Russian May Have Solved Poincare Conjecture

nev4 writes "Reuters (via Yahoo News) reports that Grigori Perelman from St. Petersburg, Russia appears to have solved the Poincare Conjecture. The Poincare Conjecture is one of the 7 Millenium Problems (another is P vs NP, also covered on /. recently). Solving a Millenium Problem carries a reward of $1M, but apparently Perelman isn't interested..." nerdb0t provides some background in the form of this MathWorld page from 2003.

Re:Duplicate?

[Disevidence]

RTFA. He published another paper on it recently.

Math?

[hunterx11]

1,000,000 USD is about equal to 560,000 GBP, not 5.6 million GBP.

$1 million USD?

[Anonymous Coward]

From the article:

A reclusive Russian may have solved one of the world's toughest mathematics problems and stands to win $1 million (560 million pounds) -- but he doesn't appear to care.

Heh. Last I checked, $1 million dollars was not quite equal to 560 million (British) pounds. (560 thousand, sure ...)

In an article on mathematics. Of all things.

Re:He'd post AC

[SYFer]

Hmmmm. Is your IP 18.72.0.3?

Re:Duplicate?

[terrymaster69]

According to Wikipedia [wikipedia.org], his proof of this surfaced around 2002 and he was lecturing on it in 2003. I guess it's not new news per se, but a Millennium prize problem is a big deal no matter how you look at it.

The Millenium Problems

[shadowmatter]

Since a great deal of discussion and awe comes up anytime one of the millenium problems is mentioned (solved?) on Slashdot, I'd just like to say that any layman interested in learning more about the millenium problems should run to his/her library/bookstore and pick up The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time [amazon.com]. Although, perhaps, for the layman, the end may become a bit tricky (the problems are explained simply in order of increasing difficulty), it's a book worth sticking with, and ultimately worth a read.

- sm

Re:Yes but...

[Anonymous Coward]

Makes sense, as I have no idea what the question is.

Hm... Let's see what the article tells us about it:

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.

Ah. Poincaré understood to ask a simple question like "what is six multiplied by seven" in such a profoundly stupid way that it puzzled the world ever since if and why the answer was 42...

Hopefully he has better luck than de Branges

[DeepRedux]

A few months ago Louis de Branges published his proof of the Riemann Hypothesis [purdue.edu] on the internet. This is also a Millennium problem. Apparently, no mathematician has read it [lrb.co.uk].

It is not that de Branges is unqualified: he settled Bieberbach's Conjecture [wolfram.com]. Interestingly, much of the validation of de Branges work on Bieberbach's Conjecture was done by a team at the Steklov Institute, referred to in the MathWorld link in the article.

Re:Riemann hypothesis reportadly also solved

[Anonymous Coward]

That's a great link, with a wonderful human-readable summary of the 7 problems.

For those too lazy to click:

Seven baffling pillars of wisdom

1 Birch and Swinnerton-Dyer conjecture Euclid geometry for the 21st century, involving things called abelian points and zeta functions and both finite and infinite answers to algebraic equations

2 Poincaré conjecture The surface of an apple is simply connected. But the surface of a doughnut is not. How do you start from the idea of simple connectivity and then characterise space in three dimensions?

3 Navier-Stokes equation The answers to wave and breeze turbulence lie somewhere in the solutions to these equations

4 P vs NP problem Some problems are just too big: you can quickly check if an answer is right, but it might take the lifetime of a universe to solve it from scratch. Can you prove which questions are truly hard, which not?

5 Riemann hypothesis Involving zeta functions, and an assertion that all "interesting" solutions to an equation lie on a straight line. It seems to be true for the first 1,500 million solutions, but does that mean it is true for them all?

6 Hodge conjecture At the frontier of algebra and geometry, involving the technical problems of building shapes by "gluing" geometric blocks together

7 Yang-Mills and Mass gap A problem that involves quantum mechanics and elementary particles. Physicists know it, computers have simulated it but nobody has found a theory to explain it

Re:Wake me...

[RedWizzard]

Wake me when someone verifies his work. I can claim to solve anything, but it doesn't mean much unless the community says I'm right. Right off the bat it seems fishy: no journal submission, just a web post? No referee? And he's not answering questions about his work? He's either a genius or a nutcase, possibly both.

The claim has been around for a while. From the referenced MathWorld article:

Almost exactly a year later, Perelman's results appear to be much more robust. While it will be months before mathematicians can digest and verify the details of the proof, mathematicians familiar with Perelman's work describe it as well thought out and expect that it will prove difficult to locate any significant mistakes.

