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. At Euler Circle, we provide research opportunities related to class curriculum for students who have demonstrated the ability to work hard on challenging problems. Many of our problem sets include open problems that are closely related to the class material.

Here are the papers that I have supervised, written by high-school (and younger) students:

- Analysis on Surreal Numbers, with Ashvin Swaminathan.
- N-Division Points of Hypocycloids, with Nitya Mani.
- The Asymmetric Colonel Blotto Game, with Yifan Zhu.
- Play in Candy Nim, with Nitya Mani, Rajiv Nelakanti, and Alex Tholen.
- Comply/Constrain Subtraction, by Archishman Sravankumar.
- Stability for Take-Away Games, with Sherry Sarkar.
- Investigations of Impartial Games with a Pass, by Emet Hirsch.
- Inscribed Triangles of Jordan Curves in ℝ
^{n}, with Aryaman Gupta.
- RSK-Complete Cycle Decompositions, with Agastya Goel.

Several others are in progress, including some that started out in Euler Circle classes. 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.

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