Applying to Math PhD programs from Weak Undergraduate [duplicate]

Question

First, some information about me. I’m a senior at a small liberal arts school at the US (I will be graduating in 3 years because I took a bunch of college courses in high school), and my heart is set on a PhD in mathematics. My ultimate goal would be to work as a professor of mathematics and do research.

I am becoming extremely concerned that my application is too weak to have any hope of being accepted to a sufficiently strong graduate school to have any chance whatsoever of actually entering academia. My primary concerns are twofold:

First, the mathematics courses offered at my university are (so it seems to me) woefully inadequate. Upon graduation I will have taken literally every single math course offered (excluding stuff like business calculus). So far I’ve taken numerical analysis, mathematical statistics, graph theory, modern geometry, and linear algebra and I am currently taking abstract algebra and intro to proofs. I took Calc 1-3 and differential equations in high school. Next semester (my last before I graduate) I will be taking complex analysis, and two independent studies in real analysis and number theory. My greatest concern here is that the level of the courses I am taking is far below what might be expected of a prospective PhD student. I say this because all of the math courses I’ve taken (with the exception of mathematical statistics) have been extremely easy for me. As in, I hardly needed to pay attention in class and almost never had to study (I received A’s in all but geometry and mathematical stat). My abstract algebra class (infuriatingly) de-emphasizes proof writing in favor of “computations and concepts”

Second, and also the impetus for writing this post, is my subject GRE (more precisely, my lack thereof). I was registered to take it today, but I lost my ID and they wouldn’t let me sit the test. The next one is not until April, i.e., after my grad school applications are finished. All of the schools I am applying to list the GRE as optional, but I was counting on a good GRE score to show admissions committees that I do actually have some competence in undergraduate mathematics, but that is now out the window. I know that this point is squarely my own fault, and I accept that, but it still bothers me tremendously and is a great source of concern.

So as not to make this post a complete downer, I’ll conclude by discussing what I see as my biggest asset, my research experience. Since the summer of 2021 I’ve been working continuously on a research project in graph theory under the advisement of one of my professors. I’ve generated a paper with (I think) some very nice results, and I’ve done this essentially by myself with my advisor providing some occasional feedback. I presented a poster on this project at MathFest earlier this year, and I will be giving two more talks on it at conferences next month and in January.

So, in light of all the above information, I’d really appreciate some brutally honest assessments of my chances, as well as any advice you may offer about what can be done to improve those chances. I accept that there is a very real possibility that I will be rejected from all the schools I apply to, but I am not willing to let that be the end of it.

If that does end up happening, I will take another year to try to shore up my application as much as possible (via research and/or auditing graduate classes). If, god forbid, I am rejected everywhere two years in a row I suppose that will be a sign to call it quits.

Edit: it’s also worth mentioning I have a great deal of experience tutoring anything Calc 2 and below. This experience is both private and through my university.

Answer 1

Frankly, I think you are underestimating yourself and your chances. I'd guess you'll be fine if you have good letters of recommendation and a decent GPA. The courses you list are typical for a math program, especially at a liberal arts college. That was my path, actually. It had some advantages over a bigger place in better opportunities to interact with faculty.

Any research will be a plus, but not absolutely essential for math. An extra year is probably a waste of time. The first years of doctoral study are course heavy in any case.

But don't restrict your applications to only the top few universities. Cast a broad net.

See the answer for the US to this question: How does the admissions process work for Ph.D. programs in Country X?

I'll warn you, however, that one problem you may wind up with is that all too suddenly the courses seem hard and you have to work to get good grades. This often happens to math students who seemed to coast through early courses. If you learn how to learn when it is hard, you'll do better. That takes a lot of practice.