the troublesome parts of the proof are in the second half of Perelman's second preprint. For instance, Fields medalist Shing-Tung Yau said in 2019 that[30]
>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. About this I have no doubts [...] 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 [...] 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.
