Less a problem than an adventure

For the typical high-schooler, math is a plug-and-play activity. You study a problem, figure out how to find the answer, and then apply what you’ve learned to a test. Creativity isn’t part of the process.

That was how Eliot Hodges, a graduating mathematics concentrator in Dunster House, experienced the subject before College. He enjoyed math — and did well on exams — but unlike some of his future classmates, left it behind when he left the classroom.

It was music that held his attention.

The third of four musically inclined siblings, Hodges studied cello — taking lessons, attending summer music programs, participating in youth orchestras, and competing in solo competitions. With his siblings, he founded InTune String Ensemble, a nonprofit that raises funds for different children’s organizations. After collecting door-to-door donations, the group realized they could raise more money playing gigs and busking around Denver. “When we started we were quite young,” Hodges said. “I think that factor helped a lot.”

During his senior year, just as he was deciding that he didn’t want to pursue music professionally, Hodges enrolled in a remote linear algebra class. He didn’t realize it at first, but the course was proof-based — focusing less on pure application and more on theorems and other mathematical statements. “I remember pulling up some practice exercises for the midterm and not knowing how to do a single one,” he said. To study, he ended up copying down the solutions to try to teach himself to write proofs. He liked that sense of exploration.

“You’re not really being taught how to do the problems,” Hodges said. “You really have the opportunity to be creative in how you think about problems, and I really fell in love with that approach.”

During a gap year, he studied as much math as he could and took a few classes through the University of Colorado at Boulder, becoming especially interested in number theory, which is a branch of pure mathematics devoted to the study of whole numbers — and especially prime numbers — and how other numbers can be built by multiplying primes, along with the patterns that emerge from that process.

“In a way, prime numbers are the mathematical atom, which is why they are so ubiquitous,” he explained, “and despite being so fundamental to math, some aspects of the primes are still very mysterious.” He liked how simple questions about the primes opened into complex problems.

“Sometimes you are banging your head against the wall, and then you realize that if you step back and move two feet over you can find an open door.”

Hodges didn’t sleep much during his first year at Harvard, but he did learn a lot of math — including through a number theory class with Gerhard Gade University Professor Barry Mazur. He was struck by the elegance of concepts like Fermat’s last theorem, which an elementary school student could comprehend but required 300 years of mathematical development to prove. “Often, solving problems in number theory requires applying ideas from different mathematical fields,” he said, “And you have to be very creative with how you bring everything together.”

After a grueling first year, he vowed to tone down his quantitative courseload to two math classes, plus research. One of the math classes had to be what Hodges called a “vegetable.”

“A vegetable class is one that is good for me,” he explained. “It would expand my horizons and be very useful later on.” The other kind of class was a “chicken nugget,” where he came in liking the material.

The combination of depth and breadth helped his research. Often, he explained, it’s harder to identify a novel problem than it is to actually solve it. “Once you have a greater understanding of what you want to say about the mathematical objects you’re studying, the work is, one hopes, more straightforward,” he said. “A lot of the hard work is in conceiving your ‘mathematical thesis statement.’”

For one project, his goal was to calculate the distribution of a certain kind of random group with an additional piece of algebraic data called a pairing. “The naïve generalization of known methods for approaching the problem didn’t work,” he said, but once he stopped thinking about the pairing as a function and thought about it as an identification of the group with its dual — a mirror image of the group — he was able to apply known techniques to solve the problem. “Sometimes you are banging your head against the wall, and then you realize that if you step back and move two feet over you can find an open door.”

It helped to have support and guidance from William Caspar Graustein Professor of Mathematics Melanie Matchett Wood, who mentored Hodges for three years and advised his thesis along with Benjamin Peirce Fellow and NSF Postdoctoral Fellow Ashvin Swaminathan.

Hodges also kept the door open to cello, taking lessons with Professor Kee-Hyun Kim of the Parker Quartet and enrolling in “MUS189R: Chamber Music Performance” each semester. Some of his favorite campus memories include playing with his sister Eloise Hodges ’22 and performing end-of-semester recitals with Christian Chiu ’25, his roommate and an accomplished pianist. The two share a background in intense musical study and a desire to focus on having fun playing together. “The recitals feel very triumphant,” he said, smiling, “even if they go badly.”

After graduating, Hodges will pursue a one-year M.A.St. in pure mathematics at the University of Cambridge as a Churchill Scholar before enrolling in a mathematics Ph.D. at Princeton.

He hopes that his time at Cambridge will help him master techniques from other mathematical fields that he can apply to number theory. The more math he can learn, the more creative he can be. “The most exciting part,” he said, “is that you might discover a connection that other people haven’t thought about before and contribute something meaningful to the field.”