How to correspond with academics as a young researcher in high school?

Researchers tend to be very busy advising their own students, and often don't have much free time to commit to supervising others. Recognize that you are asking someone who is very busy to spend some of their limited time on something that (1) may or may not be interesting to them, (2) is not part of their "official" professional responsibilities, and (3) is likely to take time away from their "official" professional responsibilities. (That is why "talk to your advisor" is a common response; they aren't brushing you off, so much as they are redirecting you to someone who is personally responsible for helping you, and has committed to investing that time in you.)

For this reason, emails to researchers asking questions that they don't feel personally responsible for answering, often go unanswered.

You also wrote that you believe the answers to your questions are not going to be immediately obvious, even to an expert:

I have frequent contact with a professor at my local university, who has taught a class on this subject but has not published any papers or done any research in this area. He and I worked on a few of my questions and were able to solve them, but he couldn't figure out any of my other ones. Although that doesn't preclude "common knowledge to any expert", I am pretty sure it isn't.

which suggests that you are asking for a fairly substantial time investment.

And you are asking for an open-ended commitment in your email:

I was wondering if perhaps we could start a short correspondence to help me understand what I'm writing about, and to gain more knowledge of class field theory.

I would be extremely reluctant to say "Yes" to this without any clue of how advanced you are in the topic, what you are expecting from me, and whether it would be an interesting or useful correspondence for me.

It would, however, be appropriate to send a short, actionable email, on a question that is directly related to the researcher's published papers. For example:

Hello Dr. *****,

My name is *****. I'm 14, and I live in South Carolina. I am working on a research project related to Hilbert class fields of global function fields.

I was wondering if you could answer a question about your paper, [name of paper]. [Specific question that you are looking for an answer to.]

With gratitude, ****

P.S. Would it be OK if I asked some further follow-up questions on this subject?

See

• When asking research-based questions, what are some good practices to maximize the rates at which people reply to emails?
• Increase chance of reply when e-mailing an eminent researcher?

for additional suggestions

For questions that are not related to a specific paper, but are more general questions about the field or your own work (that you and your advisor are unable to answer), I suggest Mathematics Stack Exchange. People are more likely to invest time in answering a stranger's questions for no professional benefit on a platform where that's what they do.

The sentence "I understand the topic, most of the current papers and literature I've read on it, and I have done an experiment on a certain cause/effect." does not inspire confidence. How on earth would one do an "experiment" on a "cause/effect" in class field theory? I'm a professional mathematician who knows something about that subject, and I'm scratching my head right now.

I think you're focused too hard on "respect". I personally don't deal with undergraduate students as equals, and I almost never interact with kids who are younger than that. I think you should dial back on your expectations.

The suggestion that you speak to your mentor (which you've admitted you have) is a good one. While sometimes faculty are willing to answer technical questions from strangers (if they have time and the questions are sufficiently interesting), it is unrealistic to expect them to devote a lot of time to mentoring you. If you've really exhausted the resources available at your school, I would talk to your teachers and the professor who is already mentoring you and ask for introductions to people who could help you. Approaching strangers by email is rarely a useful strategy.

Good luck with your research project, it sounds really cool. I'm a math professor and occasionally get emails from people asking me for help, and in fact recently entered a brief email correspondence with a high school student who asked me some questions. So I think I may be able to offer some useful insights in connection with your question.

Is there something I'm doing wrong in my emails?

No, I think the email is very well written. I can't think of any way it can be improved (except maybe tone down the flattery a bit, but who knows, that might appeal to some recipients). Certainly, as you may already realize, lying about your age or misrepresenting your circumstances to try to get people to give you a more helpful answer than they would be inclined to give otherwise is a very bad idea, and is likely to backfire in some unpleasant and potentially damaging way.

In any case, what can I include in my correspondence or generally do to be respected as a young researcher?

As I said, I can't think of anything else you can do in your correspondence that you're not already doing. But if I'm reading between the lines of your question, I get the sense that you have an expectation that your doing the right thing should somehow guarantee the positive outcome you are wishing for. That is not the case.

Let me explain: the reason professors aren't taking you up on your offer to enter a correspondence has nothing to do with you doing something wrong, and everything to do with the fact that professors are very busy people, who aside from having almost all their professional time occupied with their teaching, research, and other duties that simply cannot be ignored, are also constantly getting bombarded with unsolicited emails from complete strangers asking for advice, help, or trying to interest them in various projects. No matter how exciting or interesting each individual offer is (and trust me, most of them aren't), there simply isn't enough time in the day to give each one the amount of attention it is asking for.

In addition to time being a very scarce resource for a professor -- much more scarce than my 14-year old self could have ever imagined, so I will assume you will find it difficult to imagine as well and hope that you will not find this assumption condescending on my part -- there is also eye strain and physical fatigue to consider. Myself and most of the professors I know spend much too much of our days typing on a keyboard and staring at screens. It is unhealthy and leads to physical discomfort and sometimes pain or even injury, and yet we continue to do it, because (a) much of our work duties that we simply have to do involve those activities; (b) we really enjoy our work and are passionate about it; and (c) after finishing our work we also enjoy doing other things that normal people do on a computer (like facebook, chatting with friends, posting on StackExchange etc).

Now let's go back to your situation. When viewed in the context of what the life of a professor looks like as I described above, do you see how an offer to enter an email correspondence with a 14-year old to help him or her understand an advanced area of mathematics, which is something that would be very difficult to explain even in person to an adult with plenty of background, simply isn't appealing, and is not something that the typical math professor would consider an efficient use of his/her time and other limited resources? It's really not your fault, it's just the way things are.

Let me conclude with some positive advice to offset the somewhat pessimistic opinion above. I think there are in fact plenty of people who would be happy to talk to you in person, at least for a limited time, if you could find an opportunity where you are both physically present in the same place. And moreover in many parts of the U.S. there are all kinds of programs catering to talented youth who are interested in mathematics. I'm not very knowledgeable about this subject (and specifically about South Carolina) so I'll leave it to others to comment about, but generally speaking I think you are on the right track, and in particular the idea of talking to professors at a local university near you, as you seem to already be doing, should be a very good way to help you and connect you with useful resources. Good luck!