Invariant Rings, Frobenius, and Differential Operators

本专题将探讨交换代数中的几个问题，交换代数是研究多项式方程组解的领域。该研究将产生具体的信息，这些方程的解集的性质。多项式方程有着广泛的应用;一种卓有成效的研究方法是通过研究多项式函数的解集，形成了所谓的交换环。这提供了一个巨大的灵活性，在研究解决方案集在各种设置，并确实交换代数继续发展一个迷人的互动与几个领域，成为一个越来越有价值的工具，在科学和工程。该项目的一个关键组成部分是研究生的培训与研究计划的主题。研究的重点是有关的问题，局部上同调，微分算子，并具有有限Frobenius表示类型的属性。局部上同调通常为一些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;我们将研究与某些不变量环相关的解集。微积分中遇到的微分算子在多项式方程的解集上具有良好的一般性，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，首先介绍了微分算子的研究，被证明是一个非常强大的属性与几个应用程序。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的知识价值和更广泛的影响审查标准的支持。

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.