So, you’re a high-school student, you love mathematics, maybe you’ve won some competitions or attended a math camp. You want to step it up to the next level and do some original research. That is wonderful, and we commend you for your enthusiasm. But first, you must know what to expect from math research. (Spoiler alert: it’s really hard! Even for professional mathematicians.)
First: do not do a math research project in order to improve your résumé for college. That would be really stupid: an exceptional high-school student should expect to spend around 1000 hours over the course of two years on research in order to have a reasonable (but far from certain) chance of producing a publishable paper. Even the least imaginative person can surely think of many other ways of spending 1000 hours that will help far more with college admissions. If that is your goal, go away, before you make all involved parties miserable.
You may be familiar with high-school “research” in biology, where a high-school student works in a lab over a summer, mostly doing menial tasks like cleaning test tubes, and is rewarded with a paper to publish or to present in science fair competitions. There is a major cultural difference between biology and mathematics. In biology, everyone who is involved in any way on a project is listed as a coauthor, so sometimes author lists run into the hundreds. (See this paper for an extreme example.) But in mathematics, only those people who made a significant technical contribution are coauthors. There is no mathematical equivalent of cleaning test tubes. (We can erase our own chalkboards, thank you very much.) So, you simply cannot expect to work for a few months over the summer and have something to show for it. It just doesn’t work that way.
In order to do research in mathematics, you must be prepared to set aside at least 10 hours a week for two years. This must be a high priority. It would be a good idea to schedule actual hours when you will work on your research project each week. At the very least, this exercise will give you an idea of whether you really do have the time to be working on mathematics research. Generally, taking up mathematics research involves replacing other activities, rather than adding to the pre-existing ones.
The time you set aside for mathematics research must be of the highest quality. It is not to be time spent multitasking or socializing, or when you are likely to be tired. Schedule your research time before time for your homework or other similarly unimportant activities. Work out when or if you will do your homework after you have chosen your research hours. Your homework will expand to fill in all gaps in your schedule, so that you will not have time to think about mathematics. But if you choose to do your homework, you can put it in the gaps between your research times, and it will still get done. We expect mathematics research to be your most important activity, and it should be treated with the respect it deserves. It is not possible to do mathematics research “whenever you are not busy with something else,” for you can always arrange to be busy with something else. “I had too much homework this week” is never an acceptable excuse for not working on your research project: if you’re doing the activities of a professional mathematician, then you lose the right to make high school-level excuses.
Okay, we get it: you’re young, you’ve been wildly successful in life so far, and you believe that you can cheat the system: work less, but still produce superior results. If so, you are wrong. It will take much time just to learn the relevant background, before you even start working on an original problem seriously. Expect to spend several months learning the background material, mastering techniques others have used on related problems. Only then will you be ready to start thinking for yourself.
After that, the next task is to identify a suitable problem. The problem you start with will likely be flawed in some way: perhaps trivial for some stupid reason, perhaps impossibly difficult. However, there will surely be good problems lurking very close, but finding one is challenging, and you should also expect to spend a substantial amount of time tracking one down.
One can never guarantee success in research. Even if you are very diligent and do everything right, you might still not solve a problem or get a paper. Most research ideas are dead ends, and while we will help you find one that has better chances of success than usual, we promise nothing. Failing to solve problems is also a key part of the research experience.
There is no easy way to do mathematics research. We will not tell you how to solve a problem just because you have been working on it for a while and we pity you. (Even if we wanted to, we couldn’t: we don’t know how to solve them either. Isn’t that the point?) Of all possible fields of study to do research in, pure mathematics is the one with the highest start-up costs. Very smart people have been thinking about mathematics for thousands of years, and doing research means proving a theorem that no one else has ever proven before. Doing research in mathematics will be by far the most difficult thing you have ever done in your life, and probably the most difficult thing you will ever do.
Of course, as with many difficult things, there are great upsides to doing research in mathematics. Learning mathematics is a wonderful thing, by far the most interesting subject there is. Even without doing any research, the theorems and proofs you learn, worked out ages ago by other people, are enough to elicit great happiness for anyone willing to put in the effort to understand them. And there is no greater sense of accomplishment than what you get after solving a problem of your own, especially when you are the first to do so. Even for established mathematicians, those moments are rare, but they are worth all the despair and feelings of inadequacy that come when nothing is going right, which is most of the time.
Now, if you are still reading this and are thinking “ah, this is a good way of getting into college,” then you have learned nothing at all so far, and you are a terrible candidate for doing research in mathematics. One simply has to be able to read better in order to have any hope of learning the relevant material. If that is your situation, go away and leave us alone.
If, however, you have more noble goals and wish to study mathematics deeply for its own sake, for its own rewards, then we are happy to help you. Welcome to the club!