January 2003

The Science of Conserving Culture

The Gateway Hypothesis of Substance Abuse

Combining the Liberal Arts, Medicine and Business

Confronting Famine Abroad and Obesity at Home

Integrating Teaching and Research in Mathematics

Challenging a Prominent Hypothesis

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Bryn Mawr College
A newsletter on research, teaching, management, policy making and leadership in Science and Technology

Integrating Teaching and Research in Mathematics
By Jennifer Fisher Wilson

Panama Geer

Panama Geer’s mathematical research was inspired by an ant colony. Watching ants traveling one after another from their nest to a food source, she wondered about their behavior: How do ants know how to travel in a straight line when they have minimal vision and no auditory abilities?

"Ants don’t have a mathematical notion of a line, and yet they’re able to organize in this way," Geer says. "The geometry of that intrigued me." She set out to learn how the discrete parts (individual ants) of a complex system (the ant colony) are able to collectively describe a straight line, a circle or a curve.

Kristopher Tapp’s inspiration came when he picked up a textbook on differential geometry in graduate school. "I loved the material. It made a lot of sense to me," Tapp says.

Geer and Tapp have demonstrated a talent for both innovative research and creative teaching, which they are now developing at Bryn Mawr on three-year Keck Postdoctoral Research/Teaching Fellowships. The fellowships enable newly minted Ph.D.s to gain experience in teaching while pursuing advanced research

Interconnected Systems

Geer received her Ph.D. from New York’s Rensselaer Polytechnic Institute in the spring of 2002. A double major in math and studio arts (sculpture) at Wesleyan University, she taught high-school math at the Hotchkiss School in Connecticut for two years before pursuing a graduate degree. A contributing factor in deciding to attend graduate school, Geer says, came from struggling to answer students’ questions about how one uses math in the real world. She realized that wanted to learn more about the novel applications for mathematics.

Along these lines, Geer has developed a course titled "Emergence" that focuses on nontraditional ways to better understand how the natural world works through mathematics. The interdisciplinary seminar explores interactions within various complex systems — such ant colonies, immune systems, earthquakes, traffic jams, or the inner workings of cities or economies — by studying them from the "bottom up," she says. This approach is based on the assumption that the behavior of a complex system emerges from the interactions of its individual parts.

"Emergence is a different way of approaching things," Geer says. It differs from the traditional geometric modeling approach, for example, in which scientists work from the top down, first designing a mathematical model of a physical object or system and then developing an image based on the model, she explains. This traditional approach assumes the image is simply a representation that supports the exact geometric model needed for analysis, Geer points out.

Geer is particularly interested in geometric shapes that emerge as a consequence of interactions in natural systems — for example, the way lines are generated by the foraging behavior of ant colonies through the movement of individual ants. She has used ant behavior and other natural-systems approaches to simulate the immune response and formation of memory T-cells in the human body.

Geer believes that her course on emergence, to be taught next spring, will appeal to a wide range of Bryn Mawr students — not just math and computer science majors, but also students working in philosophy, architecture, biology, economics and other fields. "It’s a great way to study how different ideas interconnect," Geer says.

Curved Spaces

Kristopher Tapp

Kristopher Tapp says his Keck fellowship at Bryn Mawr "feels like I’m coming home." He earned his Ph.D. at the University of Pennsylvania in 1999 and worked as a visiting professor at Haverford College for a year before moving on to the State University of New York at Stony Brook on a fellowship. Tapp says he is excited to be back in the Philadelphia area, particularly at a liberal arts college.

"Liberal arts colleges are exciting places because the students are strong and the emphasis is on teaching," Tapp says.

Tapp is known for his creativity and versatility as a teacher. While at SUNY, he gave presentations on "discovery learning" in undergraduate mathematics, developed a course on mathematical thinking, and helped organize a creative-teaching seminar. As a graduate student, Tapp taught a one-year mathematics course for home-schooled children in Philadelphia, in addition to various college courses on algebra, calculus and statistics.

Tapp is enjoying the opportunity to teach at Bryn Mawr. His advanced-level course on multivariable calculus, which concentrates on three-dimensional figures, is one of his favorite to teach because it connects closely with his research on visual geometry, Tapp says.

Tapp was hooked on math when he traveled abroad in his senior year at Grinnell College to participate in the Semester in Mathematics Program at the Technical University of Budapest. He studied with eminent scholar-teachers at Eötvös University and the Mathematical Institute of the Hungarian Academy of Sciences.

The experience inspired Tapp to pursue a doctoral degree at the University of Pennsylvania, but he struggled initially and almost quit. "A lot of students had a lot more background in math than I did, and I wondered whether I should do something else," he says. Tapp’s doubts vanished when he was exposed to differential geometry, which became the focus of his graduate studies.

Tapp’s research centers on curved spaces, such as the universe, which is believed to be a three-dimensional curved space. He is especially interested in positively curved spaces that are multidimensional, but Tapp says that our ability to study higher-dimensional spaces is rooted in our visual intuitions about the three-dimensional universe.

Now that Tapp is back in the Philadelphia area, he’s enjoying the opportunity to collaborate again with his graduate-school colleagues at the University of Pennsylvania. Collaboration in mathematics can be difficult because the field is so diverse and complex, Tapp explains. Even within subspecialties of mathematics, it can be challenging for mathematician to truly understand another’s work, he says.

"Although some fields become so super-specialized that cross-communication becomes difficult, I still believe in finding every way we can to build bridges," Tapp says. Tapp and Geer are seeking ways to build bridges between mathematical specialties by making the courses they teach more accessible to a wider range of students, and by involving undergraduates in their multidisciplinary research projects.

About the Author

Jennifer Fisher Wilson is a contributing editor for The Scientist. She writes frequently about science and medicine for various publications, including Lancet Neurology, Science and UCLA Magazine.

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