That was in April 2003. It's now over a year later again and it hasn't been disproven.

Re:Duplicate?

[Y2]

Yes, yes, but googling "Grigori Perelman" gets you a 17-month-old article at Wolfram saying, "Poincare conjecture solved, this time for real."

Maybe what we have here is just the impending lapse of the Clay Math. Inst.'s required two years of scrutiny...

Re:Racist title

[Fjornir]

Dude... You missed this [slashdot.org] comment...

Chill out. It was a joke.

So, to quote Trek, "Double dumbass on you."

Re:Poincare Conjecture link sucks!

[Anonymous Coward]

Ok, imagine an apple with a rubber band around it. You can slide the rubber band down slowly, without breaking it, and have it shrink to a point, touching the apple all the while. This is called being 'simply connected.'

Now imagine a donut. It's got a hole in the center. Imagine a rubber band passing through that hole - think of two connected loops. This is not simply connected.

You can't shrink the donut's rubber band to a point, because of how it loops through the donut. You'd have to crush/tear the donut [or break the rubber band] to get it down to a point.

Poincaré's conjecture deals with this, but in one extra dimension.

Re:He'd post AC

[Phragmen-Lindelof]

If I remember correctly, the American Mathematical Society has about 30,000 members; I would guess that 5%-10% of them "become specialized enough to create anything significant." 1500 "martyrs" is a moderately large number; how many martyrs (not crazy suicide bombers) do you hear about in a year?

I agree that Ricci flows are very specialized; I believe Hamilton and Perelman are the experts (with possibly Yau, Tian, Donaldson, or a few others). Many mathematicians get a lot of enjoyment from "solving a difficult (mathematical) puzzles" (i.e. mathematical research); getting solutions published is also fun.
Hormel has come to our campus to recruit math majors for 20-30 years. One faculty member asked a Hormel recruiter what mathematics the recruits would be doing; he was told "None; math majors just make really good, hardworking employees". A few specific examples of companies who have hired BS, MS or PhD graduates in Math: Ernst & Young, the Bank of Toronto, Hormel, Boeing. Money is not the motivating factor for mathematicians who do "real" research; they could get much higher paying jobs in "industry".

Good article here

[mrthoughtful]

For an accessible math article on this, try http://mathworld.wolfram.com/news/2003-04-15/poinc are/

Re:Look at his method for solving this!!!

[Tony-A]

translate matheese-to-english and summarize in a way we can understand?

Disclaimer: MS Mathematics and I'm not about to claim I understand it, but this is Slashdot so here goes anyway.
There are a couple of fundamental ways of viewing something like a circle or a sphere, that can be generalized to an arbritrary number of dimensions. These ways are now known to be equivalent except for one lone holdout. What makes 3 so special that it can hold off our best mathematical minds?

This conjecture was subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere. The generalized statement is now known as the Poincaré conjecture, and it reduces to the original conjecture for n = 3.

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

manifold. a space that is locally Euclidean.
compact. every open cover has a finite subcover.
So a compact manifold is like a bounded chunk of Euclidean space.
The surface of the earth as a sorta spheroid is a compact manifold.
The surface of the "flat earth" is a compact manifold if there is an edge you would fall off of
Just looking at you immediate surroundings, you cannot tell which you're on.

Two objects are homeomorphic if they can be deformed into each other by a continuous, invertible mapping. Like a donut and a coffee cup are homeomorphic. So there exists f:DONUT->COFFEE-CUP (and if there's one there's many more).

Not content to leave things well enough alone, mathematicians start playing with the functions.
f:X->Y and g:X->Y
A homotopy between two functions f and g from a space X to a space Y is a continuous map G from (X,[0,1]) -> Y such that G(x,0)=f(x) and G(x,1)=g(x).
Two mathematical objects are said to be homotopic if one can be continuously deformed into the other.

Seems obvious and it should be easy to prove but intuition is not very reliable and should doesn't imply does.

f:UNIT-INTERVAL -> Euclidean-2-space. f is continuous. The image ought to be 1-dimensional. However, there are continuous functions which have 2-dimensional images.

Cantor's Perfect set. Uncountable number of points but has measure zero. Measure is a generalization of length. The measure of the rational points on a line is zero, but that's only countably infinite.

Triangle A B C. Bisectors of angles ABC and ACB are equal length. Prove the triangle is isocoles. It's provable but I've never managed it.

Four-color theorem. Finally proved with very many special cases solved by computer.

Euclid's fifth postulate. Despite a few people who thought they'd proved it, I think the current state of affairs is that if any of the geometries has a problem, then the other two geometries also have a problem. However all the geometries are "locally Euclidean".

