Local Cohomology, Differential Operators, and Determinantal Rings

该奖项涉及交换代数中的几个问题：交换代数是一个研究多项式方程解集的领域，所研究的问题最终将产生关于解集性质的信息。多项式方程出现在很多情况下；事实上，交换代数继续与多个领域发展迷人的相互作用，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培养与研究项目相关的研究生。Levasseur和Stafford描述了各种特征为0的经典不变环上的微分算子环；其中一个项目与PI最近与Jeffries的联合工作有关，研究经典不变环上正特征情况下的微分算子环，另一个是基变化下微分算子的行为。研究这些问题的关键工具来自于局部上同调理论，以及局部上同调模中整数扭转的研究。其他项目涉及汉克尔行列式环，它是经典不变理论的行列式环的近亲。研究了具有正素数特征的Hankel行列式环的f正则性；这有一个完全等价的微分算子表达式。这个问题从PI与Conca， mostafazadehard和Varbaro的联合工作中自然产生，在那里证明了这些环具有理性奇点。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.