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Given most American universities only accept applications in only one field, I'm trying to figure out what are the advantages/disadvantages of applying to a CS theory program vs an applied math program given one's interest lies somewhere in both departments.

To be more specific, my areas of interest in decreasing order are 1. Combinatorics (Both algebraic and extremal), 2. Optimization (Both convex and combinatorial), 3. Probability theory, randomized algorithm, and information theory.

I don't exactly know on what or with whom I want to work which makes applying to graduate programs a huge headache. So far my understanding is applied math programs are more flexible given CS theory groups are usually very small and focused. On the other hand, I feel a CS degree would fare better in industry if one happened to venture that path.

So to reiterate my question, for someone who doesn't exactly know what he wants to do but generally interested in aforesaid topics, which is better? CS Theory or Applied Math

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    $\begingroup$ Apply to both, of course. Why limit your options before you have any? $\endgroup$ – Jeffε Oct 31 '11 at 12:49
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    $\begingroup$ When applying, make sure the local math department culture interprets "applied math" more broadly than "differential equations". This is less common than it used to be, but is still worth watching out for. $\endgroup$ – Neel Krishnaswami Nov 2 '11 at 9:24
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    $\begingroup$ @JɛffE Most of the schools restrict applications to a single department (Berkeley, Cornell, Princeton, ...) Also, I've heard (from TCS faculties) admission to theory programs is much more competitive than applied math due to their size and it's almost impossible to get in without publications. So I'm contemplating given one shot at most of these schools, is there any benefit that justifies the risk of applying to theory programs as opposed to the applied math. $\endgroup$ – user972432 Nov 2 '11 at 12:07
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    $\begingroup$ I've never heard of such a restriction! It might be something imposed on the individual departments by the university to simply campus-level paperwork. I suggest writing the departments you're interested in directly and asking them how to apply to more than one program. If they blow you off, you didn't want to go there anyway. (On the other hand, the restricted departments may have decided that they only want students who are sure about what they want. Idiots.) $\endgroup$ – Jeffε Nov 2 '11 at 13:16
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My two cents are that in my university we've had both mathematics PhD students working on computer science questions (and faculty in the math department with interests in computer science), as well as some computer science students working primarily on purely combinatorial problems.

You might be right that it's sometimes easier to work on CS questions as a math student, rather than on pure math questions as a CS student. Keep in mind that at least in the first two years these two kinds of programs might be fairly different in content. As a math student you will be expected to take core math courses as real analysis, complex analysis, topology, algebra, etc. Combinatorics is usually not part of this core. For a CS program there will be a core CS requirement, which usually involves taking some mix of theoretical and more applied courses. While the core in a math program is fairly standard and strictly enforced, the core in a CS program tends to depend a lot on the program, and the requirements might be more flexible.

However, all that is not really of primary importance (although it will be loads of work) and is all over within the first two years. I understand it's hard to know what you want to work on before you're in grad school, and many students change their fields. Nevertheless, I would encourage you to look at the faculty pages of schools you are considering, see what professors are working on, and write several emails to faculty and students. PhD level studies are much more about personal relationships and personal drive than they're about a program as a whole. Good programs at the PhD level in my view are distinguished by a strong faculty, and an energetic research culture, rather than by curriculum. You should inquire from faculty and current students about questions like the level of collaboration between math and CS departments. And you should really try to find faculty that have a mix of interests that appeals to you. It's a good idea to write to them to express your interest as well.

As far as industry jobs, I'm not sure there is a huge difference between a CS theory degree and an applied math degree. But I am not very knowledgeable about this.

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    $\begingroup$ As for academic jobs, definitely some CS schools (not necessarily the top ones) will wonder about your ability to teach CS classes if you have through-and-through math training. For industry jobs, it depends, but for places like Google/Yahoo/M$, it helps if you understand software/hacking at some level and can demonstrate it (even if your degree is in applied math). I don't think the degree limits you per se, but it might require you to be creative in demonstrating your skills $\endgroup$ – Suresh Venkat Oct 31 '11 at 16:23
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First, I don't think it is true that at most universities you can only apply to one department or the other. I know many people who have applied to both math and CS departments, particularly at MIT where lots of theoretical computer science is done in the math department.

There are also several joint programs between math and CS departments that seem well suited to your interests. Some that come to mind are the ACO programs at CMU (here) and GAtech (here). At MIT, it is reasonably easy for you to take an adviser from either department, so it does not make a big difference whether you are in EECS or applied math.

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  • $\begingroup$ As you mentioned specific programs, I think places like MIT or Berkeley are ideal for someone who hasn't specialized in a particular field as they have huge programs and one can always find someone who matches her interest no matter what it is. But ironically it's almost impossible to get in to those places without publications which requires specialization at the first place! $\endgroup$ – user972432 Nov 2 '11 at 12:23
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I am a PhD graduate student in applied math who faced this exact problem last year. At my university, the applied math track offered much more flexibility in terms of course requirements. The CS track required various theory courses, which I wanted to take, but also required courses in networking, operating systems, and other things that held no interest for me. The applied math track basically allowed me to mix and match courses from either department with almost unlimited freedom. I actually am taking more CS theory classes than I would have been allowed to as a CS student.

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  • $\begingroup$ But you're taking the networking and OS classes anyway, of course. Right? Right? $\endgroup$ – Jeffε Nov 2 '11 at 13:22

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