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I am planning on pursuing a ph.d. in mathematics, however, really my research interests often lie in the intersection of mathematics, logic, and theoretical computer science. Thus, in many of the universities I would like to apply to, I do not have very many common research interests with the mathematics faculty, but instead with the computer science or philosophy faculty.

Is it fine to mention that my primary research interests are actually outside the mathematics department in my letters of intent, should I try to word it to sound more like I am simply interested in interdisciplinary collaboration with other departments, or would even that harm my chances of getting accepted?

I suppose another question I have implicitly is: "...Or would it be better for me to apply for a ph.d. in Computer Science instead if the person at the university who most matches my research interests is in the computer science department?", or at least, what other factors should I consider to determine which way I should go (Math, or Computer Science) for a given university?

For what it's worth, my undergraduate degree is in mathematics with a minor in computer science.

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    I think you've answered your own question. Other research interests and cross-disciplinary pursuits can be fine (as in, here's my work in math, but I've also done some things in computer science), but if they make it seem like you would be better off in another department... maybe that's the department you should be applying to.
    – Jeff
    Dec 7 '16 at 18:09
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    Have you checked whether any of the faculty have more than one appointment, or if they already mentor students from other departments? Depending on the university and subject, the lines between departments and between graduate programs may not be the same.
    – Bryan Krause
    Dec 7 '16 at 18:14
  • @BryanKrause, in some cases I have found universities that specifically pride themselves on such interdisciplinary research opportunities, so I'm not as concerned there, but for the other schools I think that's a good idea, thanks! Dec 7 '16 at 18:18
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    I think if there is a particular professor you are interested in working with, it is totally appropriate to contact them ahead of time (see previous Academia.SE questions if you need guidance). It would be appropriate to ask what types of graduate students they mentored in the past (e.g., to a CS professor: "have you mentioned math PhD students in the past") and if so, which program they think is a better fit for you if you came to their lab - maybe they would see your math background and suggest the coursework of the CS PhD would be a better addition to your training, for example.
    – Bryan Krause
    Dec 7 '16 at 19:33
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    "Interdisciplinary" is one thing. "Not really interested in working with any of the faculty in this department" is quite another. Dec 7 '16 at 23:36
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You don't need to contact the other professors right away, but you could do it now, and at some point before deciding where you're going to go, yes, good idea.

Yes, do check for cross-department affiliation. Careful with this -- sometimes a department will list Prof. Other-Department on their list of faculty members, but later will hedge that if you try to have Prof. O-D be your official advisor. If it was just done to inflate your department's faculty list, things could get a little challenging. This is not necessarily an unresolvable problem, but it could be helpful to ask the department about this sooner rather than later. Also see if there are one or two faculty members in your target department who could be natural committee members for your thesis committee.

Should you skip all of that and apply to a computer science department instead? To decide this, consider:

  1. What the PhD exams cover in each department

  2. What teaching assistantships each department has, and what your duties would be

  3. Are there other courses in the target department you'd be interested in taking

  4. Where do you think you'd feel most at home

At the end of this process, you may end up applying for a couple of math or applied math programs and a couple of CS programs.

I looked at your profile and have an additional comment. A long time ago I was familiar with CS at UW-Madison, which was, and probably still is, strong in theoretical CS, and had lots of TA-ships. Such a program would have been a reasonable choice, but you would have had to choose one or two additional areas of CS to do an exam in. That might have been an argument for choosing the math program as your home department. I'm just mentioning this one as an example of how you might apply the considerations in my proposed list.

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