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My situation is fairly rare from what I can gather (at least in the U.S), so I just wanted to see if I could get some input from people who know much more than I do. I just finished my Bachelor's in Pure Mathematics, but will first be completing a Master's in math before moving onto my PhD. I had several reasons for this, with the main one being that I ultimately want to end up at the most competitive PhD program I can achieve.

My undergrad experience had many unexpected road bumps (including COVID and breaking my leg) which prevented me from doing as well as I wanted to, so my application coming out of undergrad was not extremely competitive. I took all of the honors courses offered as well as several graduate courses and did alright (about 3.7 GPA), but besides that no research or other notable efforts.

That being said, I somehow got into a top 20 math school for my Master's, so I feel like I am in a good position to potentially move up the rankings for a PhD. There have been many questions on this site asking about how to get into a top PhD program as an undergrad, but what should I be focusing on as a Master's student to achieve my goal? Since I will be in a position of higher expectations than an undergrad, my guess is I will have to focus essentially entirely on research and classes. I was just wondering if anyone had any advice for my position, or things I can be proactive in working towards when I start this fall.

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First, let me say that top 20 is already "good enough" in some sense, if your goal is to have a successful academic career. While it is true that statistically, more PhD graduates from MIT go on to successful careers than PhD graduates from Carnegie Mellon, it's not clear how much of this is causation vs. correlation: PhD programs select for many of the same traits that make one a successful academic later on.

There's also the issue of fit. I went to a PhD program ranked around 10, and I knew someone who transferred to a program ranked around 25, because there was a specific researcher he wanted to work with.

We shouldn't exaggerate in this direction. If you were at a program ranked, say, 75, I'd be telling you to do everything you can to transfer somewhere more prestigious. But top 20 is already good, and if you perform well as a Master's student, there's a good chance you can be admitted to the PhD program at the same institution without much fuss.

Anyway, let's say after reading the above, you still want to climb up the rankings. Okay, fair enough. The most important thing you can do is write a Master's thesis, and throw yourself enthusiastically into doing a great job on it. Try to solve the problem early and write it up, so you can ask your advisor about follow-up problems. The first thing PhD admissions committees will read is the letter from your MS thesis advisor, and you want this letter to say something like "this student is noticeably stronger than the PhD students here, and belongs at a more selective program".

(Keep in mind, the PhD students at a top 20 program are going to be strong, so professors won't say this about you lightly. If you don't achieve at a level where you're outpacing the local PhD students, there's no shame in that, but it may be difficult to move up.)

You should also excel in your courses, but this by itself isn't going to set you apart much. Graduate math courses vary a lot in how lenient the grading curve is, so getting straight As doesn't send a very clear signal. However, any Bs would hurt you.

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  • This is all great advice, thank you! As for your points on rankings, you're absolutely correct. I am definitely more than happy with the school I got into, I just meant that I don't think I would've gotten into its PhD program. My guess is I would've gotten into around rank 50 programs had I applied directly to PhD, so I guess what I meant by "climb the rankings" was improve my ranking as a PhD candidate, not current prestige of school.
    – modz
    Commented May 29 at 20:45
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    I just don't think this obsession with rankings has any great connection with reality. USNWR's "math grad school rankings", for what little they're worth, put Auburn (inclusive) as your cutoff for "you really must transfer". Yet I know plenty of people with good faculty positions from Auburn. Rutgers is lower still.
    – user176372
    Commented May 29 at 20:48
  • @user176372 The main Rutgers campus is ranked 27. You're probably thinking of a branch campus. About Auburn, I have respect for that department, but I stand behind the assertion that it's better to go somewhere else if you can. The peer group of fellow PhD students will just not be as strong, which is a big deal. There is also the pedigree effect, which unfortunately matters a lot when applying for the first postdoc. Even if some people are successful from lower-ranked programs, you have to think about the average outcome, and it's certainly worse at Auburn than at Johns Hopkins.
    – user187020
    Commented May 29 at 21:07
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    You're right on Rutgers-- they have that annoying habit of putting the location right at the very end. I stand by the main point, which is that we have a lot of kids through this Stack chasing rankings rather than chasing becoming good mathematicians.
    – user176372
    Commented May 29 at 21:11
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Having some background in the area of your potential supervisor is probably the best thing. If you identify a supervisor and topic now, then spend some time learning and preparing, it will help your application. Even publish a good article on the topic, this will be even better.

Apart from that, just get the best marks possible on all courses.

Its not really like undergrad where entry exams etc entirely determine success, its more about getting someone's interest early and being able to justify funding. Getting amazing marks won't magic away the problem of not knowing what to study. I'd much rather take a good person who shows interest in what I do, than a disinterested, unprepared person who got the best marks at Harvard or something. Prove your interest by getting a paper accepted in the area of your potential PhD area (e.g. random graphs). Though I admit this is ultimately beyond many applicants, but it's a great start.

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