An Advocate for Algebra

Nora Youngs brings abstract algebra down to earth

By Laura MeaderPhotography by Gabe Souza

July 22, 2021

Nora Youngs is an intrepid mathematician. Instead of scaling a Mount Everest and specializing in one area of mathematics, she’s more of a peak bagger. Calculus. Statistics. Abstract algebra. Graph theory. Differential equations. Topology.

The view? Expansive. The advantage? Versatility.

Youngs set off on this expedition as a graduate student. She needed mastery of multiple areas of mathematics to investigate questions relating to neuroscience, now her primary field of study. Her trek revealed that some areas of mathematics are straightforward and functional, others more esoteric—beautiful, but abstract. The latter, she learned, are often viewed unfavorably by some scientists.

“There’s some idea that certain kinds of math are just for fun,” said Youngs, Colby’s Clare Boothe Luce Assistant Professor of Mathematics. “That they’re only there because we created them, because they are interesting and they do not, in fact, have any application.”

Youngs knows otherwise.

Nora Youngs, Colby’s Clare Boothe Luce Assistant Professor of Mathematics — Nora Youngs, the Clare Boothe Luce Assistant Professor of Mathematics, uses algebraic techniques as she investigates questions relating to neuroscience.

Her bottom line? For certain kinds of information one may want to extract or with certain kinds of systems, “algebraic tools are very good.”

In a recently published paper, Youngs aims to set the record straight, specifically in regards to algebra. “The Case for Algebraic Biology,” published in the Bulletin of Mathematical Biology in a special issue on education, focuses on abstract algebra, an often-overlooked tool in solving problems in biology.

In the paper, Youngs and her coauthor, Clemson University’s Matthew Macauley, dispel myths about algebraic techniques and give examples for how algebra arises in mathematical biology.

For example, Youngs uses Boolean algebra in her research on place cells, where neurons in the brain react to specific locations. In place cells, certain neurons fire only when others are active. There’s a simple “on” or “off” behavior in these neurons, a behavior, however, that’s dependent and separate. She’s found that Boolean algebra—which uses just two variables, true or false, 1 or 0—is ideal for figuring out this dependency and containment as she builds her theoretical models.

Active nerve cells, 3d rendering — Youngs uses Boolean algebra to help model the behavior of certain neurons in the brain, seen here in a 3D rendering.

Her bottom line? For certain kinds of information one may want to extract or when working with certain kinds of systems, “algebraic tools are very good.”

With one foot firmly in mathematics and the other in biology, Youngs’s paper encourages peers in both camps to use algebraic techniques themselves and to teach students how to use them. She also provides resources to get them started.

“These are things that you don’t need a biological training to introduce to your abstract algebra students,” Youngs said. “And likewise, these are also tools that a biologist can use even if they’re not trained in abstract algebra.”

At Colby, when Youngs teaches her mathematical neuroscience course, she introduces a breadth of approaches for solving problems in neuroscience. “Typically, most students wouldn’t see that kind of thing until they’re in graduate school working in [biological] fields.” By giving them a taste as undergraduates, she’s proving that abstract mathematics, while elegant and intriguing, also has applications.

Youngs’s paper also looks ahead to an unpredictable future. With data acquisition moving so fast and changing so much, she knows it’s difficult to anticipate what problems will arise.

“Will some big innovation come along that renders these algebraic methods unnecessary? It could be that the things we’re solving via these methods will be more easily solved by something else,” she said.

“But maybe not. I mean, maybe these methods will turn out to be incredibly important.”