I just started my 2nd year of master and it's already time to apply for a PhD in math, deadlines are in ~1+ month.
In most samples, students had a clear idea of what they would like to study. They write that they have studied this and that, and that they would like to be under Prof. X's supervision because they worked on his papers.
As for me, I liked most math topics that I have studied, with very few exceptions. In short, I'm undecided, exactly when I buy an ice cream and I'm facing 20 good flavours.
Most universities, on their application page, offer "undecided" among the options (algebra, number theory, ...). Although they may say "it's totally fine" (because everything is nowadays), what is the truth behind the appearances?
Stories from people who are on admissions committees or from [un]successful applicants who wrote that in their SOP are most welcome.
EDIT: I successfully got admitted to some really good US universities with "undecided" in my SOP