Flow has the useful property that it tends to make manifolds smoother and smoother. For example, if you started out with a lumpy ball, you would eventually get a smooth ball. It was hoped that it could be proved that if the initial manifold was a compact simply connected 3-manifold, then as t increased, the manifold would tend towards a 3-sphere. Unfortunately, while locally solutions to differential equations always exist, they don't necessarily exist for all time, and for some starting manifolds, eventually you would get to a road-block: a t for which M(t) could not be defined. What Perlman (hopefully) showed was that all road-blocks were of certain types, and that a surgery could be formed that would modify the manifold but not it's topological nature, and then you could again apply the Ricci Flow, until the manifold became a sphere.

Note that this method is useful beyond proving the Poincare Conjecture, as it (again, hopefully) describes all road blocks to extending the Ricci Flow, so that the same tools can be applied to any 3-manifold, and not just simply connected ones. In this manner, assuming Perlman made no mistakes (or that any mistakes can be corrected), it is possible to apply the same arguments to prove the Geometrization Conjecture of Thurston, which classifies 3-manifolds.

## Re:How does this relate to string theory? (Score:4, Interesting)

It seems to me at this Ricci Flow differential equation could be quite useful practically. For example, in pattern recognition, if a computer could build a 3d model of an object using multiple vantage points, then simplify the object to one of the handfull of object types described by Perlman using the Ricci Flow, then this simple catagorization might help in the identification of complex objects (e.g. a donut really is a donut, even if it's been heavily frosted).

Do you know if Perlman's technique for handling the singularities will help with the numerical implementation of this process? Or are these issues numerically simple to solve - but only challenging to solve in proof?

## Re:How does this relate to string theory? (Score:3, Interesting)

## The tone of the summary is typical (Score:5, Insightful)

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 (Score:2)

## Re:The tone of the summary is typical (Score:5, Insightful)

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 (Score:4, Insightful)

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 (Score:2)

## Re:The tone of the summary is typical (Score:2)

Lots of people do things for their own sake (as long as they can pay their bills and get some food). But when someone got a prize of a million dollar as a bonus (for what you enjoyed doing anyway), can you really imagine someone turning this down? Well, Perelman hasn't done this (yet), but lots of people could im

## Re:The tone of the summary is typical (Score:4, Insightful)

manage'rights holders' "explain" how, if it weren't for copyright, there would exist no art.## Maybe he... (Score:2, Funny)

## Re:The tone of the summary is typical (Score:5, Insightful)

## Re:The tone of the summary is typical (Score:2)

## Re:The tone of the summary is typical (Score:3, Insightful)

Sadly, yes, doing something for it's own sake rather than for monetary gain is frowned aponThat is not correct. Look at the hoopla around both Gates and Buffett giving way their money. Look at the adoration of Mother Teresa. Look at the army of fans for Linus Torvalds and Richard Stallman.

and sometimes viewed with fear and confusion,Sure: anything out of the ordinary will engender fear and confusion. There is a difference between suspecting that someone MAY NOT BE altrustic and "frowning upon" the

## Re:The tone of the summary is typical (Score:2)

That and also while he did the hard work, that he didn't really contribute to the full proof, which is also weird.

## Perhaps he just hates parties (Score:2)

Heck, perhaps he just enjoys math for its own sake and doesn't want to deal with all the side-effects of notoriety.Quite.

Perhaps he just hates parties. It's not like he'd be the first mathematician to do so. I and many other Slashdotters can sympathise with this, surely.

## Re:The tone of the summary is typical (Score:2)

-Eric

## Re:The tone of the summary is typical (Score:2, Insightful)

mathematiciansare amazed that someone would throw around important parts of the proof, not wait for credit and leave it to others to write it up. then again, knowing perelman they are not incredulous.in mathematics, the trend has mostly been to keep the insights of a big result under wraps until the proof is written down properly and checked for bugs. that is the way to get yourself into the hall of fame [st-and.ac.uk]. it is almos

## TFA is well worth reading (Score:2, Funny)

Quite an interesting character, this Perelman, and his proof could turn out to be a real landmark for mathematics.

I liked this bit:

Whatever he's smoking, I want some!

## Re:TFA is well worth reading (Score:3, Insightful)

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!

