# PRIMES: William Kuszmaul's Story

Before PRIMES, my mathematics background was pretty normal. In school, I've always loved math. I'd frequented a math club for several years, and taken a summer class in group theory, but I happened to mostly stay away from competition math. In the summer group theory class, taught by an MIT graduate, Chris Kennedy, I'd been exposed to how beautiful math could be, and afterwards, I felt like the end of the class left a sort of a gap inside me. So, when I heard about PRIMES, I was already looking for something interesting to try, and I saw it was just the thing.

When I applied to PRIMES at the age of fourteen, I didn't really know anything about mathematical research. However, I did know that I loved math and I loved programming. I'd been at a short class on probabilistic algorithms run by MIT Splash when I first heard about PRIMES. I was fascinated by the class, so although I didn't even know what PRIMES was, I took the teacher seriously when he suggested that everyone in the class apply. It was reading Professor Etingof's welcome message on the website that made me decide I would hate myself if I didn't.

When I was accepted, I became part of a three person research group, working with computer science and math to examine torsion in the Lower Central Series of the free algebra over the integers. Our mentor, who I would come to see as a friend, a teacher, and a role model, was David Jordan.

During the research period, we organized the research in four basic stages: learning, coding, analyzing, and writing. For the first couple of months, we just read texts about linear algebra and went over some basic group theory with David. Just seeing new material and how it all fit together really amazed me, and every day I looked forward to getting home from school and reading a section about linear algebra or commuting to MIT to talk with my mentor and see him prove Lagrange's Theorem or explain what an algebra is.

Once we understood the basic ideas set around our research, my three person research group needed to write programs which we could use to obtain data with which to make conjectures. In a lot of ways this was my favorite part. There was always code to be written and results to be gotten. If one of us got too stuck on an error, we could go to David for help. My best memories of last year were commuting to MIT for an all day coding day. After making conjectures from our code, we had to go over past papers to look for ideas for how to prove them. Reading a math paper is generally very difficult, but support from our mentor got us through. Writing our own paper and proving the theorems was both tough and rewarding.

Near the end of our research, we wanted to have a paper to enter into the Siemens competition, and it was a tight deadline to meet. We worked late into the night many nights in succession, skyping almost into the wee hours of the morning. Sometimes, when doing mathematical research, you feel like you know nothing at all, and like none of what you are doing makes any sense. It can sometimes be easy to become frustrated, seeing time spent confused as time wasted. The most important thing that David ever told me was that math research is confusing, but that the key is to grab onto the moments when you feel like you understand a little something, to the little victories, and hold them. When the paper was finally finished, we'd proven much of what we wanted to prove, and David said that our theorems were surprisingly original. It gave me an incredible feeling to have the paper come together in the final days of it being written, and I came to cherish the feeling of just putting everything in life aside and "primesing" for the rest of a day.

I recommend PRIMES to anyone who loves math or computer science. No one can afford to miss the chance to see new math unfold before their eyes, and perhaps more valuable the experience of trying to uncover new results that seem completely unattainable. Throughout PRIMES, my mentor was incredibly helpful and understanding. As the year went on, I had a chance to develop a personal relationship with others who love math just as much as I do. At PRIMES, when you think something is completely impossible or a problem is unsolvable, guidance, and an unparalleled enthusiasm from mentors and professors alike, is what makes the program so fantastic. I am thankful that I was given the chance to work with brilliant professors like Pavel Etingof in one to one situations and at the same time, the opportunity to see what they love taken to a whole new level.

William Kuszmaul, together with Surya Bhupatiraju and Jason Li, worked on the project  Lower central series of associative algebras in characteristic p under the supervision of Pavel Etingof and David Jordan.

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