Local Cohomology, Differential Operators, and Determinantal Rings

该奖项涉及交换代数中的几个问题：这是一个研究多项式方程解集的领域，将被研究的问题最终会产生关于解集性质的信息。多项式方程出现在许多情况下；事实上，交换代数继续与几个领域发展着令人着迷的相互作用，成为科学和工程中越来越有价值的工具。这里的重点是与微分算子有关的问题，特别是在不变量环的背景下；微分算子可以被认为是微积分规则对多项式解集的扩展，而不变量环是在各种变换下保持不变的多项式的集合。获奖工作的一个关键部分是研究生在与研究计划相关的主题中的培训。Levasseur和Stafford描述了各种特征零的经典不变环上的微分算子环；其中一个项目是研究正特征情况下经典不变环上的微分算子环，另一个是基变化下的微分算子的行为，这与PI最近与Jeffries的合作有关。这方面的关键工具来自局部上同调理论和局部上同调模中的整数挠度的研究。其他项目涉及Hankel行列式环，这是经典不变理论中行列式环的近亲。我们将研究具有正素特征的Hankel行列式环的F-正则性；这在微分算子方面具有完全等价的表述。这个问题自然源于PI与Conca、Mostafazadehfard和Varbaro的联合工作，在这些工作中，证明了这些环具有有理奇异性。这一裁决反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.