My main interests lie in commutative algebra. I work on a number of themes in this field. The basic object I study is a commutative Noetherian ring R. Some of the topics I have worked on concern linkage theory (also called liaison), the structure of various algebras associated with an ideal (especially the Rees algebra, R[It], and the associated graded ring), homological problems, local cohomology, symbolic powers of ideals, Cohen-Macaulay and Gorenstein rings, and various closures of ideals, including the integral closure.
In recent years, most of my efforts have been in developing and understanding, jointly with Melvin Hochster and others, a closure operation called tight closure. The basic idea is to reduce problems concerning rings containing fields to the problem in positive characteristic p and there to take advantage of the Frobenius morphism which raises elements to the pth power. I have also been working in recent years with Ian Aberbach and Adela Vraciu on problems connected with tight closure theory.
I have become very interested in the Hilbert-Kunz function, the function which mimics the usual Hilbert function by replacing powers of ideals by Frobenius powers of ideals. A recent preprint (2003) with Moira McDermott and Paul Monsky proves a result which shows more structure of the Hilbert-Kunz function in normal rings.
I am also working on several problems concerning the homological theory of modules over Noetherian rings. Basic problems concern what the vanishing of Ext and Tor modules mean, and when vanishing occurs. A focus for this work is provided by several long-standing conjectures of Auslander-Reiten and others. I am doing this work with Graham Leuschke, Liana Sega, and Adela Vraciu.
The combinatorical properties of generic initial ideals and the relationship of generic initial ideals to graded resolutions of ideals is also a topic I am currently working on with David Eisenbud and Bernd Ulrich. We are especially interested in the structure of ideals with linear presentations.
Finding the best possible bounds on the heights of determinantal ideals in regular rings has been the subject of several recent papers I have written jointly with Eisenbud and Ulrich, and there is much left to do on this topic.