When applying for the PhD program in mathematics, usually, one is not required to specify in what field (e.g., PDE, dynamical system, etc.) he/she intends to do. However, I don't know whether it will be disadvantage that one does not have a specific field in mind at all.
I ask this question because some people suggest the students who are applying to the graduate school should talk to or connect with the professor who is in the school he/she intend to apply for. But if one does not even have a specific interest in mind, how can he/she talk to a professor about his/her application? (Even if the student is interested in analysis, say, there are lots of sub-field in analysis.)
So, here is my question:
Does one need a very specific field in mind to apply a PhD program in mathematics? Would this be thought as advantage or disadvantage of an application?
At least for most graduate programs in pure mathematics in the US, there's no need to have a specialization in mind when applying. [This may be very different in other countries.]
It's valuable to demonstrate in your application that you have studied some serious mathematics, by discussing undergraduate research or advanced coursework. However, there's no implication that you intend to focus on the same fields in graduate school.
It's common to indicate an interest in a few possible specialties, usually at a level of detail ranging from "algebra" to "analytic number theory". If you are completely undecided, then that could come across negatively, by suggesting a general lack of enthusiasm. However, being too specific is also problematic. Matching a possible advisor too closely comes across as pandering, while being specific without matching anyone makes it look like you aren't a good fit for this particular department.
Overall, the general feeling is that incoming graduate students don't know enough to make well-informed decisions about specialization, and that anything they say is a little unreliable because their interests may shift as they learn more. From that perspective, it's not worth worrying about this too much.
As for talking with professors, at least at the schools I'm familiar with this will not increase the chances of admission (the decisions are made by a committee). After you've been admitted, it's important to talk with faculty and try to gauge the chances of finding a suitable advisor. However, that can and should wait until February.