Re:He'd post AC

[NonSequor]

If we weren't talking about Erdos, I'd agree with you. The thing about him, is that he wasn't just a great mathematician, he was a great collaborator. In addition to that he was generally good natured and his many quirks were (mostly) endearing. He brought out the best in the people he worked with. Erdos didn't need money because he was held in such high esteem that he could go anywhere and people would be willing to pay for his meals and give him a place to sleep just for the opportunity to work with him.

Re:Duplicate?

[radicalaxis]

Um, unless I have a huge blind spot, the article says no such thing. In fact, this article [scotsman.com] makes it clear that the latest article he has published was in March 2003, and although a further paper is forthcoming, it is believed that the first two papers contain a correct proof.

As far as I can tell, it seems the fuss is rather about the distinguished mathematician (math popularizer, rather) Keith Devlin saying that he thinks it is correct... but as far as I can tell, he has no special authority on the problem and hasn't looked it over in the details

Re:He'd post AC

[Anonymous Coward]

I'm not certain, but I think Bruce Kleiner is doing a significant portion of the checking of the proof.
(This is what I heard in the U of U math lounge.)

Re:He does not make more than $4-5k/year

[mrak018]

few hundred? Rotfl.
Here in Russia he probably earn no more than 1 or 2 hundred.

Our scientists has 0 money and infinte amount of time to work, because our scientific institutes give them office space and not enough money to spend it for anything other than food.

Re:Confused

[xoran99]

A better analogy would be to continuously move a circle on the surface until it becomes a point. In the case of a donut, you could draw the circle through the middle hole and around again, so you can't "shrink it to a point" my continuously moving it anywhere; it goes around the donut anywhere you put it. With a sphere, though, you can continuously move the circle to a "pole," where it becomes a point. This property is called simple connectivity.

It's pretty easy to see that all simply connected 2-manifolds (in 3 dimensions, at least) are homeomorphic to the shell of a sphere, i.e. they may be stretched and contorted to look like it. The question answered here is whether the same is true in the next dimension.

Billions - Spanish / English

[Guiri]

Well, there are actually differences between numbers in different languages: 1 Billion in english is 10^9, while 1 Billion in spanish is 10^12.

Cheers

Re:$1 million USD?

[Anonymous Coward]

bullitB is certainly not correct. I've yet to hear million defined as thousand, even in the UK.
The UK 'million' is the same as the US 'million'.
The difference he may have been, in a twisted way, trying to recall was billion - a US billion is 10^9, a UK billion 10^12.
But a million is not, never had been, and hopefully never shall be 10^3

Re:Any relation to Yu Perelman

[Anonymous Coward]

Grigori Yakovlevich Perelman is a son of Yakov Isidorovich Perelman, autor of "Zanimatelnaja Matematika" among other books.

Re:Billions - Spanish / English

[pjt33]

Nonono. You're confusing English with American. 1 billion in English is 10^12, whereas 1 billion in American is 10^9. However, to ensure everyone is confused the Houses of Parliament switched to using American billions in the 1970s.

Re:Billions - Spanish / English

[njj]

No, a British billion is 10^12, too (although the American definition seems to have largely supplanted the British one in popular usage). We have a (now obscure) word `milliard' which means 10^9.

Presumably this means that Bill Gates is actually a `milliardaire' when visiting the UK.

For a nation which often claims to be the greatest country in the world, I must say that American billions are a bit, well, small...

Re:Billions - Spanish / English

[Ibn al-Hazardous]

Actually, IIRC the difference is between the European and US'ian definitions of Billion, Trillion etc. So, the Brittish Billion equal the Spanish and the Swedish Billion, while the one the talk about in the US is 10^3 times smaller. That's what made the parent funny... :)

Now, all I wonder is - which standard does the Canucks follow?

MODERATORS - plagiarism / karma whore

[Anonymous Coward]

The author of the parent post claims to explain the conjecture from the point of view of a "MS Mathematics". This would be fine if the explanations had not been copied directly from MathWorld.

Quote from the parent post:

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Now please compare this with the middle paragraph from http://mathworld.wolfram.com/PoincareConjecture.ht ml [wolfram.com]. The one that starts with "The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), [...]"

This is just an example. Other paragraphs can be found in MathWorld's pages about the Poincaré conjecture [wolfram.com], definition of manifold [wolfram.com] and compact manifold [wolfram.com], homeomorphic [wolfram.com], etc.

Now I don't mind if some useful information is posted on Slashdot. But some obvious plagiarism like that without crediting any sources definitely deserves some Overrated treatment...

