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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?

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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.

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To add to what the Anonymous Mathematician said (so again, this has to do with United States)

If you have not already decided on a specialisation, you should certainly not try to shoehorn yourself into one just for the sake of applying for graduate school. If you have already decided on a specialisation, however, you should certainly specify it and try to talk to faculty members before and/or during the admission process. This is not so much to improve your chances at the school per se; this is mostly to find out whether that school would be a good fit for what you want to study.

When I was applying for graduate schools I had just one such experience. I was already quite sure about what I wanted to study, and so contacted some faculty members at places where I applied to who may be good advisors. One of them told me that (a) I have a pretty strong application and they will probably admit me (b) But he is personally busy with his existing students and won't want to take another one (c) There's no one in the same department that he sees as having similar enough interests to be a good advisor for me and (d) The department is running a bit low on money so unless I get a fellowship somehow I will have a hellish teaching load.

In the end I took an offer from another school.

For emphasis, however, what I mean by specialisation is very narrow: "partial differential equations" is still too broad. For my advice above to apply you need to be able to say which type of PDEs you are interested it (elliptic, parabolic, evolutionary, dispersive, transport, kinetic theory, fluids, and/or optimal control to list a few) and to be able to hold an informed (not necessarily expert) discussion on the subject and why you want to study it.


For other countries the system can vastly differ; this is especially so for degree programs which require a Masters or equivalent degree for admission. Many of those programs require submission of a research proposal and admission is contingent on the research being likely to be able to be conducted at said university and that a suitable advisor can be found for your research proposal. But for these kinds of degrees the opposite of the expectation described by Anonymous Mathematician holds: the incoming students are expected to be sufficiently educated (by the Masters degree) to make informed decisions and are also expected to have a good idea what is involved in pursuing PhD research and know what they want to work on. But you can general tell the expectations by reading up on the qualifications for admission and on what the departments/universities expect on the application form.

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    The experience you describe with the professor who disclosed the issues you might have at his school make this an excellent answer--those kinds of questions might not even have occurred to a person to think about when they are looking for a program. – msouth Nov 13 '14 at 2:14
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In France the picture is very different from the US, since you start right away your PhD research (so you must have an adviser from the beginning, and an idea what you will be working on).

If one includes the M2 (second-year master) in graduate studies, then anyway most of them are specialized, so by picking one you do choose a specific field. You can switch for your PhD, but for some fields with many prerequisites it is usually a bad idea.

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