CAREER: Differential Operators and p-Derivations in Commutative Algebra

理解多项式方程的解是贯穿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--上，人们可以通过笛卡尔坐标几何地想到这些解集。出于各种原因，考虑更奇特的数字系统上的多项式方程是很重要的。例如，计算机数据和算术是建立在两个数字(0和1)的系统上的。这个项目的主要研究目标是开发新的工具来研究多项式方程组的小规模行为，无论是在熟悉的数字系统(特征为零)上还是在更奇异的数字系统上(正特征或混合特征)。具体地说，PI的目的是将特征零光滑簇的微分算子理论的某些方面，如完整性和Bernstein-Sato理论，推广到特征零奇异簇和正特征簇。PI还将应用Buium和JoYal的p-导数概念来研究混合特征中的奇点。该项目的主要教育目标是增加高中生和内布拉斯加州日益壮大的拉美裔社区成员对数学的参与，促进研究生之间的国际学术关系，特别是在美国和墨西哥之间，建立传统上以新出现的问题为主题的不同数学领域之间的联系，并通过传播教学材料直接和更广泛地培训研究生。这些活动将在一系列活动中进行，包括建立一个高中数学圈，跨主题的研究生研讨会，以及来自拉丁美洲的高级和初级数学家的访问。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.