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I am about eight months away from completing a Bachelor's degree in Mathematics. I am what you would call a "non-traditional" student - I am an adult who decided to change careers and I'm pursuing my current degree online (read: I work for a living at the moment and have adult responsibilities). My school, despite having a physical campus, has zero support in the realm of undergraduate research or mentorship (this is no program for any student looking to do research). I've tried connecting with a number of my instructors, but as the program is exclusively online, it seems that they are not particularly interested in helping.

I hope to attend graduate school where I would also study Mathematics. It doesn't necessarily have to be a Ph.D. program, though I know that some schools do not offer Masters's degrees - I would opt for on-campus learning in either case. Since I have no support, I have no idea how to prepare for the application process nor do I have any primer for how to jump into the world of mathematical research. I have a superlative GPA and have aced all of my math classes, but this seems trivial when reading about how difficult it is to get into a graduate math program. I recently applied for a short-term research program for undergraduates (outside of my school), for which I was soundly rejected (I can only assume the many things about my background that seemed inadequate). I've tried perusing papers to get a sense of the research landscape, but all of it seems well beyond my abilities to dissect on my own. I'm in a really low place and it seems that things are quite hopeless. Does anyone here have any insight that might point me in the right direction?

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    If there aren't any nearby universities with graduate programs, you can also connect with graduate students online. For example, I'm organizing an online math mentoring program for Fall 2020 (sites.google.com/view/twoples), where undergrads are paired with grad students to work on a reading project for the semester. This is based on the Directed Reading Program that is run at many universities.
    – Stephen McKean
    Commented Jun 19, 2020 at 15:52
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    This is a pretty common question - you're not alone. See for instance academia.stackexchange.com/questions/8026/… and the other questions that appear in its "Linked" sidebar.
    – Nate Eldredge
    Commented Jun 19, 2020 at 16:30
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    You might like to look into postbac programs. These are intended for just this situation - you've finished a bachelor's degree but without sufficient coursework / research experience / faculty relationships to be ready to apply to a PhD program. They typically last one year.
    – Nate Eldredge
    Commented Jun 19, 2020 at 16:32
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    @GrayLiterature Yes, I am very open and able to move. It's a very interesting suggestion that you bring up. I think I had the impression that I should only be contacting institutions once you are 100% set in your research path (with research/publications already under your belt). It sounds like you're saying that there may be some schools that are willing to work with people who seem driven/bright enough to add something to their program. Is that a fair assessment?
    – Drew Dias
    Commented Jun 19, 2020 at 19:06
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    @DrewDias The expectation of having already been published prior to applying to graduate school is a bit outlandish, but perhaps not so much if you were applying to Ivy league and were exposed to research very early on. What I am saying is that you want to try and connect with professors prior and then apply to the program. The reason being is that you could apply to a math program, get in, and then realize they do not have any research areas that suit your interest.
    – GrayLiterature
    Commented Jun 19, 2020 at 19:08

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Strictly speaking, you don't need undergraduate research to get into grad school in math. Most undergraduates (even most undergrads who do get accepted to PhD programs in math) are not able to do really interesting research, and are a bit embarrassed by the work they did as undergrads. However, mentorship of a faculty member is crucial. For one thing, they need to write letters of recommendation and really be able to judge your potential. But getting mentorship in a fully online program is hard. So, I echo the advice given in the comments to look into post-bac programs. Where I got my PhD, we had a non-traditional student who had done such a program and it helped her a lot.

Speaking of which, you might ask around to figure out which PhD programs have a history of accepting non-traditional students. I got my PhD at Wesleyan University in Connecticut. When I was a first-year grad student, there were others who had been high school teachers for years, some who had master's degrees, one (myself) who had skipped years of education and was super young, immature, and annoying, and one in his 70s who had a full career and grand kids, but wanted a PhD in math. So, that might be a program to look at for any reader who identifies as a non-traditional student.

Another alternative would be to apply for master's programs first and PhD programs later. It's usually easier to get into master's programs. We've sent many students to Miami University of Ohio, which has a good terminal master's program.

Since I have no support, I have no idea how to prepare for the application process nor do I have any primer for how to jump into the world of mathematical research.

It's good to take graduate level courses if you can, since success in those demonstrates likely success in your grad school qualifying exam courses. It's essential to take the math subject GRE seriously. That's one of the main ways universities will determine if you really learned in undergrad, or just got good grades because of grade inflation. It's good to cast a wide net. Even if you only apply for master's programs, I encourage my students to aim for 8-10. For a non-traditional student, 20 might be more appropriate. It would be good to write and rewrite your personal statement to really market your life experience as a positive for the program, and for the other grad students there who you will interact with.

I've tried perusing papers to get a sense of the research landscape, but all of it seems well beyond my abilities to dissect on my own. I'm in a really low place and it seems that things are quite hopeless. Does anyone here have any insight that might point me in the right direction?

I think this is not the best use of time. Don't expect yourself to already be able to do math research as an undergrad, especially one without a mentor. You can learn all that in grad school. Don't get discouraged! Focus on how you've been successful in the undergrad program. Trust that you have the potential to be successful in research, once you have proper mentorship and an environment to do it.

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You mention that you connected with a number of instructors and it appears to be of no avail. Try contacting Academic Advising, a Dean of your Major or the head of the Mathematics Department.

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