Polynomial Interpolation, Symmetric Ideals, and Lefschetz Properties

该奖项为与代数几何有关的交换代数的研究提供支持。在这个框架内，交换代数研究多项式方程组，其解形成几何对象，如曲线和曲面。寻找通过一组给定点的曲线或曲面的过程通常称为内插。多项式插值在数据分析、数值分析、计算机图形学和数学建模等科学学科中有着广泛的应用。这个项目特别关注在底层数据表现出对称性的情况下的高次多项式内插。更广泛地说，它的目标是使用交换代数中的工具来分析具有对称性的多项式方程组。除了这些贡献，首席研究员还将带领本科生小组进行暑期研究，协调所在机构的本科生研究中心，指导研究生和博士后学者，并组织支持来自不同群体的数学家的活动。PI将研究产生当前兴奋的交换代数的三个主题：理想的符号幂及其在高阶多项式插值中的应用，对称理想的同调性质，以及由Hodge-Riemann关系加强的代数Lefschetz性质。理想的符号幂包括在给定的代数簇上消失到更高阶的多项式。该项目将探索符号幂理想的代数性质，这些理想被赋予了编码潜在簇的对称性或其他组合信息的附加结构。进一步的对称理想类的同调性质和计数性质也将被阐明。此外，还将研究分次Artin Gorenstein代数，作为光滑射影代数簇的上同调环的代数模拟。当一个光滑复射影簇的每个上同调环都满足Lefschetz定理和Hodge-Riemann关系时，该项目旨在确定哪些Artin Gorenstein代数满足类似的代数性质。该项目由代数与数论计划和既定的激励竞争研究计划(EPSCoR)联合资助。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 award provides support for research in commutative algebra, with connections to algebraic geometry. Within this framework, commutative algebra investigates systems of polynomial equations whose solutions form geometric objects, such as curves and surfaces. The process of finding a curve or surface passing through a given set of points is commonly referred to as interpolation. Polynomial interpolation finds widespread applications in scientific disciplines such as data analysis, numerical analysis, computer graphics, and mathematical modeling. This project specifically focuses on higher order polynomial interpolation in situations when the underlying data exhibits symmetry. More broadly, it aims to analyze systems of polynomial equations equipped with symmetry using tools from commutative algebra. In addition to these contributions, the principal investigator will lead groups of undergraduate students in summer research, coordinate an undergraduate research hub at their institution, mentor graduate students and postdoctoral scholars, and organize events that support mathematicians from diverse groups.The PI will investigate three topics in commutative algebra generating current excitement: symbolic powers of ideals with applications to higher order polynomial interpolation, homological properties of symmetric ideals, and the algebraic Lefschetz property strengthened by the Hodge-Riemann relations. Symbolic powers of ideals encompass polynomials vanishing to a higher order on a given algebraic variety. The project will explore algebraic properties of symbolic power ideals endowed with additional structure encoding either symmetries of the underlying variety or other combinatorial information. Homological and enumerative properties for further classes of symmetric ideals will also be elucidated. Furthermore, the investigation will turn to graded Artinian Gorenstein algebras, serving as algebraic analogues for the cohomology rings of smooth projective algebraic varieties. While every cohomology ring of a smooth complex projective variety satisfies the Lefschetz theorems and Hodge-Riemann relations, the project aims to identify which Artinian Gorenstein algebras satisfy analogous algebraic properties.This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).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.