Scott Taylor Untangles Math Concepts for Students of All Ages
The associate professor will use a National Science Foundation grant for knot-theory research
Whether you consider yourself a whiz at math or not, it connects to everything we do. A set of numbers can describe last night’s basketball game, the beat of your favorite song, and even a pretzel.
In the case of that pretzel, a mathematical field called knot theory can chart its twists and turns. The theory delves into various types of looping structures, exploring their geometry and topology. Colby Associate Professor of Mathematics Scott Taylor focuses on knot theory in his research, elaborating on the properties of tangled subjects.
Recently, the National Science Foundation (NSF) awarded a research grant to Taylor and Maggy Tomova, dean of the College of Sciences at the University of Central Florida. The three-year grant will fund their knot-theory research, as well as collaboration among Tomova’s and Taylor’s students and a month-long summer camp for local grade school students.
The camp, which began in 2019 and then was paused in the wake of the coronavirus pandemic, is designed to illuminate mathematics, a subject that, for many, can feel opaque and foreboding.
“So many elementary school kids only ever experience math through worksheets,” Taylor said. “No wonder they have a hard time with math, and no wonder they find it to be joyless.” With the help of the NSF grant, he will reconvene the camp this year in July for approximately 20 students in Waterville.
Knot theory has important relevance to biological questions involving molecular chemistry and DNA, which can become snarled, disrupting cell function. But for Taylor, the potential real-world uses are perhaps secondary to a pursuit he considers joyful.
“Applications are, of course, a big driver in mathematics, but so is just the intrinsic love of the subject: the beauty of it, the great fun of expanding people’s ability to think carefully and rigorously and quantitatively, together with the creative aspect,” he said.
At the camp for grade-schoolers, Taylor and his collaborators have tied mathematics to a specialized outdoor game using balls and frisbees, generating data from coordinates on the field. Music, with its quarter and half notes, became a way to understand fractions. In another exercise, kids tie-dyed T-shirts and then used gridlines laid over the shirts to analyze the color distribution on their creations.
The NSF grant will enable both Tomova’s graduate students and two summer research students under Taylor to contribute to the camp. Tomova’s students will mentor Colbians as they work on problems related to knot theory.
Knot theory has been around since the 1800s, pioneered by German mathematician Carl Friedrich Gauss and furthered by Scottish physicists William Thomson (Lord Kelvin) and Peter Tait. A knot can have many permutations, lending itself to intriguing questions in math and physics. Think of the way a knotted rope’s form can shift by being compressed, twisted, or stretched out, without the knot or the length of the rope actually changing.
While knot theory has traditionally concerned itself with closed loops, Taylor and Tomova, who met as graduate students at the University of California, Santa Barbara, are also studying knotted graphs. Knotted graphs have branches that are entangled with each other. The properties of these entanglements help elucidate the structure of three-dimensional space itself, Taylor said. He and Tomova are also working on methods to predict properties of knots and knotted graphs when they are added together.
Drawing or tying knots is a far easier way to convey some of these concepts than just talking or writing about them. That’s why the in-person collaboration with the NSF grant funds is so important, Taylor said.
“There are so many channels for learning and interpreting mathematics besides just the page or the screen,” he said, pointing to body language, facial expressions, tone of voice, the whiteboard, and the ability to build physical knots. “Having access to all these different dimensions is really what makes math happen.”
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