Symbolic Powers and Lefschetz Properties: Geometric and Homological Aspects

这项研究是关于代数几何所激发的交换代数中的问题。在一系列广泛的科学努力的核心是无处不在的解多项式方程的需要。一个互补的目标是找到多项式方程，例如，其图形通过一组给定数据点的方程。这个过程被称为多项式内插，在数据科学、数值分析和代数几何的界面上是一个基本的挑战。研究人员将把交换代数和计算代数的方法引入多项式插值的高阶版本。例如，当数据点表现出内在对称性时的情况将被阐明。此外，本课程还将深入了解这一主题与同调代数中新兴技术之间的相互作用。这项基础研究的更广泛影响在于研究生的参与和培训、软件开发以及初级数学家的招聘、保留和专业发展。该研究项目集中在两个引起当前兴奋的主题：多项式插值和代数Lefschetz性质。前一个主题将通过符号幂理想的透镜来分析，符号幂理想可以被认为是在给定的代数变体上消失到特定阶数的多项式的集合。后一个主题构成了硬Lefschetz定理的代数抽象，在数学的几个领域有着壮观的应用。这两个主题之间的相互关系将得到彻底的探索和利用。一个特殊的研究方向是代数Lefschetz性质在同调代数中的应用，特别是对分次自由分解的应用。其他方向还包括应用于包含问题，该问题涉及由理想定义的普通拓扑和符号拓扑。这项工作的各个方面展示了与反射群理论、超平面排列、凸几何和微分分次代数的关系。该奖项反映了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。

This research concerns problems in commutative algebra motivated by algebraic geometry. At the heart of a wide array of scientific endeavors is the ubiquitous need to solve polynomial equations. A complementary goal is to find polynomial equations, for example, an equation whose graph passes through a given set of data points. This procedure, termed polynomial interpolation, is a fundamental challenge at the interface of data science, numerical analysis, and algebraic geometry. The investigator will bring methods from commutative and computational algebra to bear on aspects of a higher order version of polynomial interpolation. For example, the situation when the data points exhibit intrinsic symmetry will be elucidated. Additionally, a deeper understanding of the interactions between this topic and emerging techniques in homological algebra will be pursued. The broader impact of this fundamental research lies in the engagement and training of graduate students, software development, and the recruitment, retention, and professional development of junior mathematicians. The research project focuses on two topics which generate current excitement: polynomial interpolation and the algebraic Lefschetz properties. The former theme will be analyzed through the lens of symbolic power ideals, which can be thought of as sets of polynomials that vanish to a certain order on a given algebraic variety. The latter theme constitutes an algebraic abstraction of the Hard Lefschetz Theorem with spectacular applications to several areas of mathematics. The interrelations between these two topics will be thoroughly explored and exploited. One particular direction of investigation is on applications of the algebraic Lefschetz properties to homological algebra, specifically to graded free resolutions. Other directions include applications to the containment problem relating the ordinary and symbolic topologies defined by an ideal. Aspects of this work exhibit relationships to the theory of reflection groups, hyperplane arrangements, convex geometry, and differential graded algebras.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.