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What Timothy Fargus ’02 (Stafford Springs, Conn.) finds most attractive about his senior honors thesis is that he is working on a topic that he chose for himself. Fargus, a mathematics major, is studying knot theory in this yearlong project. “It’s something that I found to be interesting and wanted to pursue further,” he says.
“In mathematics there are certain topological constructions that are called knots,” Fargus explains. “These are closed loops which can be manipulated in space using any of three special moves, called Reidemeister moves. There are polynomials that can be derived for these knots, arguably the most important of which is the Jones Polynomial, which is one of the topics I will be studying. Each knot has what’s called a ‘knot group’ associated with it, but that group isn’t interesting. What is interesting is the group associated with all of space minus that particular knot, called the ‘fundamental group,’ and this is the other topic that I am researching. I am hoping to take these two topics and possibly explore some new avenues that may or may not connect the two ideas together.”
Louis Zulli, assistant professor of mathematics, is Fargus’ thesis advisor. “I chose Professor Zulli as my adviser because he is well-versed in knot theory,” says Fargus. “The math community is sufficiently small that you can get to know all of the professors and their specialties fairly well, and you can discover who would be the best to suit your particular needs. Professor Zulli is available to me whenever I need help with a problem.”
Zulli says that the possibility exists that one day Fargus may make a significant contribution to knot theory or a related area of mathematics. “But, even though knot theory is an especially accessible area of advanced mathematics, it does have a hundred-year history, so there is much learning to be done before one has enough tools and experience to tackle most problems,” Zulli explains.
Fargus is excited about this field of study because there are relatively new areas in knot theory. “The Jones Polynomial is arguably one of the most important parts of knot theory, and it was only discovered about 15 years ago,” says Fargus. “This, to me, signifies that knot theory is like a new frontier to be traveled, where there is actually progress to be made.”
Even if Fargus doesn’t obtain a new result in the course of his thesis study, Zulli believes that the experience of learning even a bit about an advanced topic like knot theory is extremely valuable, especially since Fargus’ future plans include graduate school in mathematics.
“My goal is to expose him to a ‘knot theory sampler,’ with some very recent topics and some more classical material,” says Zulli. “It’s the experience of learning that’s valuable, and it’s that experience that will help Tim make a successful transition to grad school, where he’ll have to learn a lot of difficult material in a short period of time.”
Zulli says that Fargus seems quite suited for a future in mathematics. “I’m very impressed by his eagerness to learn and his independence,” says Zulli. “These traits, in combination with a natural aptitude for mathematical thought, should help Tim make the transition from ‘math major’ to ‘mathematician.’ Tim also seems to have a talent for thinking visually and geometrically, which should help him with knot theory and other topics in topology and geometry.”
Fargus conducted mathematical research this summer through Lafayette’s Research Experience for Undergraduates, participating in a study of abstract algebra topics with students from other top schools. “I greatly enjoyed the atmosphere and the feeling of making a big deduction,” says Fargus.
Fargus has received Dean’s list honors, is active in College Theater, and is chaplain of Phi Kappa Psi fraternity.