## Re:TFA is well worth reading (Score:2)

## Re:TFA is well worth reading (Score:3, Insightful)

Want to make a lot of money, do something the generates a lot of money. I can understand your point of view, but get real...

## Re:TFA is well worth reading (Score:3, Insightful)

Want to make a lot of money, do something the generates a lot of money. I can understand your point of view, but get real...Innovation in math and science generates more money than any movie.

Consider something obviously fundamental to the way we live, like calculus or Fourier transforms.

It is very foolish to think that the direct and immediate monetary rewards a person receives are any real inidcation of the value their work provides to society.

## Re:TFA is well worth reading (Score:2)

## Re:TFA is well worth reading (Score:2)

## You misunderstand free markets (Score:2)

The demand for comedy is higher than the demand for mathematical proofs. The recompense for either has absolutely nothing to do with merit, even if you believe a mathematical proof has more innate merit than comedy. BTW, if you do believe that, please define for us exactly how a mathematical proof is better (has more value or merit)

## On the contrary... (Score:5, Insightful)

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 waveand 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:On the contrary... (Score:2)

Ah. No... The money/capital isn't

generated. It's simply moved from one place to another, from low performing areas to higher performing areas. Only the governments can print money. Are you trying to tell me that money invested in the telecoms industry inherently has more merit than money invested in the entertainment industry?What makes a mathematical proof

inher## Re:On the contrary... (Score:2)

I think you hit the nail on the head with that. If only I had mod points.

Exactly. Merely because something is learned doesn't mean it's valuable. All

## Re:On the contrary... (Score:2)

## Re:On the contrary... (Score:2)

## Re:TFA is well worth reading (Score:2)

Fair enough -- if making stupid people laugh is considered more important by society than fundamental mathematical discoveries, then it should be more highly compensated. It is. What's your problem with that?Simply put, I believe that society is wrong. It is wrong to value the contribution of Adam Sandler as greater than that of Grigori Perelman

One day, probably many years from now, Adam Sandler will be a footnote in some obsolete database, and Perelman will be famous for his contribution to human kno

## Re:TFA is well worth reading (Score:2)

Simply put, I believe that society is wrong. It is wrong to value the contribution of Adam Sandler as greater than that of Grigori PerelmaBut you have to realize that most of societys concerns are immediate needs. We need food, fuel, sex and something to make the time pass by. These things are valuable because they are needed in huge quantities. Adam Sandler might not be a great comedian, but his skill serves a huge need. Mr. Perelman might be making a great contribution to math and science, which perhaps## Re:TFA is well worth reading (Score:3, Insightful)

## Re:TFA is well worth reading (Score:2)

Since he lived a long time before the advent of recording technology, the type of success available to a popular musician (or more properly their record label) today just didn't exist.

a perpetual copyright system would only serve to

## Re:TFA is well worth reading (Score:2)

## Re:TFA is well worth reading (Score:5, Interesting)

## Re:TFA is well worth reading (Score:2)

## Re:TFA is well worth reading (Score:2)

PS: Even the experts who know the mushrooms in their area have been known to get sick or die when they go to another country and assume that the species which look the same are the same.

## Recognition = Worry (Score:4, Insightful)

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.

## The prize is important (Score:2, Insightful)

## name change? (Score:5, Insightful)

## Re:name change? (Score:2)

## Re:name change? (Score:5, Informative)

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?Things that are proven, are called theorems. They do depend on axioms, but those are defined as true. Sciences about the real world that can't put up axioms (because that'd require ex facto knowledge about the real world), so they can never be conclusively "proven". Hence well call them theories, like theory of gravity, theory of evolution. A few we've called "laws" as well because they have been so extensively tested, but it is not proven in a strict formal sense.

## Re:name change? (Score:2)

## Re:name change? (Score:2)

There is no place for observation or rough suppositions like in physics or biology.This is mildly incorrect. Theories also consist of a body of associated hard to solve problems. These problems often turn out to be observations and guesses or as you put it, "rough suppositions". For example, "Fermat's Last Theorem" has for centuries been a part of number theory even though it was proven only ten years ago.

## Re:name change? (Score:2)

## Re:name change? (Score:2)

## Has anyone read the actual article? (Score:5, Informative)

## Practical consequences of the proof? (Score:2)

## Re:Practical consequences of the proof? (Score:2)

Even if the PC has no direct bearing on some of these fields, the techniques used in the proof will probably end up deeply influencing their research methods.

