How to start doing undergraduate research in mathematics when I have limited support from my professors?

Question

I have no research experience and I'm in my junior year of college. I'm very interested in doing research in any field at this point (combinatorics or linear algebra would be especially preferred) and I fully intend to apply for math grad school during my senior year.

I have a decent amount of coursework (abstract algebra [groups, rings, fields and special topics], analysis [real, complex, functional, measures], linear algebra, topology, combinatorics, number theory, partial differential equations, and more). I want to do research this summer (even without funding if need be) and throughout my senior year (I was declined to every REU that I applied for, unfortunately) and I plan to work alone since I have no other options (that I know of).

There aren't any student projects going on at my own college (except for one on wheels) and my professors (5 total though one is leaving and another will be on sabbatical) only seem willing to help if I provide a research topic and if that topic is related to their own expertise (I don't fit in this category) which means that I've had no help so far. I don't know where to look since most journals and blogs seem to be deeply invested into very complicated projects where my ability is likely not high enough to understand. There is no graduate program at my college so there are no grad students.

My questions is this: Where can I find potential research projects or at least a start? How does one decide on what they want to do? I know people from other colleges but they were simply given projects to work on so they didn't know how to find on otherwise.

Answer 1

I think this can only be explained with a personal story. I did my undergraduate at a small liberal arts institution and because of that there were not many professors to choose from. However, I knew who I wanted to work with since his general area (dynamical systems) always interested me, although we had never met (since he was on sabbatical for awhile).

I sent him an email listing 3 different projects I had thought of, and he emailed me back saying he wouldn't do those, but he had a project I should do (climate mathematics, never even thought about it before). I was kind of taken aback, but in the end I thought, well this is the best I am getting, and took the project.

When we finally met the following year, I was still a little skeptical, but as we started working through the problems I found out how interesting the subject was. I learned a lot through his mentorship, and gave chalkboard talks on different papers / analysis results (working out of Courant/Hilbert and Wittaker) every few weeks and a few months in I was very fond of the project and was spending all of my time on it just because of how interesting it was, and the story ends with it turning into a good publication.

Turns out the professor had the idea planned out during sabbatical. He was working with a research team there where many ideas were created, and the papers I worked from were recent results by newly minted PhDs he was working with. He had even published a paper on how to teach undergraduates climate science: this man had though it through!

There are a few morals to the story. First of all, when you're young in your research career, you don't know what is interesting yet, and a quick read of the literature is not enough to gauge this. Also, if plenty of people are researching a subject, it's probably pretty interesting and worth a try. Lastly, for research with undergraduates, the good mentors would like you to go with their ideas because they have a plan for how to teach it / work through the problem.

All together, I would say find a mentor you respect and trust and just ask them to take you under their wing. It may not be the project you had in mind, but you will learn a lot more this way, and you'll be surprised how interesting the project is.

Answer 2

It sounds like it would be easiest to do research with one of those professors. I'm not sure why you say you don't fit in the category of one of the professors expertise when you seem to have a general range of knowledge. I would look up your professors literature, checking for what's popular and what's interesting to you. Use those ideas while looking at related papers and try to come up with a problem to explore. Often they won't want too grandiose a problem when working with students because they want you to be able to do some of the work. Dave Renfro offers good advice. In flipping through materials likely to have student works, you can get an idea of what's appropriate.

If the professors are hard to approach, show them you've done your pre-research and maybe even attempted some things. If you need help getting to that last stage, perhaps come back to SE with more detailed information, or to some mathematics forums asking for some hints/direction.