Poincaré Conjecture Proved

[Skeptoid]

According to this article from Mathworld, it looks like the Poincaré Conjecture has been proved. The Poincaré Conjecture is one of the problems the Clay Mathematics Institute included on its list of $1-million-prize problems.

From the article:

Perelman's results appear to be ... 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.

According to the rules of the Clay Institute, any purported proof must survive two years of academic scrutiny before the prize can be collected.

 

[Denise]

Whoosh! Did anyone hear that? Anyhow, I hope that this thread generates some responses as I have no idea about this whole thing!

 

[Fade]

OMG

 

[Denise]

OMG

Care to elaborate?

 

[Thumbo]

Care to elaborate?

Having read the link, I think I understand what has been proven, but I'd be very grateful if some kind mathemetician could explain, in layman's terms, why it is important.

 

[gnome]

Having read the link, I think I understand what has been proven, but I'd be very grateful if some kind mathemetician could explain, in layman's terms, why it is important.

I could try to explain the use of proving obscure n-dimensional topology theorems (or just exotic theorems in general) if you like, or were you looking for something more specific to this case?

 

[garys_2k]

Um, uh, hooray!

 

[LucyR]

I remember reading about this several years ago.

I think the conjecture is that the points on a 3-dimensional manifold can be mapped one-to-one onto the surface of a 3-dimensional sphere. There is a generalised version to n dimensions. The cases for some values of n have been demonstrated.

Edited to add: Sorry, just read the link. Should have done so earlier. My explanation doesn't help much.

 

[clk]

Here's an explanation of it at the Clay Math site:
http://www.claymath.org/Millennium_Prize_Problems/Poincare_Conjecture/

I guess that he's a millionaire now!

 

[a_unique_person]

Next week, $2 Million to provide the answer for world peace.

 

[Thumbo]

I could try to explain the use of proving obscure n-dimensional topology theorems (or just exotic theorems in general) if you like, or were you looking for something more specific to this case?

Something more specific to this one, please. I know it must be considered important because a significant prize was offerred for its proof. I just don't have any understanding of why it is so considered.

 

[DrMatt]

Often, in math, a lot of subsequent math is broken down into cases of the form "IF this conjecture is proved true, THEN", "IF this conjecture is proved false, THEN", and "REGARDLESS of whether this conjecture is true or false".
Actually proving such a conjecture often has a ripple effect on related mathematics, allowing every such trifurcation to be simplified into a single branch of reasoning, and thus expediting a lot of other results.

 

[gnome]

Something more specific to this one, please. I know it must be considered important because a significant prize was offerred for its proof. I just don't have any understanding of why it is so considered.

Some problems loom important among mathematicians just because they defy proof for so long. This one's been bouncing around for a hundred years or so.

There are some math problems that serve no known purpose whatsoever, but have been nagging mathematicians for so long that to solve them would mean instant fame--do you remember Andrew Wiles?

 

[aggle_rithm]

Manifolds, etc.

Quote:
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.

--------------------------------------------------------------------

...wouldn't this be true of any complicated proof? Seems a little self-evident. Plenty of well-thought out proofs have had difficult-to-find mistakes that invalidated them.

Weren't manifolds also instrumental in proving Fermat's Theorem? I'll have to look that up.

 

[Thumbo]

Some problems loom important among mathematicians just because they defy proof for so long. This one's been bouncing around for a hundred years or so.

There are some math problems that serve no known purpose whatsoever, but have been nagging mathematicians for so long that to solve them would mean instant fame--do you remember Andrew Wiles?

Yes indeed.

As a kid and into my student years I was fairly deep into recreational mathematics - I still have all of Martin Gardner's collections of his Scientific American columns somewhere - and like, I suspect, many others here I spent quite a few hours trying to devise a proof just too big to fit into a book's margin.

I never went deeply into formal mathematics as my brain seems to work more happily with "almost certainly" - which works great when working with technology - instead of "definitely, absolutely, here's the no-escape-clauses proof" you need to do maths. So I ended up being content to believe that Fermat's last theorem was true for practical purposes, even though it may not have been proved rigorously.

Nevertheless, I was delighted when I read that it had been proved: then hugely disappointed when it became clear that I would never and could never understand the proof. Bummer. Well, I'm old enough to be philosophical about it, but I still wish there was a proof that I could understand, instead of having to take it on trust.

I had much the same reaction to the proof of the four color theorem. Glad it has been proved, sad that I have to take it on trust.

The questions posed by both Fermat's Last theorem and the four color theorem are easy for even non-mathematicians to understand, and to wonder (when I first heard them) that such simple questions did not have answers.

Would it be a reasonable simile to say that for mathemeticians the question posed by the Poincare conjecture is like the four color and Fermat's last theorems were for laymen? That is, a simple, easy to understand question that looks like it ought to have a simple, easy to understand answer, but turns out to be really, really hard?

 

[gnome]

As a kid and into my student years I was fairly deep into recreational mathematics - I still have all of Martin Gardner's collections of his Scientific American columns somewhere - and like, I suspect, many others here I spent quite a few hours trying to devise a proof just too big to fit into a book's margin.

My pet project was the Goldbach Conjecture. Spent hours of time working with it on my computer. Got nowhere, of course.

I never went deeply into formal mathematics as my brain seems to work more happily with "almost certainly" - which works great when working with technology - instead of "definitely, absolutely, here's the no-escape-clauses proof" you need to do maths. So I ended up being content to believe that Fermat's last theorem was true for practical purposes, even though it may not have been proved rigorously.

Nevertheless, I was delighted when I read that it had been proved: then hugely disappointed when it became clear that I would never and could never understand the proof. Bummer. Well, I'm old enough to be philosophical about it, but I still wish there was a proof that I could understand, instead of having to take it on trust.

I like to believe that Fermat really did have a "marvelous" proof, and I suspect the problem of finding it will continue to be pursued, perhaps even more now that we are certain he was right about the theorem.

I had much the same reaction to the proof of the four color theorem. Glad it has been proved, sad that I have to take it on trust.

This one never really nagged at me. A few minutes with a piece of paper and some markers and I could feel its truth...

The questions posed by both Fermat's Last theorem and the four color theorem are easy for even non-mathematicians to understand, and to wonder (when I first heard them) that such simple questions did not have answers.

Would it be a reasonable simile to say that for mathemeticians the question posed by the Poincare conjecture is like the four color and Fermat's last theorems were for laymen? That is, a simple, easy to understand question that looks like it ought to have a simple, easy to understand answer, but turns out to be really, really hard?

I think you've hit the nail right on the head there. To mathematicians in the field of topology, I imagine the theorem is quite easily understood, compared to the proof. But this goes somewhat over the head of even a simple B.A. such as myself.

 

[whitefork]

Recent developments: http://www.boston.com/news/globe/he...entury_old_math_problem_may_have_been_solved/
but doesn't appear to have any new information.

 

[phildonnia]

Most of those problems are over my head. I sort of understand the Riemann hypothesis, and since I majored in CS, I am intimately familiar with the P=NP problem.

And I think everybody's done some work on this angle of the P=NP problem in their spare time.