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In general, how does one boost PhD admission profile?

In my context, same question but besides getting a master's in pure math. (See here and here)

Here's what I got so far:

  1. Employment not specifically in research for soft skills boost (I have specific career goals in case I am not good enough for academia)
  2. Employment specifically in academic research (I've seen job posts that look for research assistants) for research experience here I think?
  3. Employment as a research analyst in finance
  4. Tests e.g. GRE Subject
  5. Self-study

What else?

  1. work for a professor for a research project like here I think?

  2. Or pay a professor to supervise my own research projects like here I think?


My context:


What about the GRE Subject?

  • I was suggested by one of my professors to study for GRE Subject, but I think that was in the context of his applying to master's having had only a bachelor's.

On one hand:

  • The GRE Subject may make up for low grades or the fact that I haven't had complex analysis, number theory, graph theory and abstract algebra. Is that right?

On the other hand:

P.S. Sucks if I don't stand a better chance for boosting admissions through self-study. Sigh. Can't believe there's a dichotomy between doing what is helpful for graduate school and what is helpful for graduate applications, but then again professors aren't the only stakeholders.

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    If you have not had the mentioned courses despite having a master's degree, then you need to somehow convince the admission committee that you know the material anyway. Those are completely essential basic courses for graduate school in pure math. And while the GRE might not show that you have research potential in itself, it is a prerequisite for being able to do research to actually know the basics. – Tobias Kildetoft Dec 14 '16 at 10:11
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    Yep, I can see you have exactly the attitude admissions committees are looking for in someone switching from applied to pure math. – Tobias Kildetoft Dec 14 '16 at 19:02
  • I'm a current Math PhD student. As Pete says, complex analysis, algebra are very important if you wish to get into graduate school, and, frankly, if you want to feel you belong once you're there. For instance, I went straight from BA with otherwise strong background but essentially no complex analysis— and I have definitely felt that hole many times over the last three semesters. (But: on the third bullet point, the Math GRE is 50% calculus and 25% DEs, or something absurd like this; algebra, number theory, and complex analysis, you'd be lucky to get four extra problems between them.) – Eric Stucky Dec 15 '16 at 8:51
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I have a master's in applied math and want to get a PhD in pure math. I'm applying to universities that don't require GRE, GRE Subject and IELTS/TOEFL for fluent English speakers.

From the context of the rest of your question, it seems clear that you are mostly applying to pure math PhD programs in the US. But in that case: more than half of these programs require or strongly recommend the GRE math subject exam. So I don't understand why you would restrict yourself in this way. It doesn't make a lot of sense to plan your graduate education and future career around not wanting to take a certain exam.

My bachelor's grades are bad and my non-pure math master's grades are bad.

Well that's bad, sorry to say. Having higher grades in pure math in your master's program will help a bit, but grades are much more meaningful and competitive for undergraduate students than for graduate students. (For that matter, what constitutes an unsatisfactory grade is much different for graduate students. In my math graduate program, a student needs to maintain a 3.0 GPA in order to be in good academic standing.)

I've had a lot of research projects in master's (mostly by group)

Group research projects will not be weighted highly by most pure math PhD programs.

and cannot get very strong letters of recommendation from main research project (sort of a thesis, also by group) supervisors.

And that's also bad, sorry to say.

On the one hand, the GRE Subject may make up for low grades or the fact that I haven't had complex analysis, number theory, graph theory and abstract algebra.

Number theory and graph theory are completely optional classes in most American programs. Complex analysis and abstract algebra are not. Not having these courses, especially as a master's student, and wanting to get a PhD in pure math is once again bad, I'm sorry to say. Complex analysis and algebra are courses that most PhD students would take (or place out of) in their first year in the program. If you haven't had undergraduate courses in these, you would be starting behind most students with bachelor's degrees. That you haven't taken these courses as a master's student is yet more concerning: it seems to undermine your stated interest in a pure math PhD program.

The keys to a strong application for pure math PhD programs are:

  1. Good GRE scores.
  2. Good grades received in courses which the committee will recognize as either good preparation for graduate courses or the equivalent versions of basic graduate courses.
  3. Strong recommendation letters.

So I'm sorry to say that your application sounds like it will not be strong on any of these fronts.

