My current research is in Voting Theory and involves exploiting some of the geometric interpretations developed by Don Saari at UC-Irvine. The basic problem arises when there are more than two candidates in an election. Depending on which procedure you use to determine the winner, you can get dramatically different outcomes, even if no one changes their preferences.
In addition to the obvious applications to political science, there are also applications in some very surprising areas, including computational biology and computer science. The main appeal for me is that there is some really interesting mathematics involved which has the added bonus of being comparatively accessible to undergraduates.
Much of my work has focused on the problem of electing committees where voters have preferences for the overall composition of the committee that cannot be reduced to preferences on individual candidates. There are the twin problems of developing reasonable means for voters to express their preferences without giving a complete ranking of all possible committees and of determining an appropriate decision procedure based on these preferences.
I also have several publications related to Dodgson's Method, which was proposed by Charles Dodgson (aka Lewis Carroll) while on the faculty at Oxford. This is a very reasonable sounded method that picks the candidate "closest" to the Condorcet winner when one does not exist. But as is often the case in voting theory, there are unexpected consequences to this approach.
By training, I am an algebraic topologist. My dissertation dealt with elliptic cohomology theories, and I did some work on the stable splittings of classifying spaces of finite groups.
You can find a complete list of my publications and presentations on my curriculum vitae.