The investigator proposes to work on questions that arise from tight closure theory, local cohomology, and the theory of intersection multiplicities. Tight closure theory was developed by Hochster and Huneke, and leads to the notion of F-regular rings. This class of rings includes well-studied examples such as determinantal rings and normal monomial rings, but several questions about F-regular rings remain unanswered. The theory of local cohomology has applications to fundamental questions such as determining the minimal number of equations needed to define an algebraic set. Local cohomology modules often have useful finiteness properties, and the investigator proposes to continue his work on Lyubeznik's conjecture, which states that local cohomology modules of regular rings have finitely many associated prime ideals. The work on intersection multiplicities will focus on the recently developed theory of Roberts rings: these rings provide a framework in which intersection multiplicities have several desirable properties. Some of the questions described in the proposal create possibilities for students to be involved in research by performing computer verifications to suggest answers and plausible approaches. The investigator will also be involved in curriculum development in areas related to this proposal, and intends to enhance opportunities for students interested in algebra in the Deep South.
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, in commutative algebra the main objects of study are certain functions on these solution sets. It continues to develop a fascinating interaction with several other branches of mathematics, and is becoming an increasingly valuable tool in engineering, coding theory, cryptography, and other applications of strategic interest.
