Invariant Rings, Frobenius, and Differential Operators

这个项目将研究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。这项研究将产生关于这类方程解集性质的具体信息。多项式方程有着广泛的应用；一个富有成效的研究方法是通过研究它们的解集上的多项式函数，这些解集形成了所谓的交换环。这为在各种情况下研究解集提供了巨大的灵活性，而且交换代数确实继续与几个领域发展出迷人的相互作用，成为科学和工程中越来越有价值的工具。该项目的一个关键组成部分是在与研究计划相关的课题上对研究生进行培训。研究的重点是局部上同、微分算子和有限Frobenius表示类型的性质。局部上同调通常为基本问题提供最佳答案，例如定义解集所需的多项式方程的最少数量；这将研究与某些不变量环相关的解集。人们在微积分中遇到的微分算子在多项式方程的解集上具有很好的普遍性，并且被证明是一个越来越富有成果的研究对象。同样，有限Frobenius表示类型，最初是为了研究微分算子而引入的，被证明是一个非常强大的性质，有几个应用。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。

This project will investigate several questions in commutative algebra, a field that studies solution sets of polynomial equations. The research will yield concrete information about the properties of solution sets of such equations. Polynomial equations arise in a wide number of applications; one fruitful approach to their study is via studying polynomial functions on their solution sets, that form what is known as a commutative ring. This offers an enormous amount of flexibility in studying solutions sets in various settings, and indeed commutative algebra continues to develop a fascinating interaction with several fields, becoming an increasingly valuable tool in science and engineering. A key component of this project is the training of graduate students in topics connected with the research program.The focus of the research is on questions related to local cohomology, differential operators, and the property of having finite Frobenius representation type. Local cohomology often provides the best answers to fundamental questions such as the least number of polynomial equations needed to define a solution set; this will be investigated for solution sets related to certain rings of invariants. The differential operators that one encounters in calculus make sense in good generality on solution sets of polynomial equations and are proving to be an increasingly fruitful object of study. Similarly, finite Frobenius representation type, first introduced for the study of differential operators, is proving to be a very powerful property with several applications.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.