Symbolic Powers and Lefschetz Properties: Geometric and Homological Aspects

符号幂和 Lefschetz 性质：几何和同调方面

• 批准号: 2101225
• 负责人: Alexandra Seceleanu
• 金额: $ 22.05万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-15 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101225&HistoricalAwards=false
• 关键词: Symbolic; Powers; Lefschetz; Properties; Geometric

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.

这项研究涉及由代数几何形状激发的交换代数问题。一系列科学努力的核心是解决多项式方程的无处不在。一个互补的目标是找到多项式方程，例如，图形通过给定的一组数据点传递的方程式。该过程称为多项式插值，是数据科学，数值分析和代数几何界面的基本挑战。研究者将带来从交通证和计算代数的方法，以在高阶版本的多项式插值方面承担。例如，将阐明数据点表现出内在对称性的情况。此外，将追求对该主题与同源代数中新兴技术之间的相互作用的更深入的了解。这项基本研究的更广泛的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。该研究项目的重点是引起当前兴奋的两个主题：多项式插值和代数Lefschetz属性。前一种主题将通过符号能力理想的角度进行分析，可以将其视为在给定代数品种上消失的多项式的一组。后一个主题构成了硬Lefschetz定理的代数抽象，并在数学的几个领域中应用了壮观的应用。这两个主题之间的相互关系将得到彻底探索和利用。研究的一个特殊方向是在代数Lefschetz特性与同源代数的应用，特别是针对分级的自由分辨率。其他方向包括针对由理想定义的普通和符号拓扑的遏制问题的应用。这项工作的各个方面与反思群体，超平面布置，凸几何和分级代数的理论展示了关系。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准来通过评估来进行评估的。

项目成果

Convex bodies and asymptotic invariants for powers of monomial ideals

单项式理想幂的凸体和渐近不变量

• DOI: 10.1016/j.jpaa.2022.107089
• 发表时间: 2022
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Camarneiro, João;Drabkin, Ben;Fragoso, Duarte;Frendreiss, William;Hoffman, Daniel;Seceleanu, Alexandra;Tang, Tingting;Yang, Sewon
• 通讯作者: Yang, Sewon

Computing rational powers of monomial ideals

计算单项式理想的有理幂

• DOI: 10.1016/j.jsc.2022.08.018
• 发表时间: 2023
• 期刊: Journal of Symbolic Computation
• 影响因子: 0.7
• 作者: Dongre, Pratik;Drabkin, Benjamin;Lim, Josiah;Partida, Ethan;Roy, Ethan;Ruff, Dylan;Seceleanu, Alexandra;Tang, Tingting
• 通讯作者: Tang, Tingting

Cohomological Blowups of Graded Artinian Gorenstein Algebras along Surjective Maps

分级 Artinian Gorenstein 代数沿满射映射的上同调放大

• DOI: 10.1093/imrn/rnac002
• 发表时间: 2022
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Iarrobino, Anthony;Macias Marques, Pedro;McDaniel, Chris;Seceleanu, Alexandra;Watanabe, Junzo
• 通讯作者: Watanabe, Junzo

Axial constants and sectional regularity of homogeneous ideals

齐次理想的轴向常数和截面正则性

• DOI: 10.1090/proc/16244
• 发表时间: 2023
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: DeBellevue, Michael;Lebovitz, Audric;Li, Yik;Lotfi, Mohamed;Mohite, Shivam;Pan, Xin;Pathak, Mrigank;Roshan Zamir, Shah;Seceleanu, Alexandra;Zhang, Sindy
• 通讯作者: Zhang, Sindy

Connected Sums of Graded Artinian Gorenstein Algebras and Lefschetz Properties

• DOI: 10.1016/j.jpaa.2021.106787
• 发表时间: 2019-04
• 期刊: arXiv: Commutative Algebra
• 作者: A. Iarrobino;Chris McDaniel;A. Seceleanu