Einstein & Poincaré

[Kardamon]

Einstein's paper "On the electrodymanics of moving bodies" contains nothing new. It was actually Poincaré who was the first to correctly state the special theory of relativity (the transformation formulas were found by Woldemar Voigt [ntu.edu.tw] in 1887, H.A. Lorentz in 1892, Sir Joseph Larmor and others)

In 1898, Poincaré attacks the distinction Lorentz and Larmor make between "local time" and "universal time": "Nous n'avons pas l'intuition directe de l'égalité de deux intervalles de temps. Les personnes qui croient posséder cette intuition sont dupes d'une illusion... Le temps doit être défini de telle facon que les équations de la méquanique soient aussi simples que possible. En d'autres termes, il n'y a pas une manière de mesurer le temps qui soit plus vrai qu'une autre; celle qui est généralement adoptée est seulement plus commode. ...Il a commencé par admettre que la lumière a une vitesse constante, et en particulier que sa vitesse est la même dans toutes les directions. C'est là un postulat sans lequel aucune mesure de cette vitesse ne pourrait être tentée. Ce postulat ne pourra jamais être vérifié directment par l'expérience; il pourrait être contredit par elle, si les résultats des diverses mesures n'étaient pas concordants. Nous devons nous estimer hereux que cette contradiction n'ait pas lieu et que les petites discordances qui peuvent se produire puissent s'expliquer facilement. ...c'est que je veux retenir, c'est qu'il nous fournit une règle nouvelle pour la recherche de la simultanéité... Il est difficile de séparer le problème qualitatif de la simultanéité du problème quantitatif de la mesure du temps; soit qu'on se serve d'un chronomètre, soit qu'on ait à tenir compte d'une vitesse de transmission, comme celle de la lumière, car on ne saurait mesurer une pareille vitesse sans mesurer un temps. ...La simultanéité de deux événements, ou l'ordre de leur succession, l'égalité de deux durées, doivent être définies de telle sorte que l'énoncé des lois naturelles soit aussi simple que possible. En d'autres termes, toutes ces règles, toutes ces définitions ne sont que le fruit d'un opportunisme incoscient." (H. Poincaré, La mesure du temps, in Revue de métaphysique et de morale 6 (1898), pp. 1-13)

In 1902, Poincare writes there is no absolute time and no absolute space: "1 Il n'y a pas d'espace absolu et nous ne concevons que des mouvements relatifs... 2 Il n'y a pas de temps absolu; dire que deux durées sont égales, c'est une assertion qui n'a par elle-même aucun sense et qui n'en peut acquérir un que par convention... 3 Non seulement nous n'avons pas l'intuition directe de l'égalité de deux durées, mais nous n'avons même pas celle de la simultanéité de deux événements qui se produisent sur des théâtres différents; c'est ce que j'ai expliqué dans un article intitulé la Mesure du temps; 4 Enfin notre géometrie euclidienne n'est elle-même qu'un sorte de convention de langage; nous porrions énoncer les faits mécaniques en les rapportant à un espace non euclidien qui serait un repère moins commode, mais tout aussi légitime que notre espace ordinaire; l'énoncé deviendrait ainsi beaucoup plus compliqué; mais il resterait possible. Ainsi l'espace absolu, le temps absolu, la géométrie même ne sont pas des conditions qui s'imposent à la mécanique; toutes ces choses ne preéexistent pas plus à la mécanique que la langue francaise ne préexiste logiquement aux vérités que l'on exprime en francais."(H. Poincaré, La science et l'hypothèse, 1902

Re:Perelman and the prize

[Anonymous Coward]

Obviously, in both cases it would have been impossible to reach such great results if the authors had had to keep a steady pace of lesser publications. But this is the rule in the academic world: "publish or perish". You must prove yourself "productive" year by year, otherwise you're out.

Umm, no. That's the whole point of tenure.

Before tenure, you do have to prove yourself, but after that you are free to spend years working on something, with no prospects of intermediate output. Sure, your salary could stagnate, but if you're in academia for the money, you're a fool.

Wiles was a full professor with all the trimmings before embarking on that 7 year project.

Riemann hypothesis

[irf]

this may also be of interest, it appears that
another one of the so called "millennium problems",
may have just been solved, that is Riemann
Hypothesis:-
http://www.vnunet.com/news/ 1157891
we are all lucky to live in such exciting times.

Re:Ironically it was Poincarré (not Einstein)

[Kardamon]

Oh, I've followed your link and now I understand why you've been modded flamebait: this is just anti-semitic bullshit.