Tying the knot
Summer research looks different for all people and departments across campus. Some students take field trips down the Hill to nurse a cup of coffee at River Road Coffeehouse while digging into some heavy reading; others work 9-to-5 in labs; and a lucky few get to build cool-looking models. At least, that’s the case for Bowen Summer Scholar Amanda Peiffer ’16, who is studying knot theory in the lab of Computer Science and Math associate professor Lew Ludwig.
Mathematical knots are three-dimensional closed loops — like the letter “O” — but they’re often twisted and tangled like a plate of spaghetti. The real-world applications of knot theory range from molecular sciences, such as DNA research, to statistical mechanics, including thermodynamics.
“It’s funny because we have toys that we play with every day. There was a week where I was using PVC pipes and Super Glue to make 3-D square knots. They were ginormous!” says Peiffer, who also uses wax wiki sticks and a computer program (developed by a previous summer scholar) to map knots.
While her work varies day to day, much of what she does involves taking three dimensional knot, or closed circuits, and turning them into a two dimensional objects. She then manipulates the pieces to see the many different ways in which the closed loop can function.
“A lot of my research is just doing hundreds of these and then seeing what happens,” says Peiffer as she shuffled through hundreds of closed loop pictures on her laptop. “We’re creating the mathematics to let the physics, engineering, and chemistry people implement it. We’re not really thinking about what’s going to come of it, but others will use our research in a bunch of different fields.”
The versatility of the mathematics behind this theory is what first attracted Peiffer, a pre-med student, to this research. While taking “Linear Algebra and Differential Equations” with Ludwig during her first year at Denison, Peiffer attended the guest lecture of Colin Adams (fellow knot theory scholar and friend of Ludwig). The presentation is a blur (she only really remembers that it was a skit, and Adams was dressed as a pirate), but she can’t forget Adams’ response to a student question, “Why does this matter?”
“The thing that really caught my attention is that they are using this in chemistry, and they are trying to figure out how can we get molecules into these knotted conformations. So, being a math and chem double major, I was like, ‘That’s awesome!’” says Peiffer, who is from Richfield, Ohio, a town 20 miles south of Cleveland.
Not only does this research marry both of Peiffer’s majors, but it also has given her the opportunity to partner with multiple experts. Earlier this summer, the knot lab was visited by Ludwig’s former summer scholar Joe Paat ’11 (now working toward his Ph.D. in mathematics at Johns Hopkins), and Hwa Jeong Lee, a knot scholar and professor at the Korea Advanced Institute of Science and Technology in South Korea.
The group spent a couple days together collaborating over the research being done in Ludwig’s lab and brainstorming new solutions and directions.
“That’s the most interesting thing I’ve done so far,” says Peiffer. “I’m only 20 years old, and I can sit at a table with people who have their doctorates in mathematics, and I have equally valid ideas. It took until day three for it to actually click that I was significantly younger than everyone in that room. I didn’t feel intimidated at all. That’s why I love Denison so much. I can’t imagine my friends having similar experiences at other schools.”
As a result of the brainstorming session, Ludwig is writing a paper with plans of publishing it this fall. Peiffer will contribute sections of the piece as well as graphics.
“I can’t believe I’m going to be published,” she adds with a smile. “But writing a paper is definitely a process in and of itself. I’m hoping to stay longer than the 10-week commitment most Summer Scholars make to help finish writing and editing it.”
While Peiffer hopes to go to medical school after graduation, she doesn’t know if knot theory has a place in her future. Maybe, she theorizes, she will be like Paat who pursued another direction in mathematics after graduation but now finds himself visiting the lab in which he worked as a student at Denison.
“Maybe knot theory is like riding a bike. I can take a break from it but come right back. We’ll see!” she says.