My primary field of research is Arithmetic Dynamics, which is the study of number theoretic properties of dynamical systems. In particular, I study number fields known as iterated extensions, which are named for their construction. Namely, if f is a monic polynomial with integer coefficients, and fn denotes the n-fold composition of f with itself, the iterated extensions are the number fields generated by the roots of fn. The roots of fn+1 are algebraic over the splitting field of fn, and thus the field defined by fn+1 contains that defined by fn. In other words, these splitting fields form an infinite tower. I have studied the disciminants of such fields and have found towers in which all the fields are monogenic.
Recently I have been looking at special points on hyperelliptic curves. A point P on an algebraic curve C is a Weierstrass point if the space of functions defined over C with poles restricted to P is larger than what would be expected by the Riemann-Roch theorem. The differences in the dimensions of these spaces (counted by the maximum order of the pole at P) are realizable as a numerical semigroup. Thus by understanding the semigroup, we may classify points on the curve.
I also have an interest in unit groups, and in particular, elliptic units. Following work of Greene and Hajir, I have written a program in PARI/GP which computes an "ideal" generating set of elliptic units in unramified extensions of imaginary quadratic fields. The code is available here: [code]. (There are a few known bugs, so contact me before using.)