6 ms·

The Poincaré Conjecture, Explained

133 points by privatdozent 5y ago

denton-scratch 5y ago
That was a marvellously lucid presentation of the subject. I can't say I understood all of it, but I love to read presentations of difficult subjects expressed clearly in (more or less) plain words. It makes me think I've learned something. There's a citation in the article to "Gardner, 1984 p. 9–10". But there's no footnote. Would that be the late Martin Gardner, formerly of Mathematical Recreations? He also had a talent for expressing hard subjects clearly, in plain words.
- pvg 5y ago
  The reference is at the bottom of this, author probably forgot to fix it when cutting and pasting from Mathworld (yes, it's the same Gardner) https://mathworld.wolfram.com/BettiNumber.html https://mathworld.wolfram.com/BettiNumber.html
wrnr 5y ago
I always liked this 8 minute explanation of the proof: https://www.youtube.com/watch?v=PwRl5W-whTs https://www.youtube.com/watch?v=PwRl5W-whTs
- go_elmo 5y ago
  Just wanted to ask if there is a similar article about or including the proof, thanks
lgl 5y ago
Also fascinating is the story of the man that eventually proved the conjecture, Grigori Perelman [0], which famously later denied the $1M Millennium prize as well as the Fields medal where he is quoted as saying: "The prize was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed." He later abandoned mathematics and returned to obscurity. [0] https://en.wikipedia.org/wiki/Grigori_Perelman https://en.wikipedia.org/wiki/Grigori_Perelman
- fakesheriff 5y ago
  There's a Russian language documentary on YouTube worth watching. It's unlikely that he's given up mathematics. Publishing, yes. Mathematics, no. If he'd finished his proof a few years later, I suspect Perelman would have followed Satoshi's lead and published as an anon. He just dumped dumped it on the arxiv, on the random. In the documentary, Gromov seemed a little miffed at Perelman. As Gromov sees it, other mathematicians spent a lot of energy helping Perelman progress and he kind of "owes" it to the community to interact and mentor. On the flip side, we should probably ask ourselves why someone like Perelman would rather be a recluse than participate in the community. The politics around his proof were particularly nasty, but I have to wonder if he sees deeper problems.
  - mathgenius 5y ago
    link?
    - deleted 5y ago
      [deleted]
    - klyrs 5y ago
      This New Yorker article was ostensibly about the controversy, but became embroiled itself... https://www.newyorker.com/magazine/2006/08/28/manifold-destiny https://www.newyorker.com/magazine/2006/08/28/manifold-desti... https://en.m.wikipedia.org/wiki/Manifold_Destiny https://en.m.wikipedia.org/wiki/Manifold_Destiny
  - wombatmobile 5y ago
    > we should probably ask ourselves why someone like Perelman would rather be a recluse than participate in the community. That's an interesting choice of language. For you to make that point at all indicates you feel it has some validity. But you didn't choose a compelling form of words to advocate your interest to other people. Why not? Perhaps you are a mathematician, and perhaps this is some expression of the difference between maths and other endeavours. The beauty of maths is that it exists independent of all other considerations. This seems to be where Perelman comes from. Perhaps Perelman goes further. It's hard to say without knowing more of his story, which Perelman doesn't give us. You, on the other hand, find the human dimension of Perelman's story interesting - so much so that you dare to conjecture what he might think. Some emotional force drove you to post about his story, presumably because you desired other people to share your interest and commune with you in a way that might resonate and drive your enjoyment further. But you didn't push it. You only just barely said it at all. But you did suggest it. I'm curious why you didn't choose a more strident form of advocacy to grab people by the lapel and drag them towards Perelman's story, where they might learn something and be entertained and enlightened. Perhaps because advocacy isn't maths? Perhaps because you have too much humility and respect to foist your own interests onto others? What might Poincare make of this curiosity?
GolDDranks 5y ago
The article seems to be full of little mistakes. A torus has Betti number β1 of 2, not 1 as is on the text. Also, β1 and genus are not "equal", they are related by β1 = 2g. Furthermore, the Euler charasteric also doesn't "equal" it. Euler charasteristic χ in terms of genus is χ = 2 − 2g, so by substitution, χ = 2 − β1.
yarg 5y ago
Shouldn't it be renamed now that it's proven? The Poincaré-Perelman theorem seems more accurate now.
- q-big 5y ago
  > The Poincaré-Perelman theorem seems more accurate now. Perelman proved a more general result: the geometrization conjecture that was initially stated by William Thurston: https://en.wikipedia.org/wiki/Geometrization_conjecture https://en.wikipedia.org/wiki/Geometrization_conjecture
jwilber 5y ago
From wiki: In August 2006, Perelman was offered the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." “ Some people are built differently, what a legend.

