Symbolic Powers and p-Derivations

这是一个交换代数的项目，与代数几何，组合数学和算术几何有关。研究中心的主题是象征权力，一个活跃的研究课题，几乎所有方面的交换代数。这个项目的目的是解决两个问题，比较几何概念的符号权力与代数概念的自然权力，并在应用程序的微分代数概念的p-导子交换代数。研究者将指导本科生的研究经验，并组织一个交换代数的研究生工作坊。这个项目的统一主题是理想的符号幂的研究。研究主要集中在两个问题：包含问题和p-导子在交换代数中的应用。虽然多项式环中的理想对应于仿射空间中在某个簇上为零的多项式，但它们的符号幂由在给定簇上为某个阶的多项式组成。这个自然的几何概念有一个来自初等分解理论的代数描述，初等分解理论是算术基本定理的理想理论版本。这是一个古老的、丰富的、无处不在的话题，然而，关于象征性权力的问题有很多，既容易措辞，又很难解决。包容问题试图比较符号幂（一种自然的几何概念）和普通幂（一种自然的代数概念）。这涉及到其他有趣的问题，如确定最小程度的多项式消失在一个给定的品种。该项目的另一部分是发现p-导子和交换代数之间的新联系，着眼于一个新的奇异性理论的混合特性，结合p-导子与微分算子的精神，F-奇异性理论及其连接到D-该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

This is a project in commutative algebra, with connections to algebraic geometry, combinatorics, and arithmetic geometry. The research centers on the theme of symbolic powers, an active topic of research with connections to virtually all aspects of commutative algebra. This project aims to resolve two questions, on comparing the geometric notion of symbolic powers with the algebraic notion of natural powers, and on the application of the differential algebraic notion of p-derivations to commutative algebra. The investigator will direct undergraduate research experiences and organize a graduate workshop in commutative algebra.The unifying theme of this project is the study of symbolic powers of ideals. The research focuses primarily on two questions: the containment problem and applications of p-derivations to commutative algebra. While ideals in a polynomial ring correspond to the polynomials that vanish on a certain variety in affine space, their symbolic powers consist of the polynomials that vanish to a certain order on the given variety. This natural geometric notion has an algebraic description coming from the theory of primary decomposition, an ideal-theoretic version of the fundamental theorem of arithmetic. This is an old, rich, and ubiquitous topic, yet there is an abundance of questions about symbolic powers that are both easy to phrase and very difficult to solve. The containment problem attempts to compare symbolic powers, a natural geometric notion, with ordinary powers, a natural algebraic notion. This relates to other interesting questions, such as determining the minimal degrees of polynomials vanishing on a given variety. Another part of the project is to discover new connections between p-derivations and commutative algebra, with an eye towards a new singularity theory in mixed characteristic that combines p-derivations with differential operators in the spirit of the theory of F-singularities and its connections to D-modules.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.