Singularities in Positive and Mixed Characteristic Commutative Algebra

正和混合特征交换代数中的奇点

• 批准号: 2200716
• 负责人: Kevin Tucker
• 金额: $ 20万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-15 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200716&HistoricalAwards=false
• 关键词: Singularities; Positive; Mixed; Characteristic; Commutative

Commutative algebra studies Noetherian commutative rings, examples of which are sets of polynomials and the ordinary integers. It is used in a wide variety of applied settings, ranging from error-correcting codes in computer science and genomics to control theory and modeling in engineering. Other fields of mathematics that use commutative algebra, and commutative rings in particular, are algebraic and arithmetic geometry, as well as complex analysis, topology, and representation theory. Execution of the projects in this grant will both advance theoretical understanding in commutative algebra and shed light on important problems from some of these related fields. In addition, the principal investigator is dedicated to promoting mathematics education, developing future generations of researchers, and assisting in building a strong STEM workforce in the US. Towards these goals, the PI will supervise, train, and mentor doctoral students and postdoctoral fellows throughout the course of the grant. The PI will also facilitate a number of seminars, workshops, and reading courses for undergraduate and graduate students.The PI specializes in the study of singularities in positive and mixed characteristic commutative algebra, and will address a number of questions arising naturally from recent developments in the theory of Frobenius splittings, tight closure, the homological conjectures, and absolute integral closure. The PI aims to exploit the connection between singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry in order to carry out research projects in three new directions. First, the PI will investigate properties of the dual F-signature limit in characteristic p 0, showing its existence and deriving new properties with applications to F-rationality. Second, also in positive characteristic, the PI will further the understanding of F-singularities outside of the F-finite setting by exploring quotients of excellent regular rings and generalizing adjunction statements. Lastly, leveraging breakthrough results on the absolute integral closure in mixed characteristic, the PI will attack open questions regarding the splinter condition and exhibit new geometric 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.

交换代数研究noether交换环，其例子是多项式集和普通整数。它被广泛应用于各种应用环境，从计算机科学和基因组学中的纠错代码到工程中的控制理论和建模。其他使用交换代数的数学领域，特别是交换环，是代数和算术几何，以及复分析，拓扑和表示理论。本基金项目的实施将促进对交换代数的理论理解，并阐明一些相关领域的重要问题。此外，首席研究员致力于促进数学教育，培养未来几代研究人员，并协助在美国建立强大的STEM劳动力。为了实现这些目标，PI将在整个资助过程中对博士生和博士后进行监督、培训和指导。PI还将为本科生和研究生举办一些研讨会、讲习班和阅读课程。PI专门研究正和混合特征交换代数中的奇点，并将解决Frobenius分裂理论，紧闭，同调猜想和绝对积分闭包的最新发展中自然产生的一些问题。PI旨在利用Frobenius图定义的正特征奇点与复杂代数几何中的奇点之间的联系，以便在三个新的方向上开展研究项目。首先，PI将研究特征p 0的对偶f签名极限的性质，证明它的存在性，并推导出应用于f -合理性的新性质。其次，同样在正特征方面，PI将通过探索优秀正则环的商和推广附加命题，进一步理解f -有限集外的f -奇点。最后，利用混合特性中绝对积分闭包的突破性成果，PI将解决有关分裂条件的开放性问题，并展示新的几何应用。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。

项目成果

Openness of splinter loci in prime characteristic

主要特征中分裂位点的开放性

• DOI: 10.1016/j.jalgebra.2023.03.025
• 发表时间: 2023
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Datta, Rankeya;Tucker, Kevin
• 通讯作者: Tucker, Kevin