Spotlight on Research: Professor Topaz discusses pattern formation in nature

Students sitting cross-legged on the floor or perched on sofas, drinking tea and laughing about mathematical topics that were far, far beyond the scope of my brain. Conversation zipping from physics to memes to calculus to computer science in one breath. A door plastered in fist-pump-worthy slogans and stickers for various social justice movements. There was an active energy permeating the atmosphere of Professor of Mathematics Chad Topaz’s office the day I came to visit.

Having recently arrived from Macalester, Topaz is the most recent addition to the mathematics department at the College. This fall, Topaz has been teaching the higher-level math courses “Computational Linear Algebra” and “Partial Differential Equations,” and he is slated to teach “Discrete Mathematics” in the spring. When he’s not teaching in Bronfman Science Center or helping the line of students spilling out of his doorway in Bascom, he also conducts his own research utilizing differential equations, mathematical modeling and topological data analysis. In short, he’s a busy man.

Suffice it to say, I was pretty lost. But when asked to explain the math-speak to a complete novice like myself, he and the circle of math-wizard students around him all perked up. They explained that topological data analysis is “squishy geometry,” where shapes are fluid dependent on the number of holes or “genii” in them, and “so a cube can be a sphere but never a doughnut.” They described differential equations to me as equations that convey how quantities change. In contrast to “functions that tell you outputs based on inputs or information given directly,” Topaz explained, the nature of differential equations is “about inference.” They’re particularly useful in Topaz’s line of work, because “things are constantly changing in nature, and [he models] it.”

As for what he specifically applies this math to, Topaz’s field of expertise lies in collective behavior, where organisms self-organize into groups or patterns. Topics of interest concerning collective behavior include biological swarms, bird flocks, fish schools, vegetation, fluid physics – anything where there can be a pattern in natural distribution. “Pattern formation is everywhere in nature,” Topaz said. “I find it amazing that, while we’re told nature tends toward chaos, there are still so many ways in which specific systems self-organize.”

As a graduate student, Topaz had an advisor who studied such patterns, and they sparked a curiosity in his own research. Topaz is currently working on analysis of vegetation patterns, which in turn requires extensive topological data analysis and numerical simulation. “I think we have this view that research is Einsteinian leaps of knowledge, when in reality, it is mostly very modest, incremental, but still important steps forward,” Topaz said.

From here, we switched gears to another recent project of Topaz’s, one involving research into gender disparities across editorial boards. Editorial boards hold significant weight in the world of academia because they determine publication content in journals. “They’re the gatekeepers of knowledge,” Topaz said. Throughout his postgraduate mathematical studies, Topaz’s advisors were all women, but in talking to his colleagues about this, Topaz knew how uncommon his case was. After a Facebook conversation one day about this very issue, Topaz decided to investigate the degree to which women were underrepresented in higher academia and subsequent editorial boards. The project was more “data-science-like” in nature, he explained, where he would use various human sources to find out the gender ratios of different boards. His findings? There was a disproportionately low representation of women across editorial boards, to an even greater degree than he had first feared. “It gets worse the higher up you go,” Topaz said. He described the increasing disparity between the male-to-female ratio in undergraduate professors and graduate, graduate and doctoral and so on. Topaz passionately believes in the importance of increasing diversity across these fields and has shared this project’s finding in a colloquium at the College, as well as at several other academic institutions.

“There are never enough hours in a day,” Topaz said, citing packed scheduling as one of the challenges he faces as a teacher. “I don’t want people to be underserved.” In between my questions, he was whirling from one student to the next, answering questions and generating discourse amongst them all. His passion for math was positively infectious. Ultimately, Topaz has a philosophy that he extends through both his teaching and his research. “Mathematics is a powerful way to examine the universe,” he said. “Unfortunately, we live in a society that has not embraced it in the way it should. I want to show people the power of a mathematical viewpoint.”