My research is primarily concerned with commutative Noetherian rings, especially local rings (i.e., rings with a unique maximal ideal). I have been particularly interested in Cohen-Macaulay rings and modules (a finitely generated graded algebra over a field is Cohen-Macaulay if and only if it is a finitely generated free module over a polynomial ring). One of the main techniques in my work has been that of attacking problems, even problems that seem to arise primarily over a field of characteristic 0, by using positive characteristic methods (the action of the Frobenius endomorphism). One example of a theorem that can be proved this way is that, given a linear algebraic group (i.e., a group of matrices) that is linearly reductive (every representation is completely reducible) over the complex numbers acting on a polynomial ring, the ring of invariants (or fixed ring) is Cohen-Macaulay.

A great deal of my research, much of it joint with Craig Huneke, has been aimed at developing the notion of tight closure. The result mentioned above on the Cohen-Macaulay property for rings of invariants and several other apparently unrelated results can be proved using this technique. Tight closure is a closure operation defined on ideals (and submodules of Noetherian modules) first over Noetherian rings of positive prime characteristic p, but it can be extended to all Noetherian rings containing a field. Other applications include the Briançon-Skoda theorem on integral closures of ideals, and the result that regular rings are direct summands of their module-finite extension algebras. In particular, I am very interested in developing an analogue for tight closure theory that could be applied to Noetherian rings that do not necessarily contain a field.

Other research interests include the behavior of local cohomology and problems concerning various kinds of multiplicity. Recent joint work with Tigran Ananyan settles Stillman's conjecture, bounding the projective dimension of an ideal generated by a given number n of forms of degree d in a polynomial ring over a field independent of the number of variables and of the field. Many related problems remain open. Here is a link for some primarily expository manuscripts related to these subjects, and here is a link for some primarily research manuscripts related to them.