The questions in this proposal are centered around 4 different but deeply connected research areas in commutative algebra: local cohomology, algebraic D-modules, F-singularities, and homological conjectures. Local cohomology is the common thread. For example, the study of Frobenius actions on local cohomology modules has been fruitful in the study of F-singularities, and the content of local cohomology, a notion introduced by Hochster and Huneke, has provided a number of new approaches to homological conjectures, and in the study of local cohomology, algebraic D-module theory has proven to be indispensable. The PI wants to study various aspects of local cohomology and their applications to both F-singularities and homological conjectures and to investigate a characteristic-free notion of holonomicity.
This project is mainly concerned with questions in commutative algebra and their applications to algebraic geometry. Algebraic sets are sets of solutions of systems of polynomial equations. Commutative algebra studies functions over algebraic sets (in commutative algebra, these algebraic sets can be over any commutative rings); while algebraic geometry studies the geometric or topological aspects of algebraic sets (in algebraic geometry, these algebraic sets are very often over a field). These two areas are closely related and are mutually beneficial. Questions in this proposal can be considered as questions about the existence of solutions to systems of polynomial equations and the topological or geometric nature of the solution sets.
