Ever since graduate school, I've been interested in model theory, a branch of mathematical logic. Basically, model theory determines which objects follow a specific set of rules (or axioms). More specifically, I've have looked at results from classification theory and stability theory and explored how those results can be extended when looking at permutation groups of objects (that is, the group of automorphisms of an object). I have several results in this area, including:
Sona sand drawings are drawings created by the Chokwe people of Africa with some interesting mathematical properties. (A good introduction to them can be found in Paulus Gerde's book Geometry from Africa: Mathematical and Educational Explorations.) These types of drawings can be seen as a subset of figures known as mirror curves. (You can find a definition and examples at Slavik Jablan's page at http://members.tripod.com/~modularity/mir.htm.) I have submitted one paper connecting mirror curves to abstract algebra called Mirror Curves and Permutations.
A few years ago, I was looking for a mathematical way to rank teams in the NFL. I came across an article by James Keener (The Perron-Frobenius Theorem and the Ranking of Football Teams, SIAM Review, Vol. 35, No. 1, pp 80-93, March 1993) and used what he calls the direct method (basically finding the principal eigenvector of a nonnegative matrix by using the power method). You can find more details in this short essay I wrote.
The mathematics of this method bring up some interesting questions involving matrix algebra and graph theory such as:
For more wide-ranging information on sports ranking and mathematics, I recommend this article by Ivars Peterson and this site explaining the theory behind some other ranking systems.
I have long been interested in the reform movement in mathematics with a special interest in the use of CAS (computer algebra systems) and visualization. I have written and administered a Lousiana Board of Regent Support Fund grant outfitting our calculus classrooms with TI-89's, CBR's , and CBL's. You can also find a paper showing some of the software I've developed in Matlab for our multivariable calculus class. You can also check out my teaching page for more information and links.
For our multivariable calculus class, I have also created a set of conceptests (multiple choice questions that test mathematical concepts rather than calculation) following the example of Eric Mazur. You can read a paper on my use of conceptests, download my collection of 89 conceptests, or visit Eric Mazur's site at galileo.harvard.edu to see conceptests from other fields.
I've done some work explaining the debunking of the Bible Code results. You can look at the slide show I've presented to Centenary undergraduates.