I am a master student of some engineering (other than EE/CS) in a decent US institution who will apply for a PhD in mathematics in Fall 2014 (possibly in PDE / applied analysis, but not sure). I finished with three A's and five A+'s in eight upper-undergrad / grad math courses, along with four more to come this fall. Since my intention for a math PhD came pretty late, I am only able to look for supervisor this semester. The sad thing, as expected, is that all three potential supervisors declined my request softly. My previous engineering research experiences are mostly experimental, with little to do with math.

So I have roughly one full semester left, and my questions are:

  1. Is it possible to gain any sort of research experience in this semester without a supervisor?

  2. If not, then how can I show my research potential by other ways?

  3. Will the three LORs solely from course lecturers be convincing, even if the lecturers are well-known, and I did pretty well in their courses?

  4. In addition, is abstract algebra a must or at least very important in math PhD application? Since I have to make a choice between Algebra II and Dynamical systems, and I don't have Algebra I.

Thank you for your responses and criticisms.

  • Your last question is better suited for a math board than here; that part of the question is out of scope for our board.
    – aeismail
    Sep 4 '13 at 15:15
  • @aeismail Thank you, I have modified the question so it may have a definite answer.
    – yl2868
    Sep 4 '13 at 15:30

Firstly, I should mention that I am a 3rd year math PhD student at Brown. So I have a hint of knowledge, but not as much as a professor who has been to more institutions, and certainly not as much as someone who has been on an admissions committee.

Firstly, many starting PhD mathematicians have no real research experience. The reason is simply that research mathematics is so far from most undergraduate mathematics. This is also why doing research is so hard without a supervisor (for that matter, it's hard with a supervisor). Many candidates have done REUs, which look generically good but often don't usually result in a publication or anything.

What I recommend to you is that you:

  1. Identify what you're interested in and pursue it. You say you maybe like applied analysis or PDE. Maybe see what some people are doing, read some of their stuff, backtrack if you need to. In this sense, you can just do research without waiting for someone else's permission. I'd like to mention that there is a polymath project you might be interested in, Polymath 7 on the Hot Spots Conjecture (here is a link to the most recent progress page). It's slowed down, but they've done some interesting things and the polymaths are generally good about leaving a clean trail of breadcrumbs.

  2. Ask about showing research potential and getting letters of reference. Instead of worrying about what's the absolute best thing, I think you should worry about doing the best that you can do. You've tried to set up some research with professors and that fell through. You haven't done math research before, and you're probably not going to get far in the next 4 months. In particular, it's unlikely that you're going to come up with a result great enough to inspire someone you haven't worked with to write you a letter of reference. This is all to say that you should get your letters of reference from those best able to recommend you. If these are lecturers, then so be it. But I hope that you've been attentive, proactive, and inspired in those classes. For that matter, if you're inspired, you might be able to ask your dynamical systems professor for an interesting project - approachable projects exist. It might not be new research, but at least it would be something you, and others, could talk about. As a final note, you should talk to your math professors and ask them for advice - I suspect that at least one of them will be able to say something, and they're more familiar with your situation.

  3. Have you done any research in your 'some engineering?' Conceivably, a great letter of reference could come from your advisor/a great teacher/I don't know these details. You should talk to them about this, and get their advice.

  4. Determine what schools you're interested in applying to (apply to many), and their requirements. Brown's Applied Math department does not require algebra. But they do (essentially if not officially) require the math GRE, which does require algebra (though for less than 1/4 of the test). On the other hand, Brown's Pure Math department (the one I'm in) has a severe algebra requirement. Interestingly, PDE is done in both, though in different directions. Many schools require algebra, or require it in the sense that they will have qualifying exams and one of those is in algebra. My undergrad, which I'd say has an applied bent, had quals in algebra and both real and complex analysis. So you could consider taking algebra if you needed to or wanted to, based on looking into what universities you want to apply.

    However, algebra II without algebra I sounds incredibly perilous, especially as the fall semester is upon us already, or is about to be. Perhaps possible with a lot of pre-work. If you are going to attempt to hop into algebra II, I recommend getting a copy of Dummit and Foote's algebra book (clear exposition at a low level).

    Finally, plan on taking the math subject GRE. It's only offered 3 times each year, and you've missed the summer one. You should take it both times this fall, and those signups might already be occurring.


To supplement what @mixedmath aptly observed: your experiences in engineering departments will have been misleading about the expectations of mathematics grad admissions, almost universally. That is, it's not so much "research record" as getting up-to-speed on the very basics of the vast established literature in mathematics. Usually this is done by coursework, but not necessarily, though having a paper trail is useful. Taking the GRE subject test is a sign of awareness of how the game is played, if nothing else.

I am frequently surprised at the apparent possibility in engineering and CompSci to do meaningful research at an undergraduate level. Perhaps "research" refers to a different thing than it does in mathematics... I think we have a severe shortage of labels, insofar as there seem to be only two valid/understood descriptors: "coursework" and "research", with some belief that the former is just gatekeeping and unrelated to the latter.

In any case, unless one tries awfully hard, the very most-basic topics in mathematics are used and useful throughout mathematics, despite much gossip and mythology to the contrary. E.g., the central notions of abstract algebra arise everywhere... although, sure, one can struggle to "get by" by reinventing crappy versions of "wheels" without ever knowing that the technology had been perfected 100 or 200 years ago.

Yes, it is toooo often true that courses and exams are conducted in a nearly-punitive fashion, and perversely emphasize separation from other subject matter, but one should try to ignore this exaggeration and caricature.

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