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I have been looking into several different PhD programs in mathematics, and I am trying to keep my options open. After finishing a doctorate, I am open to working either in academia or industry, although I would prefer a teaching job or similar at a liberal arts college. Additionally, I am currently in the US on a student visa in an undergraduate program.

I would say I have a pretty strong resume, and my GPA is quite good (at around 3.7 on the 4.0 scale). I have research experience in combinatorics and data science, as well as independent reading courses, workshop participation and experience at a summer program for prospective PhD students. I have also given several presentations at student conferences.

I have looked at plenty of different schools in the United States. I am from a small state school, and I believe that applying to Ivy Leagues etc, I would probably not stand a chance. Even some top 50 programs I am quite sure I won't get into, although no risk no reward, so I will definitely try.

After talking to my student advisor, she told me that I should definitely have a few safety schools on my list, and that I will do. I started looking into it, and I noticed there are a few R2 universities with mathematics PhD programs that seemed to have professors interested in similar research as I am.

There was just one main concern: if I want to get a postdoc or similar after finishing PhD studies, would going to a R2 university completely kill my chances of doing this? I have been heavily advised against it by some of my peers.

If it matters at all, I am currently mainly interested in research in discrete mathematics and algebraic topology.

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First of all, I don't give individual advice for prospective math PhD students without saying that the job market for permanent positions (including teaching positions) is terrible and likely to stay that way. There are significantly fewer jobs than highly qualified candidates.

With that out of the way... You can get pretty much as good an education at a program ranked around 60 as at a program ranked around 20. It's true that, at the lower ranked program, you will likely have to study some more advanced topics independently with your advisor that you might have been able to take a 2nd or 3rd year graduate course for at a higher ranked program. But, given the job market, the qualifications of the professors are going to be pretty much the same. You'll be able to get pretty much just as good an advisor who, if you're capable, will help you work on just as good a project, and the advisor will have pretty much just as good a network to help you promote your work and find a postdoc.

Another difference is the expectations of the culture. At a program around 60, a student who gets a research postdoc is exceptional. The professors will not generally push you to do that level of coursework and research until you demonstrate that kind of capability, and your peers will not be expected to do that level of research. Some people are more likely to make an effort beyond what their peers do; others are not. I don't know you.

However, you should also realize that, if you are capable of earning a PhD with research that will get you a research postdoc, then every program ranked around 20 will be happy to admit you. Of course it's somewhat hard to judge the potential of an undergraduate student, but if the higher ranked programs all turn you down and you do get a postdoc after your PhD, then those programs have all made a mistake. Sometimes that mistake is systemic and out of their control; perhaps all your letters of recommendation underestimated your potential. This means, though, that failing to get admission to a top 25 program is an indication that admissions committees don't think you're likely to succeed to that extent.

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    I think you're right about 60 vs 20, but it's dubious that admissions committees, even in aggregate, can reliably predict who is and is not likely to have the potential to succeed. It might be true that they are pretty good at identifying who "fits," but that's not the same thing as identifying talent. If there weren't systemic skews in evaluation and opportunity, mathematics wouldn't be as homogeneous as it is. – Elizabeth Henning May 12 at 1:56
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Although the reputational rankings may give you a rough idea, a lot depends on very specific individual factors. It is impossible to predict exactly how things will turn out for you.

I suggest that you should look at how graduates from the programs you are interested in have done after completing their PhD’s. Have recent graduates been placed into good post docs? Going back a decade or so, are graduates finding tenure track positions?

An easy way to do this is to use the mathematical genealogy website to look up graduates from a particular year and then see where those graduates are now by googling them. You’ll probably find a wide range of outcomes. Ask yourself whether you would be happy with all of those outcomes.

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