Research Interests
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, CohenMacaulay 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 p^{th} 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 HilbertKunz 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 HilbertKunz 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 longstanding conjectures of AuslanderReiten 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.
