The Poincaré Conjecture, Explained | Hacker News Reader

The Poincaré Conjecture, Explained

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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

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.

link?

This New Yorker article was ostensibly about the controversy, but became embroiled itself...

https://www.newyorker.com/magazine/2006/08/28/manifold-desti...

https://en.m.wikipedia.org/wiki/Manifold_Destiny

> 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?

> The beauty of maths is that it exists independent of all other considerations. This seems to be where Perelman comes from.

Okay, so why publish at all? Why not just sit in mathematical nirvana in a cave, having discovered the math-God?

I think many do this, though it’s impossible to tell of course. I do know a couple of mathematicians (one sometimes discussed on HN) who only circulate proofs, or conjectures, among themselves.

Because you wish for others to share in its beuty.

> But you didn't choose a compelling form of words to advocate your interest to other people. Why not?

I thought it was a good question and the wording was compelling enough.

I thought it was a good question too. I was prompted to think about Perelman the person by it.

It's the style of the remark that struck me:

> we should probably ask ourselves why

> 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.

So, I’d just like to point out that this is a very naive perspective on the philosophy of mathematics. Math is an intrinsically social activity, because you need to lead other mathematicians through your proofs/arguments.

> 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.

Proving the Poincaré conjecture wasn't enough? Why would he owe more than that? It's significantly more return on the investment of energy than anyone expected.

I think it's important to remember that the Millennium Prize problems were chosen partly because they would drive the development of new mathematics (it’s not as though the Poincare conjecture or Fermat’s last theorem are useful things). Ideally, when someone solved one of these problems, they would train a new generation of mathematicians with their toolbox. Gromov benefitted from this - his work was built using technology that people had spent their entire careers exploring and clarifying. Perelman didn’t really do anything like that, so it’s fair someone like Gromov would feel Perelman didn't pay it forward.

Of course, this whole debacle could have been avoided if Princeton had given Perelman tenure after he proved the Soul Conjecture. To an extent, I agree with Perelman’s opinions of the mathematics community, I just felt Gromov’s opinion was a bit more defensible than you were giving it credit for.

> it’s not as though the Poincare conjecture or Fermat’s last theorem are useful things

Useful for what? A lot of people see mathematics as and end unto itself.

Did you seriously not read the rest of my comment, which was about how mathematics is practiced as an ends unto itself?

Yes, of course I read your comment. I have read it again now. I still don't see how it is about mathematics being an end unto itself.

What I mean by mathematics being an end unto itself is that mathematical results have intrinsic value. That is, they have value in and of themselves, regardless of how they may or may not be used.

Perhaps you thought I meant that mathematical results could be used to discover more mathematical results? No, that is not what I meant.

So, I think the point you’re missing is that how results are proved is just as important as the existence of a proof (in fact, I’d say it’s often more important). That’s why people are often excited about new proofs of old theorems, and why the Simon’s foundation chose certain problems to be “Millennium Problems” - the techniques developed to solve the problem would, ideally, drive progress in that area of mathematics.

The Poincaré conjecture is interesting because it was a simple statement that ended up being very hard to prove. How someone proved it, and understanding why such an innocuous statement is so difficult to prove, is far more interesting than knowing that the obviously-true-sounding statement is true.

> I suspect Perelman would have followed Satoshi's lead

Or perhaps they are the same person?

https://www.reddit.com/r/CryptoCurrency/comments/nez1n1/sato...

There is a very large number of people who hold similar views. The fact that these two characters each hold that view is very weak evidence for the claim that they are in fact two characters from one person.

Tiny bit of nitpicking - he declined the prizes. I know that's what you're saying but phrasing it that way ('which denied') makes it sound like he was denied the prizes by someone else. He just didn't want them.

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.

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.

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

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.

Shouldn't it be renamed now that it's proven?

The Poincaré-Perelman theorem seems more accurate now.

> 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

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

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

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