Student Research at the Highest Level: Avenues Senior Publishes Peer-Reviewed Mathematics Paper

When most people think of high school mathematics, they think of problem sets, exams, and formulas to master.

At Avenues, it can look very different.

This year, Avenues senior Kylan published a peer-reviewed research paper in Mathematics Competitions, the official journal of the World Federation of National Mathematics Competitions—an international publication read by mathematics educators, competition organizers, and advanced student mathematicians worldwide. It’s an accomplishment that is rare at any level, and especially uncommon for a high school student. It requires not only advanced mathematical ability, but also persistence, curiosity, and the ability to communicate complex ideas clearly.

Kylan’s paper focuses on mathematical invariants, a powerful problem-solving concept. Invariants are quantities that remain constant even as other elements change—an idea that, when recognized, can unlock elegant solutions to complex problems. Drawing from Olympiad competitions at the regional, national, and international levels, Kylan not only analyzes existing problems but also contributes original ones, demonstrating the depth and versatility of this approach.

But beyond mathematics itself, this achievement represents something larger: the power of student-driven research.

At Avenues, students are encouraged to move beyond mastering content and into asking their own questions. Through the Avenues Institute for Advanced Mathematics (IAM), students work closely with faculty mentors to pursue advanced study, conduct original research, and engage deeply with mathematics as a creative and investigative discipline. The goal is not simply to learn what is already known, but to explore what is not yet fully understood.

In Kylan’s case, that meant engaging deeply with a concept often encountered in elite mathematics competitions and pushing it further by testing ideas, refining arguments, and ultimately shaping their piece of scholarship that could stand up to peer review. It requires not only advanced mathematical ability, but also persistence, intellectual curiosity, and the ability to communicate complex ideas clearly—skills that are essential for success in university research and beyond.

Kylan’s work was guided by Dr. Patrick Galarza, Director of the Avenues Institute for Advanced Mathematics, whose mentorship creates opportunities for students to engage in authentic, high-level mathematical exploration.

Moments like this reflect a broader belief at Avenues: rigor is not defined by the amount of homework or the difficulty of a test, but by the level of thinking they are invited and supported to do. When students are trusted with meaningful questions, supported by expert mentors, and given the time and space to pursue their ideas deeply, they are capable of extraordinary things. 

Congratulations to Kylan and Dr. Galarza on this remarkable achievement. We look forward to seeing where Kylan’s mathematical journey takes him next as he begins his studies majoring in Mathematics at Stanford University this fall.

Read the article here (page 34)