I love mentoring students on mathematics research projects, and I have accumulated a good collection of interesting research-level problems suitable for hard-working and enthusiastic high-school students. Students who have been successful in the classes will be good candidates for doing research.
So far, I have written two papers with high-school students:
- Analysis on Surreal Numbers, with Ashvin Swaminathan.
- N-Division Points of Hypocycloids, with Nitya Mani.
Several others are in progress. I hope for many more to come.
Before you ask about doing research, read this article.
Warning: Research is hard, and it’s not for everyone. People have been doing mathematics for millennia, and doing research means doing something that no one has ever done before. Before you can even start to do research, you have to learn a lot of background, and this is likely to take several months. Expect to spend around two years working very hard on a problem before you have something publishable. If that doesn’t sound like your cup of tea, that isn’t something to be ashamed of. It’s perfectly reasonable to spend your time learning mathematics that has already been developed by other people.