Less a challenge than an exploration

For the average high-school student, mathematics is a straightforward task. You analyze a problem, determine how to resolve it, and then apply your knowledge in an examination. Imagination is not part of the equation.

That was the experience of Eliot Hodges, a graduating mathematics student from Dunster House, prior to attending College. He appreciated math — and performed well on tests — but unlike some forthcoming classmates, he set it aside once he departed the classroom.

His interest lay predominantly in music.

The third of four musically talented siblings, Hodges studied the cello — taking lessons, enrolling in summer music workshops, joining youth orchestras, and competing in solo contests. Alongside his siblings, he founded InTune String Ensemble, a nonprofit organization that raises funds for various children’s causes. After canvassing for donations, the group discovered they could generate more revenue through performances and busking in Denver. “When we began, we were quite young,” Hodges remarked. “I believe that factor was beneficial.”

During his final year of high school, just as he was concluding that he didn’t wish to pursue music as a career, Hodges signed up for a remote linear algebra course. Initially, he didn’t realize that the class was proof-oriented — emphasizing less on direct application and more on theorems and other mathematical declarations. “I recall pulling up some practice problems for the midterm and not understanding how to tackle even one,” he said. To prepare, he ended up transcribing the solutions in an attempt to guide himself on how to write proofs. He enjoyed that feeling of exploration.

“You aren’t exactly shown how to solve the problems,” Hodges mentioned. “You genuinely have the chance to be inventive in your approach to problems, and I truly fell in love with that methodology.”

During a gap year, he immersed himself in as much mathematics as possible and took several courses at the University of Colorado at Boulder, developing a keen interest in number theory, which focuses on the study of whole numbers — particularly prime numbers — and how other integers can be formed by multiplying primes, along with the patterns that arise from this process.

“In a sense, prime numbers are the mathematical building blocks, which is why they are so prevalent,” he elaborated, “and despite being so essential to mathematics, some aspects of primes remain quite enigmatic.” He appreciated how simple inquiries about primes could lead to intricate challenges.

“Occasionally you find yourself frustrated, and then you realize that if you take a step back and shift your perspective, you can discover an opening.”

Hodges had little sleep during his first year at Harvard, but he did gain a wealth of knowledge in mathematics — including through a number theory class with Gerhard Gade University Professor Barry Mazur. He was captivated by the beauty of concepts like Fermat’s last theorem, which an elementary student could grasp yet took 300 years of mathematical advancements to validate. “Often, resolving problems in number theory entails integrating ideas from various mathematical domains,” he stated, “and you must be quite inventive in how you coordinate all these concepts.”

After a demanding first year, he committed to reducing his mathematical course load to two classes, along with research. One of these math classes had to be what Hodges termed a “vegetable.”

“A vegetable class is one that is beneficial for me,” he clarified. “It would broaden my perspective and be very advantageous later on.” The alternate type of class was a “chicken nugget,” which he entered with a fondness for the subject matter.

The combination of depth and variety enhanced his research. Often, he noted, it’s more complicated to pinpoint a novel problem than it is to resolve it. “Once you possess a deeper understanding of what you wish to convey about the mathematical entities you’re investigating, the task becomes, hopefully, more clear-cut,” he mentioned. “A considerable amount of the heavy lifting involves formulating your ‘mathematical thesis statement.’”

For one assignment, his goal was to compute the distribution of a specific type of random group with an additional algebraic component called a pairing. “The simplistic generalization of established methods for tackling the issue didn’t succeed,” he stated, but once he ceased viewing the pairing as a function and regarded it as a correspondence between the group and its dual — a mirrored counterpart of the group — he was able to leverage known strategies to resolve the challenge. “Occasionally you find yourself frustrated, and then you realize that if you take a step back and shift your perspective, you can discover an opening.”

He benefited from the support and mentorship of William Caspar Graustein Professor of Mathematics Melanie Matchett Wood, who guided Hodges for three years and oversaw his thesis along with Benjamin Peirce Fellow and NSF Postdoctoral Fellow Ashvin Swaminathan.

Hodges also maintained his connection to the cello, taking lessons with Professor Kee-Hyun Kim of the Parker Quartet and registering for “MUS189R: Chamber Music Performance” every semester. Some of his cherished memories on campus include performing with his sister Eloise Hodges ’22 and participating in end-of-semester recitals with Christian Chiu ’25, his roommate and a talented pianist. The two share a common background in intensive musical training and a focus on enjoying their performances together. “The recitals feel very victorious,” he said, beaming, “even if they don’t go perfectly.”

Upon graduating, Hodges plans to pursue a one-year M.A.St. in pure mathematics at the University of Cambridge as a Churchill Scholar before embarking on a mathematics Ph.D. at Princeton.

He aspires that his time at Cambridge will equip him with techniques from various mathematical disciplines that he can apply to number theory. The more mathematical knowledge he acquires, the more inventive he can be. “The most thrilling aspect,” he expressed, “is the possibility of uncovering a connection that others haven’t considered and contributing something significant to the field.”