133 points by privatdozent 5y ago

That was a marvellously lucid presentation of the subject. I can't say I understood all of it, but I love to read presentations of difficult subjects expressed clearly in (more or less) plain words. It makes me think I've learned something. There's a citation in the article to "Gardner, 1984 p. 9–10". But there's no footnote. Would that be the late Martin Gardner, formerly of Mathematical Recreations? He also had a talent for expressing hard subjects clearly, in plain words.

pvg 5y ago
The reference is at the bottom of this, author probably forgot to fix it when cutting and pasting from Mathworld (yes, it's the same Gardner) https://mathworld.wolfram.com/BettiNumber.html https://mathworld.wolfram.com/BettiNumber.html

wrnr 5y ago

I always liked this 8 minute explanation of the proof: https://www.youtube.com/watch?v=PwRl5W-whTs https://www.youtube.com/watch?v=PwRl5W-whTs

go_elmo 5y ago
Just wanted to ask if there is a similar article about or including the proof, thanks

lgl 5y ago

Also fascinating is the story of the man that eventually proved the conjecture, Grigori Perelman [0], which famously later denied the $1M Millennium prize as well as the Fields medal where he is quoted as saying: "The prize was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed." He later abandoned mathematics and returned to obscurity. [0] https://en.wikipedia.org/wiki/Grigori_Perelman https://en.wikipedia.org/wiki/Grigori_Perelman

fakesheriff 5y ago
There's a Russian language documentary on YouTube worth watching. It's unlikely that he's given up mathematics. Publishing, yes. Mathematics, no. If he'd finished his proof a few years later, I suspect Perelman would have followed Satoshi's lead and published as an anon. He just dumped dumped it on the arxiv, on the random. In the documentary, Gromov seemed a little miffed at Perelman. As Gromov sees it, other mathematicians spent a lot of energy helping Perelman progress and he kind of "owes" it to the community to interact and mentor. On the flip side, we should probably ask ourselves why someone like Perelman would rather be a recluse than participate in the community. The politics around his proof were particularly nasty, but I have to wonder if he sees deeper problems.
- mathgenius 5y ago
  link?
  - deleted 5y ago
    [deleted]
  - klyrs 5y ago
    This New Yorker article was ostensibly about the controversy, but became embroiled itself... https://www.newyorker.com/magazine/2006/08/28/manifold-destiny https://www.newyorker.com/magazine/2006/08/28/manifold-desti... https://en.m.wikipedia.org/wiki/Manifold_Destiny https://en.m.wikipedia.org/wiki/Manifold_Destiny
- wombatmobile 5y ago
  > we should probably ask ourselves why someone like Perelman would rather be a recluse than participate in the community. That's an interesting choice of language. For you to make that point at all indicates you feel it has some validity. But you didn't choose a compelling form of words to advocate your interest to other people. Why not? Perhaps you are a mathematician, and perhaps this is some expression of the difference between maths and other endeavours. The beauty of maths is that it exists independent of all other considerations. This seems to be where Perelman comes from. Perhaps Perelman goes further. It's hard to say without knowing more of his story, which Perelman doesn't give us. You, on the other hand, find the human dimension of Perelman's story interesting - so much so that you dare to conjecture what he might think. Some emotional force drove you to post about his story, presumably because you desired other people to share your interest and commune with you in a way that might resonate and drive your enjoyment further. But you didn't push it. You only just barely said it at all. But you did suggest it. I'm curious why you didn't choose a more strident form of advocacy to grab people by the lapel and drag them towards Perelman's story, where they might learn something and be entertained and enlightened. Perhaps because advocacy isn't maths? Perhaps because you have too much humility and respect to foist your own interests onto others? What might Poincare make of this curiosity?

GolDDranks 5y ago

The article seems to be full of little mistakes. A torus has Betti number β1 of 2, not 1 as is on the text. Also, β1 and genus are not "equal", they are related by β1 = 2g. Furthermore, the Euler charasteric also doesn't "equal" it. Euler charasteristic χ in terms of genus is χ = 2 − 2g, so by substitution, χ = 2 − β1.

yarg 5y ago

Shouldn't it be renamed now that it's proven? The Poincaré-Perelman theorem seems more accurate now.

q-big 5y ago
> The Poincaré-Perelman theorem seems more accurate now. Perelman proved a more general result: the geometrization conjecture that was initially stated by William Thurston: https://en.wikipedia.org/wiki/Geometrization_conjecture https://en.wikipedia.org/wiki/Geometrization_conjecture

jwilber 5y ago

From wiki: In August 2006, Perelman was offered the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." “ Some people are built differently, what a legend.