[EDITED to make the question more suitable for MO. See meta.mathoverflow.net for discussion about re-opening.]
According to Wikipedia, Shing-Tung Yau expressed some doubts about Perelman's proof of the Poincaré conjecture in his 2019 book The Shape of a Life. Yau wrote
"I am not certain that the proof is totally nailed down. … there are very few experts in the area of Ricci flow, and I have not yet met anyone who claims to have a complete understanding of the last, most difficult part of Perelman's proof … As far as I'm aware, no one has taken some of the techniques Perelman introduced toward the end of his paper and successfully used them to solve any other significant problem. This suggests to me that other mathematicians don't yet have full command of this work and its methodologies either."
To what extent are the doubts that Yau expressed well-founded?
Here is a fuller quotation of the relevant passage from Yau's book The Shape of a Life (pages 258–260).
One problem I'm not actively working on is the Poincaré conjecture, as I'm happy to put the controversy surrounding it behind me. But I can't keep my mind from turning to that problem, upon occasion, and I still have some lingering doubts that—if expressed out loud—are likely to get me in trouble. Although it may be heresy for me to say this, I am not certain that the proof is totally nailed down. I am convinced, as I've said many times before, that Perelman did brilliant work regarding the formation and structure of singularities in three-dimensional spaces—work that was indeed worthy of the Fields Medal he was awarded (but chose not to accept). Perelman built upon a foundation painstakingly laid down by Hamilton and carried us further along the path laid out by Poincaré than we've ever ventured before. About this I have no doubts, and for that, Perelman deserves tremendous credit. Yet, I still wonder how far his work involving Ricci flow "technology" has taken us. And I also can't keep from wondering whether another approach—making use of some of the minimal surface techniques I developed many years ago with Bill Meeks, Rich Schoen, and Leon Simon—might lend some clarity to the situation.
In 2003, Perelman told Dana Mackenzie, a reporter for Science magazine, that it would be "premature" to make a public announcement regarding a proof of the geometrization and Poincaré conjectures until other experts in the field weighed in on the matter. Confirmation of this proof resided largely with outside "experts," given that Perelman receded almost completely from the mathematics scene, which is a great loss to the field. The thing is, there are very few experts in the area of Ricci flow, and I have not yet met anyone who claims to have a complete understanding of the last, most difficult part of Perelman's proof.
In 2006 or thereabouts, a visiting mathematician who was knowledgeable about this area stopped by my Harvard office to reproach me for raising questions about Perelman's work. Yet he admitted, when I asked him, that he did not entirely grasp the latter part of Perelman's argument. That's no knock on him, as that admission puts him in a rather sizable group. In fact, I don't know whether anyone else, including Hamilton, has fully gotten it, and I'd put myself in that category as well. As far as I'm aware, no one has taken some of the techniques Perelman introduced toward the end of his paper and successfully used them to solve any other significant problem. This suggests to me that other mathematicians don't yet have full command of this work and its methodologies either.
Hamilton, who's now in his seventies, has told me that it is still his dream to prove the Poincaré conjecture. That does not mean that he thinks Perelman did anything wrong. Hamilton, a truly independent spirit, is not one to follow in someone else's footsteps, nor would he be inclined to "connect the dots" of another's argument. He just may want to do it his own way and complete his life's work of the past three and a half decades.
Nevertheless, that still leaves me with the sense that this situation is not unequivocally resolved, perhaps leaving theorems of incredibly broad sweep hanging in the balance. Expressing my doubts on this subject, I know from experience, is a politically fraught proposition. But for the sake of my own questions—and for mathematics as a whole—I'd still like to be more certain of where we stand. If that makes me a pariah, so be it. In the end I care more about mathematics—the path I chose to follow more than a half century ago—than I do about what others think of me.
Best Answer
First, let me make some preliminary remarks. We sometimes like to think that "being proved" is a black-and-white property, but in fact there are shades of gray. At one extreme are things like the infinitude of the primes, whose proof every mathematician understands. But then there are results that are widely accepted but no proof has appeared. Vladimir Voevodsky has pointed out that "a technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail," and that this practice can lead to false statements being erroneously accepted as proved; recognizing this point led Voevodsky to spend much of the later part of his career on computer verification of formal proofs. Although the mathematical literature is generally very reliable, it is far from perfect; this point has been addressed in more detail in another MO question about the extent of wrong papers in research mathematics. So in vast majority of cases, "being proved" isn't about being 100% confident that there is no error; it's about whether the proof has been sufficiently scrutinized that the chances of a serious mistake are negligible.
Returning to Yau, if you look carefully at what he is saying, you will see that, technically, he does not say that he thinks Perelman's proof is wrong, or that it has a serious gap, or even that there are parts of the proof that nobody understands. He says only that he is not certain that the proof is totally nailed down, and that he has not met anyone who understands the most difficult part of the proof. He also points out that if a powerful new idea is properly digested by the mathematical community then it usually leads to the solution of new problems, and that if this has not happened with the most difficult part of Perelman's proof then it probably means that this part of the proof deserves more study.
In principle, calling for the mathematical community to devote more time to studying an important and difficult proof in order to "nail it down" and acquire a "complete understanding" and a "full command of this work" is unobjectionable. In the past, I have heard colleagues say that the original work of various Fields Medalists—Hironaka and Freedman come to mind—was very difficult to understand and that there was a need for the community to study and assimilate those groundbreaking ideas more thoroughly. In both the cases of Hironaka and Freedman, the community has indeed put in effort to study their work, and rich dividends have resulted, so this type of activity is definitely worth encouraging. Note that this doesn't mean that the original proofs were wrong or had serious gaps; it just means that the proofs moved closer to the infinitely-many-primes ideal of universal understanding, and the chance of an unnoticed significant gap or error was driven down even closer to zero.
Unfortunately, Yau chose to phrase his remarks in a "politically fraught" manner that he knew would "get him in trouble." He says things in a way that (probably intentionally) gives many readers the impression that he is casting doubt on the correctness and completeness of Perelman's proof (even though, as I said, technically he doesn't explicitly say that the proof is wrong or incomplete). The book appeared in 2019 but the most recent conversation he cites was from 2006. He makes no mention of recent research in the area which does in fact apply Perelman's ideas to solve new problems.
It should therefore not be surprising that the consensus of the mathematical community is that Yau's remarks do not pose any serious challenge to the conclusion that Perelman's proofs—especially of the Poincaré Conjecture, which involves fewer technicalities than the Geometrization Conjecture—are correct. There were at least three separate efforts which came to this conclusion. Kleiner and Lott's detailed notes say, regarding Perelman's original papers [51] and [52]:
Similarly, Morgan and Tian wrote:
There is also the account of Huai-Dong Cao and Xi-Ping Zhu, which Yau himself refereed.
On top of those three detailed accounts of Perelman's proof, there have been more recent developments. Terry Tao mentions the recent survey by Richard Bamler. Moishe Kohan mentions Kleiner and Lott's Geometrization of Three-Dimensional Orbifolds via Ricci Flow and Bamler and Kleiner's proof of the Generalized Smale Conjecture. So contrary to the impression you might form from what Yau said, the community is indeed continuing to milk Perelman's ideas and apply them to solving new problems. If there are specific technical points which Yau thinks are obscure, I am sure that other researchers would be happy to address them if Yau were to spell them out explicitly. Until then, there is no credible reason to doubt the fundamental correctness of Perelman's arguments.