This issue we ask 5 Questions of…David Pollack, associate professor of mathematics and computer science.
Q: How did you become interested in mathematics in general, and as an
academic career specifically?
DP: Mathematics was my favorite subject in school as far back as I can
remember. At that time I had no idea that one could be a mathematician, so I imagined I would be a scientist or engineer. After my sophomore year in high school I was fortunate enough to attend the summer mathematics program at Hampshire College, where I was first exposed to professional mathematicians. I realized more or less immediately that mathematics itself was the right career for me. The next summer I attended the Ross Mathematics Program at Ohio State, an incredibly rigorous mathematics immersion course that teaches students to “think deeply about simple things” by developing number theory and basic abstract algebra from the ground up. Students get the opportunity to explore questions, observe patterns, make good conjectures, and discover proofs. I wound up returning each of the next five years as a counselor and never looked back from teaching and doing mathematics.
Q: What have you been working on recently?
DP: My research focuses on questions about the arithmetic cohomology of higher rank matrix groups. These cohomology classes can be thought of as generalizations of the classical modular forms that were central to the proof of Fermat’s Last Theorem given by Andrew Wiles in the 90’s. Most recently I’ve been thinking about how cohomology classes deform in families with a varying coefficient module.
Q: For those of us not so well-steeped in number theory, what is “arithmetic cohomology of higher rank matrix groups” and why is it important in number theory?
DP: Cohomology classes are geometric invariants that often describe an obstruction to the existence of solutions to certain equations. They originally arose in the study of topological spaces, but now play an important role in many branches of mathematics, including algebraic geometry, representation theory, and number theory. One of the major themes of current research in number theory is the surprising connection between cohomology classes and some seemingly unrelated objects called Galois representations. These representations measure notions of hidden symmetries in collections of numbers. For example, the usual complex conjugation is one such symmetry, but there are many many others. Number theory has long been interested in understanding these symmetries as a key step towards understanding the behavior of the numbers themselves. Galois representations allow us to get a more concrete view of these mysterious symmetries by interpreting them as more explicit symmetries of n-dimensional space.
Q: You also oversee the Putnam Competition for Wesleyan. What is this and how do you recruit students and prepare them to participate?
DP: The Putnam competition is a national mathematics exam given each year on the first Saturday in December. It consists of 12 very difficult questions that usually depend only on knowledge of the first two years of undergraduate mathematics but that require a great deal of creativity and cleverness. The solutions are judged strictly, with little partial credit given for incomplete answers. In fact, out of a possible maximum of 120 points the national median hovers around 1 or 2 (Yes that’s 1 out of 120, not 1 out of 12).
We’ve been holding informal problem solving seminars in the fall for many years, and for the last three years I’ve been teaching a half semester course to help students prepare for the exam. The real point of the course, though, is to expose the students to some interesting mathematics and to give them a chance to try their hand at problem solving. It’s great fun to teach since the students are very enthusiastic and really enjoy attacking the problems. The audience has been growing, including several students who attend but didn’t even bother signing up for the credit. This year we had 19 students who took 6 hours out of a busy Saturday to sit and take the Putnam exam itself.
Q: I understand your brother is also a mathematics professor. What is that like?
DP: It’s great. Robert is also a number theorist, and although we started out in fairly different subfields while we were in graduate school, our research interests have been converging since then. We’ve even co-authored a paper and have an extension of that project in the works. It makes visiting him very interesting, as we often switch back and forth between playing with my two year-old nephew and discussing overconvergent modular forms!