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Her summer experience at Lafayette not only gave Jenna Bratz ’06 (Springfield, Pa.) the opportunity to explore complex mathematical theorems, she also learned what it is like to be a research mathematician.
“This summer opened my eyes,” says Bratz, who had always thought she would go into teaching after college. “Now I see what mathematicians do. I know what research is, should I want to go that route. It helped me see the other career paths I could take with math.”
She has indeed conducted the work of a mathematician. Bratz, in addition to training as a midfielder on the field hockey team, researched automorphisms of Cayley graphs to determine connections between group theory and graph theory.
“An automorphism of a graph, in mathematical terms, is a function from the graph to itself that is one to one and onto, meaning that you are moving things around but you are preserving the integrity of the original structure. One is moving the vertices around in a way that the edges remain edges and what wasn’t an edge is still not an edge,” explains Elizabeth McMahon, professor of mathematics and Bratz’s mentor.
The Cayley graph is very regular, with identical vertices. By moving and manipulating the graph with computer-generated automorphisms, Bratz tried to find patterns from which theorems can be derived.
While a group, a set of things, is not very visual, a graph, dots connected by lines, is very visual, notes McMahon.
“What we were doing was exploring the connection between these two branches of mathematics,” she says. “We were looking at these graphs that are pictorial representations of groups and asking ‘what does having the pictures tell you about the original group?’”
Bratz was part of a four-student group conducting the research in the National Science Foundation’s Research Experience for Undergraduates (REU) at Lafayette. Nick Haber of Brown Unversity, Patricia Cahn of Smith College, and Sarah Tekansik of Mesa State University teamed throughout the summer with Bratz, who participated through Lafayette’s EXCEL Scholars program.
In Lafayette’s distinctive EXCEL Scholars program, students assist faculty with research while earning a stipend. The program has helped make Lafayette a national leader in undergraduate research. Many of the more than 160 students who participate in EXCEL each year go on to publish papers in scholarly journals and/or present their research at conferences.
“We took different permutations that would generate a group and then put them into the computer program to turn the group into a Cayley graph. Then we created the automorphisms and compared the different structures of the group,” explains Bratz. “Determining these automorphisms included coloring the edges of the graphs and then discovering the ways we could change the colorings and still have a Cayley graph.”
“We also looked at the structure of these automorphisms to see how large they can get under different circumstances,” she adds.
“Some people describe mathematics as the science of patterns,” says McMahon. “Jenna and the REU group did many examples to look for patterns – looking for new relationships – and they found some. We looked at Cayley graphs in a different way, looking to find how many more of these automorphisms we could get.”
Bratz wanted to take on the project because it was a mix of geometry and abstract algebra.
“It was challenging; I had to read papers by mathematicians and try to understand what they were saying and apply it to what we were doing,” she says. “I had to learn a lot more by myself than I do in the classroom.”
“Another thing that was great about the EXCEL experience is that I was able to use what I learned in class in mathematics research,” she adds.
Practical application of the theoretical concepts learned in class is just one of the benefits of EXCEL, says McMahon.
“I had two goals for Jenna: I wanted her to enjoy what she was doing, and I also wanted her to understand what mathematics research is. Most people don’t have an idea – they can understand what a historian does but ‘what does a mathematician do?’ And I wanted her to understand how mathematicians do research,” she says.
Bratz says her summer research experience is an invaluable asset for her honors thesis, which she began this fall.
Adds McMahon, “Jenna is capable of going on to get a Ph.D. and if that’s what she wants, great. But if she doesn’t go to grad school and instead decides to become a math teacher, her work this summer will be something she can share with her students. She’ll be able to say ‘I did some math research, and here is a paper I wrote. This is what mathematicians do; this is why you’re learning this stuff.’”
A tri-captain on the field hockey team, Bratz started all 18 matches last season and was an All-Patriot League second team selection.
She also is a peer mentor, peer tutor, and member of Lafayette Christian Fellowship, Fellowship of Christian Athletes, and Alpha Phi sorority.
As a national leader in undergraduate research, Lafayette sends one of the largest contingents to the National Conference on Undergraduate Research each year. Thirty-nine students were accepted to present their research at this year’s conference.