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Assume that I'm interested in a certain broad field of mathematics (say A). Also, assume that have done a honors thesis during my BSc in a certain narrow-ish subfield (say A.1).

Suppose that I'm interested in applying to a certain outstanding university mostly because its graduate program offers, among other things, lots of good specialized courses and research seminars concerning various aspects of field A and related areas of mathematics (way more and of way higher level compared to any other university, as far as I know).

Now, suppose that (despite the breadth of the courses offered) the outstanding faculty members of this university who work in field A are interested in a subfield that is different and not very related, as far as I know, (say A.2) from the one I've done my bachelor thesis in (A.1).

Assume that I am eager to know more about this different subfield because it "feels" promising and interesting, but I'm not particularly familiar with it yet.

Would my current (but fixable during the courses) lack of knowledge about the main subfield of interest of the faculty and my different "specialization" make my statement of purpose (where I've to explain why I've chosen the program) and therefore my application weak?

In short, is it acceptable to convey the following "message" in a statement of purpose?

"I love the broad area A, my specialized knowledge of which is limited to a narrow-ish subarea A.1; and I'm ready to get the most out of the many courses and seminars that you offer. I'm not quite familiar with A.2 (the main subfield of interest of the faculty), but I'm willing to learn something new and I've a hunch that it's going to be fun."

Disclaimer (and clarification): This question was originally raised by a former classmate of mine and I post it on this board (with his approval) because I may probably find myself in a kind of similar situation in short time.

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  1. It is dangerous to super-specialize prematurely. One needs to know where one is located with respect to the rest of (related) academia. So doing general broad based courses is a perfectly fine goal.
  2. The way selection works is, if every professor in the institute of interest, finds a person who has already worked in their area of interest, then you may not be selected. But the odds of that happening is low.
  3. Professors always don't expect someone who would be productive right from day one. Most are prepared to spend some time preparing the student for the work. Showing the capacity for doing work, by citing previous work should be strong enough. Most professors can access quality of your work and make a judgement, even if the work is not in their area of specialization.
  4. Also Professors sometimes appreciate talent coming from outside their domain, because they may sometimes bring with them some interesting perspective.
  5. And lastly, one should be as honest as possible, because getting rejected for right reasons is sometimes better for you, than getting selected for wrong reasons. I mean, it could be a disaster, if one gets selected for who one is not!
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It is not at all uncommon (especially at BSc level*) for people to go into vastly different areas; their knowledge and skills in one area can often add to their abilities in another.

For simplicity in what follows, let your original field be A and your newly taught field be B.

Things can get a little hairy if you are determined to stay in A for your project (or, at least, it might not be as successful as having a supervisor with strong knowledge within A). If you are willing to make the transfer to B, and if you are willing to accept that further academia in A may be rather impeded (again, compared with originally taking a project in A), I do not see any issue with such a switch at this point.

There might be some circumstances in which "belonging to the same parent field" does not mean that A and B are even remotely related in their approaches and terminology in the modern day, and catching-up might be a lot of hard work. It is, I suppose, up to you and your friend to figure this out with available literature.

I am probably not giving you any information above what you already know - but I had to say the above just in case. What I would really like to say is the following: If you want to follow a path, follow it. Prior knowledge adds something to the group - not only will you have some unexpected tricks up your sleeve, but you may become the "go-to guy" for certain problems, and their knowledge will provide you with yet another set of tricks. Such mixing rarely comes to a detriment of either side of the table, provided you are willing to learn and they are willing to stretch you.

But if you are unsure on a path, do not take it simply because you dont feel you have a choice. I personally fell into that trap - in my case, joining a doctoral training centre in a much more general field than my own (Theoretical Physics -> Nanoscience). It turned out to be so general that most of the material was hand-wavy and useless. Now I'm back doing what I love, but it was half a year of plain old misery for me.

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