# PRIMES: Caroline Ellison's Story

I've been interested in math for as long as I can remember. In middle school, I started doing competitions, which exposed me to lots of new and interesting math topics. In high school, I continued to do math contests. I still had fun, but sometimes I wondered where the point was in rushing through easy problems in 75 minutes. After a while, all AMC problems look the same, and there's nothing particularly satisfying about doing better than usual because this time you remembered the angle bisector theorem. I wasn't sure if the accomplishment of good scores was worth the time I would have to put in to achieve them, and I wanted to do something more important somehow.

It was this desire to delve further into math that led me to apply to PRIMES, and it has been simply an amazing experience. My mentor, Giorgia Fortuna, has been wonderful. Research can be overwhelming at first, but she's always been able to break it down for me into achievable steps. Her fortitude in the face of my procrastination and other failings has been impressive, and I would have gotten nowhere without all of her contributions. Pavel Etingof and Tanya Khovanova, who both have a lot of experience with science competitions, have given me invaluable advice and ideas for my project.

At first, I was definitely nervous; I didn't think I would find anything at all, and I didn't think I could really write a paper or give a math talk. It turns out I was wrong on all counts. Coming from a math contest background, research has been very different. I still sort of expect everything to have an elegant solution that can be found in a half hour or less, and I have to remind myself that isn't always the case. Things are often messy, but when I do find something that might be interesting or useful, the fact that it exists at all among the chaos makes it doubly beautiful.

So please apply to PRIMES, because it's crazy that this opportunity even exists for high school students. Research is a lot of work, but it feels great when you find things.

Caroline Ellison worked on the project Polynomial coefficients over finite fields under the mentorship of Giorgia Fortuna.

Email us: primes@math.mit.edu