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I don't know if this question is more appropriate for the academia stack exchange, but I'm posting it here because it's more closely related to math itself.

I'm not actually an undergraduate, I'm a 10th grader. I've made a few posts about 3 weeks ago about finding research opportunities in math right now, but then I kind of forgot about them because of a reason I'll mention very soon. I live near a large university that I'll attend next year via dual-enrollment, so I think my best chance is to cold mail one of the math professors there for an opportunity (I cannot attend a math camp like SUMaC for a variety of reasons mentioned in the other posts).

So why haven't I mailed them? Well, I don't even know what I want to research, and I think that it would be a good idea to have a topic in mind in the email. In fact, I don't even know what topics there are. I'm currently working through Introduction to Abstract Algebra by Nicholson and I think I may be able to finish it in a few months (I'm currently learning about the Sylow theorems). I doubt this is even enough background to work on anything in contemporary research, but I'm not entirely sure.

Should I just be patient and learn more before approaching a professor? If not, can anyone give me advice on how to start my journey of selecting a topic?

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    $\begingroup$ Figuring out what to study is definitely something a mentor/advisor can help you with. Better to get advice from a mentor/advisor than random people on the Internet. MO is not a substitute for a mentor/advisor. I'm voting to close this question. Regardless, good luck in your studies! $\endgroup$
    – Sam Hopkins
    Commented Feb 27 at 19:45
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    $\begingroup$ How about studying the Honors Thesis by Geordie Williamson people.mpim-bonn.mpg.de/geordie/Hecke.pdf done by an undergraduate, and taking you straight into contemporary research. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Feb 27 at 20:50
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    $\begingroup$ In general, requests for personal advice are off-topic. I think that better than being dead-set on doing “research” it would be better to just be open to a wide variety of mathematical experiences depending on what is available locally. Most faculty do not direct undergraduate research, but there are often other kinds of enrichment available. $\endgroup$
    – Andy Putman
    Commented Feb 27 at 22:58
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    $\begingroup$ I entirely agree with Andy. Why the rush to do write research papers so soon? I would start just by learning reams of mathematics, and let the discovery of what really grabs you occur naturally and organically. (And get to know a lot of mathematicians to boot.) Another avenue, which I believe is practiced at U Chicago (and Harvard in the past and maybe still now) is the senior paper where learns a nontrivial subject deeply to the point of being able to write a beautiful exposition of it. $\endgroup$
    – Todd Trimble
    Commented Feb 28 at 0:31
  • $\begingroup$ The textbook you're reading (or any good textbook) surely contains some problem sets. Solve problems. Then experiment with variations on those problems, changing some of the hypotheses and seeing how your conclusions change. Try to find interesting families of alternative hypotheses and see if you get interesting families of alternative conclusions. At that point you are doing research. Probably it will duplicate research that others have already done, but so what? You're learning what research is like and you're learning to get good at it. $\endgroup$
    – Steven Landsburg
    Commented Feb 28 at 2:38

2 Answers 2

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From my perspective, we could preliminarily investigate the following pair of questions:

  1. Are there any areas of mathematics that a self-taught undergraduate can successfully learn?
  2. Which part of mathematics produces the greatest number of open problems (since there are only a finite number of mathematicians out there, the greater the number of problems is the better your chances to produce something new in that open field)?

We can certainly answer affirmatively to the first question, but we need to agree on the semantic meaning of "successfully learn" and, IMHO, it is the level of knowledge that will allow the student to write a publishable paper on any non-predatory journal.
About the last question, by just taking a look at Open Problem Garden, we see that about half of the listed problems are in graph theory and combinatorics is also one of the "easiest" areas of mathematics that you can self-learn!

Generally speaking, I would say to try to "specialize" yourself in something that you like the most and that you can manage by yourself, at least for the next pair of years, and also to start climbing the ladder of mathematical knowledge by attending courses on a possibly different path, taking your time and starting writing papers at the same time on the "simpler" topic that you independently approached years before.

This would lead you to get some peer-reviewed publications when you are going to start your solid research on the main topic that you are mastering by following the ordinary path at university. Just my two cents.

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One option is to focus on an elementary but unsolved problem, and let exploring that problem lead you to learn relevant techniques.

(1) For example, tight bounds on the number of "edge guards" to cover a simple polygon of $n$ vertices remains unresolved (as far as I know), between $\lfloor n/4 \rfloor$ and $\lfloor 3 n /10 \rfloor + 1$.

Mukhopadhyay, Asish, Chris Drouillard, Godfried Toussaint, and Abu Dhabi. "Guarding simple polygons with semi-open edge guards." In 3rd Int. Conf. on Digital Information Processing and Communications, pp. 417-422. 2013.

(2) Another example (from my own work) is:

"On coloring box graphs." Emilie Hogan, J. O'Rourke, Cindy Traub, Ellen Veomett. Discrete Mathematics, 338, 2015, pp. 209-216.

We could only show certain subsets of box graphs are $3$-colorable. But maybe all are.

There are many such accessible problems that can lead into interesting mathematics.

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