My primary research interest is commutative and computational algebra, with related interests in homological algebra and algebraic geometry.In essence, these fields find their roots in the study of algebraic equations in relation to the geometry of their solutions. Such a line of investigation goes back at least to Descartes and the idea of coordinatizing the plane.
Nowadays, though, commutative algebra and algebraic geometry study the solutions of those equations by forming an algebraic object, called a ring, which consists of the `generic' solutions. The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the equations.
The methods in use are no longer from algebra alone, but also from analysis and topology. Conversely, they have been extensively used in those fields as well, and have proven useful in other fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.
Some of my research interests include:

Blowup algebras: Rees algebras, associated graded rings, special fiber rings and Sally modules;

Linkage and residual intersections theory;

Hilbert functions;

Integral closure of ideals;

Ideal theory of graphs;

Koszul homology.