##### Math and philosophy double major is working with** Lorenzo Traldi**, Metzgar Professor of Mathematics, and **Louis Zulli**, assistant professor of mathematics

When Trustee Scholar **Joe Dudek ’09** (Honeoye Falls, N.Y.) asked how he can get more involved in the math department and get a taste of high level research, **Lorenzo Traldi**, Metzgar Professor of Mathematics, gave him a unique opportunity combining math and computer science.

This summer, the mathematics and philosophy double major is working with Traldi and **Louis Zulli**, assistant professor of mathematics, to figure out what happens to linear algebra if all of the numbers except for zero and one are taken away.

“Basically, imagine 1+1=0 and start counting. You don’t get very far,” Dudek explains. “This lack of numbers has some real life applications, particularly to computer science, so we are trying to figure out exactly how math in this field works.”

The work is an extension of a paper Traldi and Zulli wrote over the winter which extended some known results about knots to graphs in a new way. According to Traldi, this kind of research is not as precisely planned and structured as research in the laboratory sciences, so the team is pioneering different methods to get answers.

“In our project we’re studying the relationships connecting several objects that might seem unrelated — knots (closed curves in space), graphs (also called networks, these are assemblies of individual points, some of which may be connected to each other – or themselves – by straight or curved line segments) and linear algebra (that is, algebra involving vectors) over the two-element field,” explains Traldi. “We’re trying to understand how to formulate some of these relationships so that, for instance, we could use graphs to picture the linear algebra.”

Dudek has been working on software that will help visualize the mathematical

relationships they are studying and form a more defined bond between the matrices they’ve been working on and tangible graphs.

Traldi praises his apprentice and highlights Dudek’s well-rounded capabilities.

“Joe is a good student and he likes math, so he’s interested in the project and able to work independently on it,” says Traldi. “He has a broader background than some math students, as he’s interested in computer science and also philosophy. So questions like ‘What is really going on here?’ – which are important for us because we’re trying not just to solve a problem, we’re trying to understand how a situation looks from a variety of viewpoints – come naturally to him.”

Dudek is equally impressed with his mentor.

“Professor Traldi has been an absolutely fantastic adviser,” he says. “This project has been a totally mutual endeavor and I’ve felt like an equal the entire way. He has made working on this fun and interesting.”

“I think that Lafayette, particularly in the math department, is an amazing place to do research,” he adds. “Getting to interact with a fantastic department and with

the summer Research Experience for Undergraduates (REU) students has been my favorite part of the summer.”

Traldi explains how Lafayette’s EXCEL program benefits students beyond the classroom and notes that students who conduct EXCEL research over the summer enjoy some additional perks.

“I think of this project as a good example of the way the EXCEL program contributes to an undergraduate’s education,” asserts Traldi. “Joe is getting to see what it’s like to be part of a conversation among mathematical researchers, so he’s seeing the special kind of disciplined creativity that mathematicians exercise. He’s learning a bit of mathematics he hadn’t seen before, and he’s learning to look for ways to use one kind of mathematical idea to analyze another.

“A special advantage for summer EXCEL students in mathematics is that they get to be part of the community of students supported by the National Science Foundation’s Research Experiences for Undergraduates program. All in all, there are 15 or so students doing research with at least six math professors. The students give presentations through the summer, so they get the experience of trying to describe their work to others and trying to understand what others are working on. They also go on trips, go hiking, watch movies together, and so on.”

Traldi adds that the advantage for faculty is similar to the advantage for students; rather than thinking of their research as “only for us,” they perceive it as something to explain and share.

“That motivates us to think about not just the technical details that specialists will appreciate but the broader theoretical context in which our work belongs,” he says.

Dudek is still uncertain about where his academic career will take him but he plans to obtain a Masters degree and is certain that his EXCEL research will give him an edge in the process.

Dudek is a member of the Forensics Society, the Jazz Ensemble, and Haven, a group that promotes substance-free living.