# PRIMES: Noah Arbesfeld's Story

In middle school and high school, mathematics was one of my favorite subjects. As a result, I participated in numerous math competitions and olympiad-oriented programs. Some of the most satisfying moments in these competitions programs were those spent discussing interesting problems with other students. Research mathematics seemed like a way to emulate these moments in an even more satisfying, long-term manner. Also, I was simply curious to see what mathematicians actually do.

My mentor, David Jordan, was able to strike a perfect balance between providing guidance and allowing me to learn about the mathematics behind our problem independently. Much of our progress was made through discussions, which helped me to achieve a greater understanding of our problem. Beyond that, David taught me how to write and present mathematics effectively.

Through computations performed in MAGMA, a computer algebra system, we were able to obtain new data that enabled us to form conjectures. Using theoretical techniques I learned throughout the course of the project, we were then able prove these conjectures, and also verify conjectures in earlier work. Perhaps most importantly, I also gained exposure to lots of very interesting mathematics.

Mathematics research felt very different from competitions; in competitions, one has a relatively short amount of time time to solve many problems, while in research, one focuses on one problem for an extended amount of time. There is not necessarily a correct answer, and much of the time, it is not even clear how one should proceed. Needless to say, it is extremely exciting to complete the proof of a theorem. However, even though not making progress can be frustrating, it is still quite satisfying to obtain a greater understanding of the problem and the mathematics behind it.

I think that PRIMES provides a unique opportunity for high school students to gain exposure to mathematics outside of high school courses and competitions, and get a taste of what doing mathematics research is like. Even if one does not know whether they want to continue studying mathematics in university, I think it can be extremely valuable to devote the time to an in-depth project while developing scientific writing and presentation skills.

**Noah Arbesfeld**,
the 6th place in the 2009 Intel STS finals for a project in abstract algebra
done at the RSI under the supervision of Pavel Etingof and David Jordan.

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