Most generally I am deeply interested in geometry and study it using Algebraic, Combinatorial, and most recently Complex Differential techniques.
My thesis research studied the classical Hilbert scheme of Grothendieck parameterizing the closed subschemes of projective space. The approach was to use the notion of a Borel-fixed ideal -- that is an ideal in the ring of regular functions on projective space which is fixed by the action of the Borel subgroup of GL(n) consisting of upper-triangular matrices. It turns out that these ideals are most naturally described combinatorially.
Recently I have been drawn to the study of complex geometry -- that is the study of those manifolds endowed with a complex structure.
Interior derivative estimates for the Kähler-Ricci flow. Joint with Ben Weinkove; (arXiv) to appear in Pacific Journal of Mathematics
When is a trigonometric polynomial not a trigonometric polynomial? Joint with Joe Borzellino (arXiv) Mathematical Association of America Monthly.
Convergence Properties of Donaldson's $T$-iterations on the Riemann sphere. (arXiv) Journal of Experimental Mathematics
On an extension of Galligo's theorem concerning the Borel-fixed points on the Hilbert scheme. (math.AC/0512023) Journal of Algebra
The infinitely near Borel-fixed points on the Hilbert scheme. (Thesis)