How can I do undergraduate research in high school without special math programs?

Question

I'm a 10th grader and I really love math. I study it at the upper undergraduate level, participate in competitions, engage in the philosophy of math, etc. But I've been wanting to do research while I'm in high school. My original plan was to apply to a math camp like SUMaC or PROMYS, but this unfortunately is no longer an option. I've already sent an application to SUMaC; I did well on the exam and got a good recommendation letter but I don't know if that's even useful at this point.

(The main reason is that my parents don't want me to go out of state for a couple weeks out of fear that I may die or something very unlikely happening, and convincing them otherwise is impossible)

The good thing is that they are fine with programs or events that take place in-state. I'm within commuting distance of a well-respected public university that I will attend as dual enrollment next year, I can give the name if necessary but I like to keep things as general as possible. Can anyone offer some advice to me? Do such opportunities even exist? Thank you!

Answer 1

Most areas of Mathematics are now so well developed that you would need years to work yourself to the frontier of research. Even a field like Graph Theory, which could boast of some successful, accomplished amateur scientists is now difficult to enter, because you need to know so much. Having a guide is therefore extremely valuable. You will find potential guides at your local universities.

If someone from a high school would approach me, I would be worried about your potential to actually receive results. Now, based on a cold email, I would have no idea whether you would be a new Ramanujan or Dantzig or whether your perception of your ability differ widely from the reality. I would need to see some evidence of your capability to even think about talking to you. (If you talk to a candidate and then decide that it is not worth your time, it might be very time-consuming to convince them to look for someone else.) Thus, your reference letter and your test results are important information.

I am a bit skeptical about research at the high-school level. First, high school is not only about academia and you do not want to burn yourself out academically and miss on the normal socialization. Also, most research presupposes solid knowledge and the acquisition of techniques. While the history of Mathematics is full of individuals who had such raw talent that they could acquire these very quickly and become astonishingly productive, the chances that you are one of these exceptional individuals is low. But even if you are, acquiring background knowledge is important and a good test of your talents and of your interests is to learn a mathematical theory. I would recommend for a high school student to learn Linear Algebra (I liked the book by Serge Lang, but there are now probably better ones available) or elementary number theory, or graph theory. If you are thinking about applied Mathematics, consider learning a Programming Language such as Python.

The reason I am proposing this to you is that a high school student who has understood Serge Lang or who has taught themself programming and can show me good code is one that I would be interested in mentoring. If you are not one of these exceptional people then learning any of these topics is still useful, and if you are one of them, then it would show by the little effort you would have to expend on it.

I know that I am not really answering your question, because you want some type of established program. But it sounds to me that your parents have ruled out this possibility. Now, if you have already done this, then you need to identify individuals that are going to be helpful at a university, usually by common interest.

Answer 2

Mathematics is one of the fields where you actually could do research all by yourself. It is also one of the fields where there is very little interest from the academics on engaging with people who do not have an advanced background since it just means that we would have to explain a lot before we could get to anything interesting.

I do not know any suggestions on programs or other organised/supervised ways to help you pursue your interest. But my suggestion is to, speaking as someone who did exactly this when I was in high school; find a topic or problem and find out everything you can about it. That is doing research!

Do it to learn about problem solving, you will likely not find a new solution, but you will get to practice new ways of finding solutions and new ways of collecting information. And those are the central skills needed to do research in mathematics.

Answer 3

As Thomas Schwarz correctly explains, research in math is difficult without much more background than you are likely to have without advanced degrees or the equivalent. However, depending on your local circumstances you may be able to do something that is research adjacent and that will give you a better idea both of math and of research in math.

If any of your instructors has a PhD in math, or can introduce you personally to someone who does you can do something like the following.

Ask the person if you might read their doctoral dissertation and, afterwards, have some discussions with them about it. Expect it to be very difficult. You will learn what a serious research document looks like. It will probably have a lot of references to earlier work. Follow up on those references, through some academic library if the person doesn't have copies. Trace the ideas back.

Ask for an hour of their time every few weeks for a while to ask questions and get further guidance.

Researchers in many fields need to do something like this. It is called a literature search, though most such searches start with more than one paper.

Take a lot of notes. Note definitions. Note the pattern of proofs. Think about (and note) variations on those definitions that might lead to other things. Note the things you don't understand.

Expect that you won't understand everything. When you meet up with the person again, ask a lot of questions based on what you don't understand. Ask for guidance in understanding the paper, not just for answers. Try to answer their questions if asked. It is OK if you are wrong at first. But understanding can come from keeping at it.

---

Some people with a research-based masters might be able to help, but not quite to the same extent as a person with a doctorate. And the paper that they give you might not need to be a dissertation, but those are usually more complete with background references than the typical paper written for experts.