Local cohomology, tight closure, and related questions

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The PI will work on several problems in commutative algebra that are centered around local cohomology theory, the homological conjectures, and the theory of tight closure. The PI will continue work on the conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is known in various cases due to the work of Huneke-Sharp and Lyubeznik, but remains unresolved for polynomial rings over the integers. Some critical cases have been settled recently in joint work with Lyubeznik and Walther; these in turn, lead to new vanishing theorems for local cohomology modules. Hochster's monomial conjecture remains unresolved for rings that do not contain a field, such as those arising in number theory; the PI will pursue an approach to this via local cohomology. The proposed work in tight closure includes investigating the question whether weak $F$-regularity---the property that all ideals of a ring are tightly closed---is preserved under localization; this will be attacked via the notion of splinter rings. This project is concerned with questions in commutative algebra. This is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solution sets of polynomial equations, the point of view in commutative algebra is to study the ring of polynomial functions on a solution set. Most of the questions that will be investigated are questions about the nature of the solution sets. Commutative algebra continues to develop a fascinating interaction with several branches of mathematics, and is becoming an increasingly valuable tool in engineering, coding theory, cryptography, and other areas of strategic interest.