Exploring Matroid & Graph Theory to Power Algorithms

Broadly speaking, Brigitte Servatius is concerned with the intersection of combinatorics, discrete mathematics, geometry, and algebra. Her research and teaching interests also include the history and philosophy of mathematics, but she has not yet had an opportunity to design and teach these courses. It’s feasible that courses on these topics will eventually be worked into the curriculum for graduate and undergraduate math majors at WPI.

Servatius is a discrete, or finite, mathematician, and her major tools are matroid and graph theory. Matroid theory, says Servatius, is essential in developing and speeding up algorithms that are used to power the Internet and implement GPS tracking technology. These fast algorithms are also being applied in biology to predict the shape of molecules. They can be used, she says, to show how DNA knots and unknots, and how it responds to heat. Shape prediction is fundamental to biology, she says, and her work in matroid theory and rigidity is particularly suitable to molecular shape prediction.

Servatius comes alive when she speaks of her interdisciplinary work with biologists, but she’s sober about the main challenge: the fact that it’s interdisciplinary. The differing languages and methodologies of the two disciplines pose significant hurdles to mutual understanding. Servatius spent a year sitting in on a biology lecture to learn the ways in which biologists pose and solve problems. She wants to help biologists avoid reinventing the wheels mathematicians have already invented as well as understand what in biology is possible for mathematicians to measure. Servatius grasps the mathematical solutions to the biologists' problems, and the challenge is always to best communicate those solutions in a language biologists can understand.

Although progress can be slow, there is nothing about her research that Servatius finds frustrating because she views frustration as essential to her work. “Part of what you learn as a student,” she says, “is to find frustration interesting. A challenge is actually something good. If the problems were too easy, they wouldn’t be interesting. It’s only interesting if it’s hard. And if it’s hard, there’s frustration involved.”

“As a student, I think you’re much more uptight that somebody else will solve a problem before you,” Servatius continues. “There’s always this competition. You’re trying to do something nobody else has done because otherwise somebody solves it just two weeks before you and gets their name on it. That’s not the case for me anymore. I’ve solved so many problems; I’m very happy to share my problems. And I’m very happy if somebody else solves them.”

Given her interdisciplinary focus, it’s not surprising that Servatius takes umbrage at the pure/applied mathematician binary. She likes to say she does “applicable math.” Generally speaking, she defines the applied mathematician as one posed with a real-world problem who uses mathematics to solve it, and the pure mathematician as one who does math for math’s sake.

In her opinion, mathematics is both pure and applied: “It’s pure as long as you don’t know the application, but there might be an application at the time.” Take, for example, number theory, Servatius says, which for centuries has been considered pure math. Number theory has been significant to applications in cryptography; so, she says, number theorists can now consider themselves to be applied mathematicians.

Currently, Servatius is writing a book on the configurations of points and lines, and incidence structures in general. Her publishing career started with her dissertation, Planar Rigidity, in 1987. After that she published Combinatorial Rigidity (1993), The Bracing of Grids (1995), and Matroid Theory (1996).

When it comes to her other contributions to the university and department, Servatius is notably proud of her editorial work with the Pi Mu Epsilon Journal. The journal is more than 60 years old, and Servatius has been the editor since 1999. PME is dedicated to graduate and undergraduate scholarship in mathematics, and it publishes papers and poses mathematical problems of general interest but is accessible to advanced undergraduates.

Her work as an editor of a math journal and the list of books she has published gives credence to Servatius’ advice to graduate students: Choose for your PhD a problem that you want to work on for the rest of your life. If you want to become a good mathematician, you should learn diverse fields of mathematics and not specialize early.

Servatius’ road to WPI was a long one. She was born and raised in Graz, the second-largest city in Austria. Her father was a street car conductor; her mother a housewife. Having been denied the opportunity to earn any kind of degree because of WWII, Servatius’ parents instilled in both Servatius and her sister a strong sense of the transformative power of education.

Servatius inadvertently discovered her love for mathematics in high school. It was an all-girls school focused on languages, but one day she noticed the Math Olympiad, and it piqued her interest. Her math teacher advised her not to enter the competition, because it was not for people who were bad at math. Servatius ignored the teacher’s advice and made it all the way to nationals, where the top 20 students in the country compete. She was the only girl; she was 17.

After high school, Servatius went on to earn her master’s degrees in mathematics and physics from the University of Graz and became a high school teacher. She wanted to pursue a doctorate in mathematics, but the career prospects for women in Austria to be hired as university professors were slim in the 1980s without connections. However, at a math conference she had a chance meeting with a professor from Syracuse University, who encouraged her to pursue her doctoral ambitions at his school.

Servatius took him up on his offer, leaving Austria for Syracuse in 1981, and eventually attained her dream at WPI. She has been here since she finished her doctorate at Syracuse University in 1987. Like many of her colleagues, Servatius was attracted to WPI because of its small size and dedication to innovation and creativity. However, the Interactive Qualifying Project was the main attraction for her. Servatius was excited by the fact that WPI students were encouraged to do work beyond their immediate theoretical and applied concerns.