My main research area is commutative algebra, and I am particularly interested in charamcteristic p methods, such as tight closure.

Characteristic p methods yield powerful results, which can usually be proved in the characteristic zero case as well, via a standard reduction to characteristic p. Some examples of results obtained using characteristic p techniques are the following: The rings of invariants of linearly reductive groups are Cohen-Macaulay (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 subring), the Briancon-Skoda theorem about the integral closure of an ideal (this was first proved by analytic techniques), the direct summand conjecture in equal characteristic, vanishing theorems for maps of Tors, existence of big (not necessarily finitely generated) Cohen-Macaulay algebras, and many other homological conjectures.

The advantage of working in characteristic p is that one can make use of an extra structure, in addition to the usual ring structure. This extra structure is given by the map which raises each element to the p th power, called the Frobenius map . The Frobenius map endows the ring with a structure of algebra over itself. The understanding of this algebra structure is central to the applications and development of tight closure theory.

Tight closure was formally introduced in 1990 by Hochster and Huneke, but characteristic p methods were used by various mathematicians before the theory of tight closure was systematically developed. The modern language of tight closure provides a unified approach to a large class of problems, and sheds light on some questions which are still open.

Although the applications of tight closure are abundant and well understood, tight closure itself is very hard to compute, and basic theoretic questions about its properties are still unanswered. The reason for the difficulties encountered in trying to understand tight closure lies in the fact that the definition requires infinitely many tests.

Here is a detailed RESEARCH STATEMENT.

PAPERS:

1. Boundedness of Local Cohomology of Frobenius Images, Journal of Algebra , 228 (2000) , 347--356.


2. *-Independence and special tight closure, Journal of Algebra, 249 (2002), 544-565.


3. Strong test ideals, J. Pure Applied Algebra, 167 (2002), 361-373.


4. Special tight closure; (with C. Huneke); Nagoya Mathematical Journal, 170 (2003), 175-183.


5. Tight closure and linkage classes in Gorenstein rings, Mathematische Zeitschrift, 244 (2003), 873--885.


6. Tightly closed ideals of small type Proceedings Amer. Math. Soc. 132 (2003), 341--346.


7. Vanishing of Ext and Tor over some Cohen-Macaulay local rings (with C. Huneke and L. M. Sega), Illinois J. Math., 48 (2004), 295--317.


8. Chains and families of tightly closed ideals , Bull. London Math. Soc., 38 (2006), 201--208.


9. Rings which are almost Gorenstein (with C. Huneke), Pacific J. Math., 225 (2006), no. 1, 85--102.


10. Socle degrees of Frobenius powers (with A. Kustin), Illinois J. Math., 51 (2007), 185--208.


11. A length characterization of *-spread (with N. Epstein), Osaka J. Math, 45 (2008), 445--456. .


12. A new version of a -tight closure , Nagoya Math. J., 192 (2008), 1--25.


13. When is tight closure determined by the test ideal? (with J. Vassilev), J. of Commutative Algebra, to appear.


14. Drops in joint Hilbert-Kunz multiplicities and projective equivalence of ideals, submitted.


15. The canonical module of almost Gorenstein rings (with J. Striuli), submitted


16. A formula for the *-core (with L. Fouli and J. Vassilev), submitted.