Questions on Local Cohomology and Tight Closure Theory

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This project is concerned with several questions in commutative algebra. This is a field that studies solution sets of polynomial equations. Understanding solution sets of polynomial equations is of fundamental importance in many sciences, in engineering, and in other disciplines as well. Most of the questions that will be investigated deal with the nature of the solution sets: some are questions that have been unresolved for a number of years, but for which recent advances provide the likelihood of a solution; others arise naturally from new developments in the area. A number of projects are connected with local cohomology theory: this theory often provides the best answers to basic questions such as the least number of polynomials needed to define a solution set.The projects on local cohomology theory include algorithmic aspects, as well as structural properties such as support and injective dimension. There is a special focus on local cohomology modules of polynomial rings and hypersurfaces over the integers: this stems from the fact that there is a canonical homomorphism from the integers to any ring, and this makes local cohomology modules over the integers, in a sense, universal; this viewpoint has proved useful in recent work of the PI and collaborators. The project will also investigate local cohomology over the integers. The research will further develop the connections of local cohomology with prime characteristic numerical invariants such as the F-pure threshold, and will study the composition series of local cohomology modules over rings of differential operators.