The two main areas represented in this page ultimately involve the theory of Cohen-Macaulay rings and algebras. Such objects carry significant algebraic and geometric weight, but also considerable computational efficiencies (probably related!). Two foci for us are:
Commutative Algebra: Cohen-Macaulay rings and algebras, and the determination of their numerical invariants; also the algebraization of tangent and blowup algebras (see The Arithmetic of Blowup Algebras, Cambridge University Press, 1994).
Computational Algebra: Applications and development of Groebner bases techniques to the construction of integral closure of rings and ideals, primary decomposition and criptography, rings of invariants, cohomology, and numerical equations solvers (see Computational Methods in Commutative Algebra and Algebraic Geometry, Springer-Verlag, 1998; Paperback corrected edition, 2004).
A recent book mixes these two strands: Integral Closure, Springer Monographs in Mathematics Series. Springer-Verlag, 2005.