Symbolic Powers and Lefschetz Properties: Geometric and Homological Aspects

符号幂和 Lefschetz 性质:几何和同调方面

基本信息

  • 批准号:
    2101225
  • 负责人:
  • 金额:
    $ 22.05万
  • 依托单位:
  • 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

项目摘要

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.
本研究关注的问题,交换代数的代数几何动机。在一系列广泛的科学工作的核心是无处不在的需要解决多项式方程。一个互补的目标是找到多项式方程,例如,一个方程的图形通过一组给定的数据点。这个过程被称为多项式插值,是数据科学,数值分析和代数几何界面上的一个基本挑战。调查员将带来的方法,从交换和计算代数承担方面的高阶版本的多项式插值。例如,将阐明数据点表现出固有对称性的情况。此外,更深入地了解这个主题和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。该研究项目集中在两个主题,产生当前的兴奋:多项式插值和代数Lefschetz属性。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代数簇上消失到一定阶的多项式集合。后者的主题构成了一个代数抽象的硬莱夫谢茨定理与壮观的应用到几个领域的数学。这两个主题之间的相互关系将得到深入探讨和利用。一个特别的调查方向是对应用程序的代数莱夫谢茨性质同调代数,特别是分级自由决议。其他方向包括应用程序的包容性问题有关的普通和象征性的拓扑结构定义的理想。这项工作的各个方面展示了反射群理论、超平面安排、凸几何和微分分级代数的关系。该奖项反映了NSF的法定使命,并通过使用基金会的智力价值和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(11)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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:
  • 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
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:
  • 影响因子:
    1
  • 作者:
    Iarrobino, Anthony;Macias Marques, Pedro;McDaniel, Chris;Seceleanu, Alexandra;Watanabe, Junzo
  • 通讯作者:
    Watanabe, Junzo
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:
  • 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
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:
  • 影响因子:
    0
  • 作者:
    A. Iarrobino;Chris McDaniel;A. Seceleanu
  • 通讯作者:
    A. Iarrobino;Chris McDaniel;A. Seceleanu
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:
  • 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
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Alexandra Seceleanu其他文献

The projective dimension of codimension two algebras presented by quadrics
  • DOI:
    10.1016/j.jalgebra.2013.06.038
  • 发表时间:
    2013-11-01
  • 期刊:
  • 影响因子:
  • 作者:
    Craig Huneke;Paolo Mantero;Jason McCullough;Alexandra Seceleanu
  • 通讯作者:
    Alexandra Seceleanu

Alexandra Seceleanu的其他文献

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{{ truncateString('Alexandra Seceleanu', 18)}}的其他基金

Polynomial Interpolation, Symmetric Ideals, and Lefschetz Properties
多项式插值、对称理想和 Lefschetz 属性
  • 批准号:
    2401482
  • 财政年份:
    2024
  • 资助金额:
    $ 22.05万
  • 项目类别:
    Continuing Grant
Conference: Women in Commutative Algebra II
会议:交换代数中的女性 II
  • 批准号:
    2324929
  • 财政年份:
    2023
  • 资助金额:
    $ 22.05万
  • 项目类别:
    Standard Grant
Conference on Unexpected and Asymptotic Properties of Projective Varieties
射影簇的意外和渐近性质会议
  • 批准号:
    1953096
  • 财政年份:
    2020
  • 资助金额:
    $ 22.05万
  • 项目类别:
    Standard Grant
Collaborative Proposal: Central States Mathematics Undergraduate Research Conferences
合作提案:中部各州数学本科生研究会议
  • 批准号:
    1811000
  • 财政年份:
    2018
  • 资助金额:
    $ 22.05万
  • 项目类别:
    Standard Grant
Symbolic Powers, Configurations of Linear Spaces, and Applications
符号幂、线性空间的配置及应用
  • 批准号:
    1601024
  • 财政年份:
    2016
  • 资助金额:
    $ 22.05万
  • 项目类别:
    Standard Grant

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