I am interested in group generalizations: What do you get if you relax the defining properties for a group? (If you don't know what a group is, look here.)
There are many different sets of axioms people use to define groups, and you have to be a bit careful as to how you choose when you decide to make changes. E.g., a loop is sometimes described as "a group without associativity". But if you start with group axioms which do not specifically state that the identity works on both sides, relying on associativity to imply that, then dropping associativity requires you to add the requirement for the identity to be two-sided.
It is probably better to start from the other end and build up. A quasigroup is an algebraic object consisting of a set and a binary operation which meets a very simple requirement: You want to be able to "divide", and the usual way to say this is that if x denotes the operation, the combination a x b = c has a unique solution for any one of a, b, and c if the other two are specified in the set. A loop is a quasigroup with an additional requirement, that it have a (two-sided) identity element.
One of the things which interests me is the interplay between the algebraic description of an object and its combinatorial properties. I am usually interested in the case when the set is finite, which simplifies some of these descriptions. A quasigroup, for example is just the case where the operation table is a Latin Square, i.e. has each entry appearing exactly once in each row and in each column of the table. Some surprising results come from combining these views.
In algebra we usually study properties which are preserved by isomorphisms. I am interested in a generalization not only of the objects but also this characterizing mapping. An isomorphism is a function which "takes" one algebraic object to another which, among other things, satisfies that f(a) * f(b) = f(a x b) where f is the function and * and x are the operations on the two objects. An interesting variation is to allow different functions in those three places, and the resulting triple of functions is called an isotopism. Viewed in terms of the picture suggested above, the one that gave a relation between a quasigroup and a Latin Square, an isomorphism requires a permutation which works simultaneously on the rows, the columns, and the entries in the table, while an isotopism allows three different permutations.
I am also interested in mathematics education, both in the immediately practical sense of "how can we teach our classes better" and in more basic work on how people learn mathematics and what forces shape their mathematical education.
One of my primary interests at the present is the extent to which our cultural context affects how students succeed in mathematics. Data from international studies show that US kids at grades K-12 do not do very well in comparison with students from some other countries. (You can see some results at http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2001028.) It is easy to blame schools and teachers. But this is also a country where many (highly educated in some cases) people are actually proud that they can't do math, where a "nerd" is something kids don't want to be, and where toys and the media broadcast an anti-math, anti-science, anti-thinking point of view far more than in many of the countries that did well on those tests. Can we expect K-12 students to do well at something they are constantly being told they should avoid if they are to be socially acceptable? If that is not possible, what can we do to change society's image and message? If it is possible, how? What could/should be done in teaching to give the societal message less impact? (It is interesting to note that US student's abilities in math and science fall away as they progress from 4th to 8th to 12th grade, exactly as they become more aware of and sensitive to social pressures!)