CAREER: Differential Operators and p-Derivations in Commutative Algebra

职业：交换代数中的微分算子和 p 导数

• 批准号: 2044833
• 负责人: Jack Jeffries
• 金额: $ 40万
• 依托单位: University of Nebraska-Lincoln
• 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 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2044833&HistoricalAwards=false
• 关键词: CAREER; Differential; Operators; Derivations; Commutative

Understanding the solutions of polynomial equations is a fundamental problem with applications throughout the basic sciences. Over the most familiar number systems—real numbers and complex numbers—one can think of these solutions sets geometrically via Cartesian coordinates. It is important for various reasons to also consider polynomial equations over more exotic number systems. For example, computer data and arithmetic are built on a system of two numbers (zero and one). The main research goal of this project is to develop new tools to study the small-scale behavior of systems of polynomial equations, both over familiar number systems (of characteristic zero) and over more exotic number systems (of positive or mixed characteristic). Specifically, the PI aims to extend aspects of the theory of differential operators for smooth varieties of characteristic zero, such as holonomicity and Bernstein-Sato theory, to singular varieties in characteristic zero and positive characteristic. The PI will also apply Buium and Joyal's notion of p-derivation to the study of singularities in mixed characteristic. The main educational goals of this project are to increase participation in mathematics by high school students and members of the growing Hispanic community in Nebraska, to foster international academic relationships among graduate students, particularly between the United States and Mexico, to create connections between traditionally different areas of mathematics themed around emerging problems, and to train graduate students, both directly and more broadly through dissemination of pedagogical materials. These will be pursued in a range of activities, including the formation of a high school math circle, cross-topical graduate workshops, and visits from senior and junior mathematicians from Latin America.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.

理解多项式方程的解是整个基础科学应用的一个基本问题。在最熟悉的数字系统（实数和复数）中，人们可以通过笛卡尔坐标以几何方式想到这些解集。由于各种原因，考虑更奇异的数字系统上的多项式方程也很重要。例如，计算机数据和算术是建立在两个数字（零和一）的系统上的。该项目的主要研究目标是开发新工具来研究多项式方程组的小规模行为，包括熟悉的数系统（特征为零）和更奇特的数系统（正或混合特征）。具体来说，PI 旨在将特征零平滑簇的微分算子理论（例如完整性和 Bernstein-Sato 理论）扩展到特征零和正特征的奇异簇。 PI 还将应用 Buium 和 Joyal 的 p 导数概念来研究混合特性中的奇点。该项目的主要教育目标是提高高中生和内布拉斯加州不断增长的西班牙裔社区成员对数学的参与，培养研究生之间的国际学术关系，特别是美国和墨西哥之间的关系，在以新问题为主题的传统不同数学领域之间建立联系，并通过传播教学材料直接和更广泛地培训研究生。这些活动将通过一系列活动来实现，包括组建高中数学圈、跨主题研究生研讨会以及来自拉丁美洲的高级和初级数学家的访问。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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

A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic

混合特征多项式环直接被加数的一致Chevalley定理

• DOI: 10.1007/s00209-022-03035-2
• 发表时间: 2022
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: De Stefani, Alessandro;Grifo, Eloísa;Jeffries, Jack
• 通讯作者: Jeffries, Jack

Lower bound on Hilbert-Kunz multiplicities and maximal F-signatures

Hilbert-Kunz 重数和最大 F 签名的下界

• DOI: 10.1017/s0305004122000238
• 发表时间: 2022
• 期刊: Math.Proc.Camb.Phil.Soc.
• 作者: Jack Jefferies; Yusuke Nakajima;Ilya Smirnov,Kei-ichi Watanabe;Ken-ichi Yoshida
• 通讯作者: Ken-ichi Yoshida

Extensions of primes, flatness, and intersection flatness

素数、平坦度和相交平坦度的扩展

• 发表时间: 2021
• 期刊: Contemporary mathematics
• 作者: Melvin Hochster;Jack Jeffries
• 通讯作者: Jack Jeffries