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。这项研究将提供关于这类方程解集性质的具体信息。多项式方程出现在广泛的应用中；研究多项式方程的一个卓有成效的方法是通过研究其解集上的多项式函数，这些多项式函数形成了众所周知的交换环。这为在各种环境下研究解集提供了巨大的灵活性，事实上，交换代数继续与几个领域发展着令人着迷的交互作用，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研究生在与研究计划相关的主题上的培训。研究的重点是与局部上同调、微分算子和具有有限Frobenius表示型的性质有关的问题。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目；这将针对与某些不变量环相关的解集进行研究。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，并被证明是一个日益卓有成效的研究对象。同样，有限Frobenius表示类型，最初是为研究微分算子而引入的，在几个应用中被证明是一个非常强大的性质。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.