## He's turned down the money (Score:5, Interesting)

## Re:He's turned down the money (Score:2)

## Re:He's turned down the money (Score:2)

That's stupid. He

mightturn money down. Now it's announced that proof is correct and that makes him candidate for that prize.Even Russian newspapers do not have any official reaction of Perelman himself yet.

His (western) colleagues speculate that he might turn the award down. He is too far from normal life and money would distract him - so his friends say. That's speculation.

## Re:He's turned down the money (Score:2)

Is it thanks to receive some money and a medal after your peers roasted you for a couple of years?

## Re:He's turned down the money (Score:2)

## disillusioned with Academia (Score:3, Interesting)

TFA mentions he has distanced himself from others in the Math community because he has become disillusioned. I read into that my own experience, which involved professors trying to hit on me, others trying to get me to write/edit their papers and then taking the credit, others who weave tall tales with just enough truth to fool grant money providers.

One of my colleagues now believes that Science is actually performing a random walk on the landscape of Truth. Occasionally, the walk stumbles over something

## After reading TFA . . . (Score:2)

[/lie]

## A question about hypersphere volumes (Score:2)

Obviously you're going to get an extra r with each dimension, buy why do you only get another pi every other dimension?

While I'm at it, on a related subject it seems to me there are two possible ways of constructi

## Re:A question about hypersphere volumes (Score:5, Informative)

The Jacobian, or unit volume if you will, of a hypersphere [wikipedia.org] has a a highest term of sine, or cosine, which grows as you increase dimension. Specifically, for an n dimensional sphere, the highest power of sine or cosine will be sin^(n-2).

Anyway, to answer your question, integrals of sine or cosine to odd powers produce only functions of other sines and cosines. However, integrals of sine or cosine to

evenpowers produce functions of sin(x), cos(x)andx. The x part gives you your pi, but only does so every second dimension, when the highest power is even.Here's the integrals of (sin(x))^n, for various n

n=0: x

n=1: - cos(x)

n=2: x/2 - sin(2x)/4

n=3: 1/3 * (cos(x))^3 - cos(x)

n=4: (sin(4 x) - 8 sin(2 x) + 12 x)/32

## two Perelman anecdotes (Score:4, Interesting)

1) I met him at the Mathematical Sciences Research Institute in Berkeley at a workshop sometime around 1994 and he at that point had ridiculously long fingernails and was quite unkempt, even by the quite weak standards applied to research mathematicians. That was a while ago, of course and that was probably one of his first visits to the US. He gave an incomprehensible energetic talk so what most people commented on was his nails.

2) In 2003 or so, during a limited lecture tour about his proof of the Poincare Conjecture, he responded deftly and hilariously to a comment of Misha Gromov in the audience. Gromov is one of the most difficult people to have in a talk- he is a great mathematician with not much patience and has derailed or rerouted talks by many great researchers, who sometimes get quite flustered. I can't remember the exact wording of the exchange, which is too bad since it was precious, but Gromov asked something like "I don't see how that goes, I'd like to see some more details" and Grisha responded with something like "well, yes, you would" and carried on as he had intended.

## Re:I remain skeptical (Score:3, Funny)

## Re:I remain skeptical (Score:5, Insightful)

Secondly, I would invite you to write down a complete proof of some well-known mathematical fact, the Stone-Weierstrass [wikipedia.org] 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.

## Re:I remain skeptical (Score:2)

## Re:I remain skeptical (Score:4, Funny)

balls? And what does that make it? A sphere! There's some connection here,I swear.Whoa...## Re:Ellipse in Highschool (Score:2)

## Re:Ellipse in Highschool (Score:2)

## Re:High Mips, Low I/O (Score:4, Insightful)

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

## Re:1000 pages?! (Score:2)

Example: Maxwell's four equations were each about 1 1/2 to 2 1/2 pages long in a notebook when first formulated as differential equations. Expessed using the currently familiar "del" operator form they are each much less than a line of type, and have been fit onto T shirts. Sometimes along with additional text, as in:

And God said

(max)

(well's)

(equa)

(tions)

and there was light!