Determinantal Rings, Local Cohomology, and Tight Closure

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The project is concerned with several questions in commutative algebra: this is a field that studies solution sets of polynomial equations by means of studying polynomial functions on the solution sets. Polynomial equations arise naturally in a number of situations, and commutative algebra continues to develop a fascinating interaction with several fields, becoming an increasingly valuable tool in science and engineering. The focus of the project is on questions in commutative algebra relating to local cohomology, tight closure theory, and classical rings of invariants; all of these questions arise quite naturally from recent developments. Local cohomology often provides the best answers to fundamental questions such as the least number of polynomial equations needed to define a solution set. Projects in this direction include algorithmic aspects as well as structural properties; there is a special focus on local cohomology modules of polynomial rings 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 joint work with Lyubeznik and Walther. At the same time, new techniques for investigating local cohomology over the integers have been developed in joint work with Bhatt, Blickle, Lyubeznik, and Zhang; it is proposed to extend these new techniques to an algorithm. Projects related to tight closure theory include investigating the singularities of Hankel determinantal rings, a question coming from recent joint work with Conca, Mostafazadehfard, and Varbaro, and whether these rings arise as invariant rings for actions of linearly reductive groups.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.