If phrases like null lines, degenerate planes and orthogonal complements aren’t part of your daily vocabulary, you’re safely in the majority.

But Dr. Tom Wears, drawing boxes and lines on a white board, tosses out these phrases and Sabrina Walker ’17 just nods her head in understanding. To a casual observer, the pair may seem to be speaking another language—not what you’d picture a summer looks like for the typical college student.

Understanding the language is easy. Using the concepts is a whole different story. Walker, a member of the Cormier Honors College and mathematics major, is spending eight weeks of her summer doing intensive research with Wears as part of Longwood’s PRISM program. For her, the days push the boundaries of what she thought possible—both in mathematics and for herself.

Take Monday, June 1. The day started off like most do—Walker and Wears met in the morning to go over new concepts and address any lingering questions. They’re working in a field of mathematics that lies at the intersection of geometry and algebra — two fields that stay separated for most undergraduates.

But on Monday morning, Wears, assistant professor of mathematics, explained a concept that seemingly defies everything we’re taught: In this field of math, lines don’t have to form right angles to be perpendicular to each other.

“It’s not possible!” exclaimed Walker. “Lines that form really acute angles cannot be perpendicular to each other—this is challenging everything I thought I knew about geometry. It’s really astounding, and I still haven’t quite wrapped my head around it—but we’re getting there.”

Mathematics research works differently from laboratory experiments: There are long periods of solo work peppered with brief interaction. Sometimes the interaction is a mini lecture in which Wears elucidates a concept to help push Walker’s work forward. Other times, the interaction takes more of a collaborative form, as two mathematicians put their minds together to solve a particularly difficult problem.

“I thought I had faced some intellectual challenges before,” she said, “but this work is on another level. It’s really pushing my limits far past where I thought they were.”