I'm in my last year of an undergraduate degree in pure maths, doing my thesis on probabilistic combinatorics. In Australia (at least at my university), Hungarian-style combinatorics is classified as "pure".

I'm currently thinking about PhDs and where to apply. One thing that I've noticed is that combinatorics is often classified under "applied maths" (for example, this is the case at MIT).

One important reason I'd consider pursuing a PhD at an American university is for the coursework component -- I'd like to broadly improve my foundational understanding of mathematics, and I like the possibility of changing my area of focus after deeper exposure to different areas of mathematics. However, I have comparatively little interest in applied mathematics coursework (other than combinatorics, of course).

I have a submitted paper in combinatorics, my undergraduate thesis will be in combinatorics, and by far my strongest reference will be from a combinatorialist. Other than my supervisor, my strongest potential references would come from areas that everyone seems to agree are "pure". The higher-level classes in my transcript will also be almost entirely "pure".

  1. For American universities which have both a pure and an applied math program, which one does one typically choose to study combinatorics? Is it flexible? With a cursory search, I wasn't able to find a university quite so explicit about it as MIT.

    For universities that classify combinatorics as "applied":

  2. Given my situation, would my application be stronger to pure or applied maths?

  3. Can one do a PhD in pure mathematics with the (tentative) intention for the research component to actually be "applied" (combinatorics)?

  4. Can one do a PhD in applied mathematics but actually take mostly pure coursework?

  • 1
    My suggestion is to find the people whose research area is combinatorics, then the schools they are at.
    – scaaahu
    Oct 9 '13 at 7:28
  • I am not a combinatorialist. But I see a number of good pure mathematicians are working in combinatorial group theory, which is related to geometric group theory, random walks and computer sciences. I suggest you talk to some experts about these areas of mathematics and see if they are interesting for you. I strongly believe that these areas will be important parts of pure (and to some degree applied) math for a long period of time.
    – user4511
    Oct 9 '13 at 8:25
  • As in answers and comments... "combinatorics" is not an explanatory label, most of the time. Often, it is a reasonable label only in the sense that all matter seems to be made of bosons and fermions... but without explanation? To my mind, much of the best "combinatorics" is really better described by other labels... hence, you may fail to find such work under the label "combinatorics". Jun 14 '17 at 0:22

If you come here for "general advice", I am afraid you'll be disappointed.

  1. You should choose based on the people there. For each school you apply to, look through both the applied and pure departments and see where the mathematicians who work in combinatorics fit in. Most large research universities will list their faculty by research interest, or have dedicated research group webpages. For example, based on this page you probably do not want to apply to Princeton's pure math program.

    Note that many schools do not, at least on the graduate admission level, differentiate so strictly about going into applied versus pure mathematics.

    (For choosing the school, since you already have research experience, you should look up articles which you find interesting and find out where those authors currently reside.)

  2. It really depends on the school. If a school has no combinatorialist in its pure maths department, and you apply there with an intention of studying combinatorics, your chance of admission is practically 0. Similarly in reverse if all the combinatorics is done in the pure maths department for the school you are applying to.

    There is no "one size fits all" solution.

  3. This again depends on the school. Some schools allow it, some schools don't. In many departments it is expected that you find an advisor from within the department and do your dissertation research on a subject that your advisor is interested in and/or an expert on. In other departments more leeway are given to students interested in more interdisciplinary subjects to be jointly advised by two advisors (possibly from different departments with different expertise). But remember, if you are going for a PhD you will need to find an advisor in any case. Shoehorning yourself into a situation where it maybe difficult to find a professor in your own department who is willing to advise you is, in my view, generally not recommended.

  4. This again depends on the individual policies for the schools. Some departments have very strict requirements on what the students must learn for their comprehensive exams; some, not so much. Most schools with graduate programs have very clearly written information on their websites about what is expected in their degree plans. For example, here's Harvard's version for the Applied Maths degree, though it seems they are a little bit short on the exact details.

The only "general advice" I can give is this:

Don't apply to graduate schools blindly, especially since you have a confirmed research interest. Do your homework and find the experts with whom you would like to study, and apply to study with them. Write to them in advance to confirm that they are interested in taking on students, and possibly solicit advice about other possible individuals if they are not.

A good research school may not be strong in the field of research you want to do. Always, always check before applying.

  • This is a fantastic answer, thank you for taking the time. It sounds like I'm going to be doing some research and email-sending for each institution.
    – Matt
    Oct 9 '13 at 9:46

I think you are overgeneralizing from an example here; the placement of combinatorics in applied math at MIT is due to ancient historical department politics, and regarded as an amusing quirk by those of us educated at other schools. While I completely agree with Willie's answer that you should carefully research schools you want to attend, at most schools combinatorics is regarded as a perfectly fine branch of pure mathematics. If you want to do the pure math coursework, and you see yourself as wanting to do pure mathematics, then probably you should just apply for the pure math programs at places with some strong combinatorialists (Berkeley, Penn, Davis, San Diego, Michigan, Minnesota, etc.)

  • That's definitely reassuring. That short list is invaluable, thank you so much! Perhaps my generalization is understandable given that apart from Berkeley and MIT (and CMU, which seems to have an applied-focused maths faculty), all the "top" american universities (up to rank about 20 on the QS rankings) have almost no combinatorialists in their maths faculties. I had assumed they were hidden away in applied, stats or CS.
    – Matt
    Oct 9 '13 at 15:53
  • @Matt That's an extremely restrictive criterion, and false anyways. Berkeley has several excellent combinatorialists listed in the algebra research group. Anyways, you'd be better off looking at a list like: grad-schools.usnews.rankingsandreviews.com/… The US News rankings are somewhat bullshit, but at least are a starting place. Oct 9 '13 at 17:13
  • @Matt Ah, sorry, I hadn't read the "aside from Berkeley" part of your comment (though Michigan and Georgia Tech are in the top 20 of the QS list for math, and they have easily findable combinatorialists). Anyways, the point stands that top 20 in the world is not a very good criterion. Minnesota, UIUC, UCSD and Rutgers are all excellent graduate programs (top 25 in the US according the US News list) with multiple combinatorialists in their pure math faculty, and which also have the breadth to do plenty of other things if your interests change. Oct 9 '13 at 19:05
  • Thanks for the US news list! That's very helpful. I wasn't meaning to say I'm only interested in a top 20 school, but not being American I had no other way to know what schools exist and are reputable (i meant US unis in the top 20, not top 20 US unis, sorry). My supervisor is English and knows a lot more about universities there. It turns out I was wrong about that anyway, after seeing UCLA so high on the US news list I took a closer look and there are definitely combinatorialists there, though they're not listed under a research group for some reason.
    – Matt
    Oct 10 '13 at 2:41
  • Like Ben Websiter said, at times excellent combinatorialists will list something other than combinatorics as their research interest (algebra, number theory, harmonic analysis and functional analysis comes to mind). Why not ask your supervisor about a list of people whom you should approach? Oct 10 '13 at 10:39

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