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.
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.