Local Cohomology, Differential Operators, and Determinantal Rings

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The award is concerned with several questions in commutative algebra: this is a field that studies solution sets of polynomial equations, and the questions that will be investigated eventually yield information about the nature of the solution sets. Polynomial equations arise in a number of situations; indeed commutative algebra continues to develop a fascinating interaction with several fields, becoming an increasingly valuable tool in science and engineering. The focus here is on questions relating to differential operators, particularly in the context of rings of invariants; the differential operators may be thought of as extensions of the rules of calculus to solutions sets of polynomials, while the rings of invariants are collections of polynomials that remain unchanged under various transformations. A key part of the awarded work is the training of graduate students in topics connected with the research program.Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; one of the projects, related to the PI's recent joint work with Jeffries, is studying the rings of differential operators on classical invariant rings in the positive characteristic case, and another is the behavior of differential operators under base change. Key tools for these come from local cohomology theory, and the study of integer torsion in local cohomology modules. Other projects involve Hankel determinantal rings, close cousins of the determinantal rings of classical invariant theory. The F-regularity of Hankel determinantal rings of positive prime characteristic will be investigated; this has an entirely equivalent formulation in terms of differential operators. The question arises naturally from the PI's joint work with Conca, Mostafazadehfard, and Varbaro, where it was proved that these rings have rational singularities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.