“If you can’t solve a problem, then there is an easier problem you can solve: find it.”
That quote from mathematician George Pólya is one that inspires Summer Science Scholar Lucas Waite ’28. Rather than trying to solve a famous unsolved problem outright, Waite spends his days breaking one in particular into smaller questions that may gradually pave the way towards a solution.
Waite’s research with Professor of Mathematics Noah Aydin lies at the intersection of coding theory and graph theory. Coding theory studies how information can be transmitted efficiently and accurately, while graph theory examines relationships between objects. By using the visual aspect of graph theory as a lens, Waite hopes to better understand the structures underlying coding theory and develop new tools for studying them.
The majority of his work focuses on approaches to the Gilbert-Varshamov conjecture, a major open problem in coding theory. At its core, the conjecture asks what is the maximum amount of useful information that can be reliably transmitted beyond a specific threshold. Although Waite is not attempting to solve it directly, he is investigating which methods might help researchers approach it.
While the mathematics itself is highly abstract, its implications extend far beyond the room in Hayes Hall where Waite spends his summer days. Codes are used everywhere information is transmitted, from text messages to data transmitted across vast distances. Better understanding these mathematical limits could eventually contribute to more efficient methods of communication, particularly in situations where signals must travel through noisy channels (through disturbances) or over enormous distances — say, to astronauts in space.
Unlike many summer research projects that rely on laboratory equipment or large teams, Waite’s work is largely independent and requires little more than his computer, a notebook and a whiteboard. One recent day, he filled the board with streaks of red, blue, and green marker while a computer displaying meticulously formatted pages of code sat beside a digital notebook crowded with half-formed ideas and scattered notes.
The contrast captures the rhythm of experimentation and the path ideas can take. Waite’s daily routine consists of generating increasingly focused questions about difficult mathematical problems and then attempting to answer them. If a question remains too difficult, he narrows it further.
“You have to focus on small approaches — finding realistic ways to approach them by reducing them to smaller things,” said Waite, a student from Massachusetts who created the Kenyon Math Problem-Solving Club.
That mindset is especially important in pure mathematics, where major breakthroughs are rare and progress often comes in small steps. Waite said it is “detrimental” to think about solving the problem as your only goal; instead, mathematicians make progress by studying the questions surrounding it.
Perhaps the most surprising lesson he has learned through his research is that undergraduate students can contribute to unsolved mathematical problems. Because research papers are often written for experts, many students assume they lack the knowledge to participate. “It’s a false (assumption) that I held for a long time,” he said.
For Waite, the summer has provided a glimpse of the career he hopes to pursue as a mathematics professor — exploring difficult questions, finding new ways to approach them, and discovering that sometimes the most important step is identifying the smaller problem hidden inside the larger one.
“It’s what I want to spend my life doing.”
This article was written by Arden Kraunz-Brown ’28 as part of the Hoskins Frame Summer Science Writing Scholars program.