Invariant Rings, Frobenius, and Differential Operators

不变环、弗罗贝尼乌斯和微分算子

• 批准号: 2349623
• 负责人: Anurag Singh
• 金额: $ 30万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349623&HistoricalAwards=false
• 关键词: Invariant; Rings; Frobenius; Differential; Operators

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.

该项目将研究交换代数的几个问题，该领域研究了解决多项式方程的解决方案集。该研究将产生有关此类方程解决方案集的特性的具体信息。多项式方程在大量应用中出现；他们研究的一种富有成果的方法是通过在其溶液集上研究多项式功能，这形成了所谓的交换环。这在研究各种环境中的解决方案集方面提供了很大的灵活性，并且实际上是交换代数继续与几个领域建立迷人的互动，成为科学和工程学中越来越有价值的工具。该项目的一个关键组成部分是对与研究计划相关的主题研究生的培训。该研究的重点是与本地共同体，差异操作员以及具有有限的Frobenius代表类型的财产有关的问题。当地的共同体通常为基本问题提供了最佳答案，例如定义解决方案集所需的多项式方程数量最少；这将研究与某些不变性环相关的解决方案集。一个人在微积分中遇到的差分运算符在多项式方程的解决方案集上具有良好的通用性，并且被证明是一个越来越有成果的研究对象。同样，有限的Frobenius代表类型首先是为差异操作员研究引入的，被证明是具有多种应用程序的非常强大的财产。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子来评估的支持，并具有更广泛的影响。