How does one boost PhD admission profile...

You boost your PhD admission profile by improving on 1. through 3.

Anything else?

Nothing that is as important as what I've said above.

Is it possible to work for a professor for a research project?

Yes, but in pure math PhD admissions, research projects are not weighted very strongly at all, as has been discussed thoroughly on this site. Especially, the very few students who can do serious pure math research before they begin a PhD program will (with very large probability) be exceptionally strong on the metrics 1. through 3. above anyway.

Or pay a professor to supervise my own research projects?

Wait, what? Definitely don't do that.

But on the other hand, the GRE Subject doesn't seem to show ability for research...

Admission to a PhD program is not a debating contest that can be won if you find the right argument. Overall you seem to be asking how to spin a poor application into a good one. Well, you can't...or certainly, you can't if that's what admissions faculty realize you're trying to do. If you want to convince us that the metrics we're using our faulty, here's how to do it: get a PhD in math, get a job at a university with a math PhD program, join the graduate admissions committee, and argue that applications should be weighted differently or that certain metrics should not be used at all.

Even if I were to study the basics* of complex analysis, abstract algebra, number theory and graph theory I'm not sure that that translates to a much higher GRE score. Does it?

I thought you hadn't even taken the GRE Subject exam, so I'm confused when you talk about a "higher GRE score." Certainly I would advise you to take the GRE Subject Exam now (or ASAP; I think it is too late for the 2017-2017 admissions season) and see how you do. It is certainly possible to get a good score without this coursework. Whether it helps to increase your GRE scores is a kind of irrelevant question to my mind...it helps your application a lot to take these courses anyway, and it helps you to take these courses and know this material (some of it more than others, as mentioned above).

*I mean come on, of course I'm just gonna use Schaum's Outlines or a GRE Subject book instead of actually learning from a textbook,

(Probably not the point, but: although students treat "Schaum's Outlines" as being the mathematical equivalent of Cliff's Notes, many old-timers know that you could do worse than trying to learn a subject from these books. Gian-Carlo Rota famously wrote on this subject, praising several of the books in the series. When I taught linear algebra a few years ago and spoke to one of my older colleagues, he lent me his copy of the corresponding Schaum's Outline: it was very nicely done in many respects.)

Why "of course"?!? At this point you've halfway convinced me that you don't actually want to be a pure math PhD student. You seem to have one or more unstated assumptions that are strongly undermining your stated assumptions. I can't tell exactly what they are, but I suggest some introspection on this point on your part.

  • "Research projects [...] discussed thoroughly on this site"... really? I couldn't find this in some quick searches, but those searches did find three threads suggesting the opposite. (Anecdotally, when I was applying for school my professors told me the same three things you did, but also had research experiences as a fourth component. So I was curious and did the search.) – Eric Stucky Dec 15 '16 at 9:12
  • @Eric: I have trouble comprehensively searching this site as well, but I promise that it has come up before. You can find a lot of other instances by looking through my own answers (and of course you should look carefully at other answers / corroborating information on those questions). With regard to the instances you cite: only one of them is from someone I'm sure is a pure mathematician, and that person said that undergraduate research will definitely be considered, regardless of the field or relevance.... – Pete L. Clark Dec 15 '16 at 15:33
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    ...If you read it carefully, I think you'll see that it's more in support of the value of doing the research itself than explaining how strongly the research will be weighted. (That same person has expressed sentiments similarly to my own elsewhere.) One of the other answers you cite is from someone who seems to admit they are not in math. I'm not sure about the last answerer ("Dan C"), but e.g. he advises master's students to write "several papers." This would be great advice for many academic fields, but it's quite unrealistic in most areas of pure math. – Pete L. Clark Dec 15 '16 at 15:36
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    Nice answer! I wasn't reading this Stack Exchange group when you wrote it. +1 especially for Especially, the very few students who can do serious pure math research before they begin a PhD program will (with very large probability) be exceptionally strong on the metrics 1. through 3. above anyway. – Dave L Renfro Oct 9 '17 at 9:08
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    Can you spend a year taking those particular classes for a grade outside of a degree? I did something similar at a well-respected school in the US and it helped my PhD application. It just helped to reassure the committee. – Dawn Oct 10 '17 at 12:34

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