Local Cohomology, Differential Operators, and Determinantal Rings

局部上同调、微分算子和行列环

• 批准号: 2101671
• 负责人: Anurag Singh
• 金额: $ 27万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101671&HistoricalAwards=false
• 关键词: Local; Cohomology; Differential; Operators; Determinantal

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.

该奖项涉及交换代数中的几个问题：这是一个研究解决方案的解决方案集的领域，以及将要研究的问题最终会产生有关解决方案集的性质的信息。多项式方程在多种情况下出现；的确，换档代数继续与几个领域建立着令人着迷的互动，成为科学和工程学中越来越有价值的工具。这里的重点是与差异操作员有关的问题，尤其是在不变的环的背景下；可以将差分运算符视为对多项式解决方案集的计算规则的扩展，而不变的环是多项式的集合，在各种变换下保持不变。奖励工作的关键部分是对与研究计划相关的主题研究生的培训。Levasseurand Stafford描述了差异操作员在各种特征零的经典不变环上的戒指；其中一个项目与PI最近与Jeffries的联合工作有关，它正在研究差异操作员在积极特征中的经典不变环的环，而另一个项目是基本变化下差异操作员的行为。这些的关键工具来自当地的同时理论，以及对局部协同学模块中整数扭转的研究。其他项目涉及汉克尔决定性环，是经典不变理论的决定性环的近亲。将研究汉克尔决定性环的f频率。就差异操作员而言，这具有完全等效的配方。这个问题自然来自PI与Conca，Mostafazadehfard和Varbaro的共同工作，在此证明，这些环具有合理的细纹。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优点和更广泛影响的审查标准来通过评估来通过评估来支持的。