All they have to do is denounce those people that tried to steal Perelman's proof and claim it as partially their own.
Obviously he has a very strict standards for honesty, and will not accept anything from the mathematics community, unless he perceives them as honest.
I think mathematicians do not think they need to do anything because they think it is very clear by now who solved the Poincare conjecture and who did not, and there is no need to embarrass a very influential mathematician over the whole deal. But Perelman does not see it this way, and wants to be vindicated explicitly.
Note that this has been strongly disputed by certain parties mentioned in the article that do not come off that well, but the New Yorker stands by the reporting and has not issued corrections.
Perelman, by casually posting a proof on the Internet of one of the most famous problems in mathematics, was not just flouting academic convention but taking a considerable risk. If the proof was flawed, he would be publicly humiliated, and there would be no way to prevent another mathematician from fixing any errors and claiming victory.
This is both frightening and disappointing. I can hardly imagine how much advancement in the field is being held back by this attitude.
They think of this math as the math they saw in high school, which is focused on accuracy. No such thing is true. At this level, mathematics is about insights and creativity. It's like writing a novel, posting it on the internet, and having someone else fix the typos and claim it as their own.
He has never asked for anything like that, and his decision not to accept the Fields was made long before the controversy with Yau. He is a purist who doesn't see the value in that type of recognition. The work is reward in and of itself. His choice to quit mathematics, however, was based on his complaints about its politics and lack of integrity, but even this was not directed at his work on the Poincare. Since this is so far off the mark, I thought that that article you cited must be really bad. It isn't. Why don't you try reading it again?
What do you mean by long before? The paper was published in June of 2006, but an abstract came out in April of 2006. Also in the beginning of June of 2006 Yau made some statements minimizing Perelman's contribution to the solution.
Perelman got the Fields medal in May 2006 and finally refused it in June 2006 when a team of mathematicians came to his home to visit him.
So Perelman was most likely aware of the paper before he was even awarded the medal and was definitely aware of the controversy when he finally refused it.
Your other statements are attempts to dive deep into semantics in order to change the meaning of the cited article. You make it sound like his decision to refuse the medal and quit mathematics were completely separate, and based on different reasons, but the persons that interviewed him reported differently: "The prospect of being awarded a Fields Medal had forced him to make a complete break with his profession."
Regarding his comments about lack of integrity in mathematics, it is true that he kept his comments general, but that does not mean that they had nothing to do with his work on Poincare. Since Poincare was the last big thing he worked on and he worked on it for about ten years, finishing only a couple of years before the interview, and since it is by far the most important thing to happen to his career thus far, one might think that his comments on the field of mathematics might have had at least something to do with his work on Poincare.
There was an interesting BBC radio show that looked at this chap's achievements and 2006 win: http://www.bbc.co.uk/radio4/science/thematerialworld_2006082... (it's still listenable but is RealAudio, though VLC played it fine for me - I also just tested it through a US-based VPN and seemed to be OK unlike a lot of BBC content.)
I also recall seeing a Marcus du Sautoy documentary where he went to Russia to try and meet Grigori (without much success as I recall). I can't find a watchable version of that, though.
That du Sautoy documentary was lame and the bit where he goes to Russia was cringe worthy. du Sautoy must have known that Perelman would never see him. If Perelman turned down the Fields then appearing on TV was hardly going to excite him.
Ultimately that entire series was about du Sautoy going 'look at me, I'm clever'
I got the feeling it was a bit "dumbed down" for the masses (à la the Xmas Lectures since about 1990) but I was still impressed. du Sautoy is still æons smarter than I am when it comes to math, after all.. :-)
That caught my eye as well, my interpretation was that it was intentional as Perelman has retired from mathematics and is no longer a professional mathematician. I can't imagine any other reason for the odd phrasing and this being the NYT I'm going to assume some modicum of facility with the english language.
I think Perelman himself wouldn't care too much what you call him. He would say something like:
“Everybody understood that if the proof is correct then no other recognition is needed.”
(As he said before.)
You don't need a license to be a mathematician. There's this conjecture, undecided for 100 years. Whoever solves it gets the glory. He doesn't have to be a "mathematician" or an "expert". A correct proof is all it takes.
He solved it, and all that's left for us is to discuss whether he's an expert or not:)
At least in the context of the headline, you can assume any mathematician being offered a $1m prize is likely to be an expert.
In wider usage, I guess you could say "Math expert" to distinguish between students and professionals, but it's still as clear as mud because it doesn't state which field of mathematics the person is an expert in.
Obviously he has a very strict standards for honesty, and will not accept anything from the mathematics community, unless he perceives them as honest.
I think mathematicians do not think they need to do anything because they think it is very clear by now who solved the Poincare conjecture and who did not, and there is no need to embarrass a very influential mathematician over the whole deal. But Perelman does not see it this way, and wants to be vindicated explicitly.