Studies on Local Cohomology, Derivations, Integral Dependence, and Blowup Algebras

局部上同调、导数、积分相关性和爆炸代数的研究

基本信息

批准号：
1902033
负责人：
Claudia Polini
金额：
$ 15万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902033&HistoricalAwards=false
关键词：
Studies Local Cohomology Derivations Integral

项目摘要

This project is in commutative algebra, but has also been inspired by algebraic geometry. Systems of polynomial equations in several variables occur in many applications in science and technology, for instance in engineering, computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. Commutative algebra and algebraic geometry are concerned with the qualitative study of such systems of polynomial equations. This project focuses on the algebraic approach. One of the goals of the project is to understand the vectors that are tangent to solution sets of systems of polynomial equations. Another objective is to find systems of polynomial equations describing a given geometric object. The latter has applications to computer graphics, algebraic statistics, and rigidity of structures. The PI intends to involve undergraduate students, graduate students, and postdoctoral fellows in her research. She will continue to organize national and international meetings. Throughout her activities, the PI will continue to promote underrepresented groups in mathematics. The first objective of this research is to relate degrees of vector fields on projective space to invariants of curves, or varieties, that they leave invariant. This is a difficult problem that has been studied for over a century, mainly from the point of view of complex analysis, dynamical systems, algebraic and differential geometry. The PI will investigate this question using tools from commutative algebra. The second objective is to find criteria for a variety in projective space to be a set-theoretic complete intersection. When there is only one non vanishing local cohomology module the PI believes that the property of being a set-theoretic complete intersection is encoded in the structure of this module. The third objective is to prove a numerical characterization of integral dependence of modules using a notion of multiplicity that arises in intersection theory. Theorems of this kind have a bearing on equisingularity theory, in fact they lead to fiber-wise numerical conditions for a family of analytic spaces to be Whitney equisingular, hence topologically trivial. The last objective is to study the implicit equations defining the graph and the image of rational maps between projective spaces.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.

该项目处于交换代数，但也受到代数几何形状的启发。多个变量中的多项式方程系统在科学和技术的许多应用中都存在，例如在工程，计算机科学，加密图，编码理论，机器人技术，模式识别和理论物理学中。交换代数和代数几何形状与此类多项式方程系统的定性研究有关。该项目侧重于代数方法。该项目的目标之一是了解与多项式方程系统的解决方案集相切的向量。另一个目的是找到描述给定几何对象的多项式方程系统。后者在计算机图形，代数统计和结构的刚度上有应用。 PI打算让本科生，研究生和博士后研究员参与她的研究。她将继续组织国家和国际会议。在她的整个活动中，PI将继续促进数学中代表性不足的群体。这项研究的第一个目的是将投影空间上的向量领域的程度与它们留下不变的曲线或品种的不变性相关联。这是一个多世纪的困难问题，主要是从复杂分析，动力学系统，代数和差异几何形状的角度来看。 PI将使用交换代数的工具调查此问题。第二个目标是找到射影空间中各种各样的标准，成为一个固定的理论完整交集。当只有一个非消失的局部共同体模块时，PI认为作为设定理论完整交叉点的属性是在该模块的结构中编码的。第三个目标是证明使用相交理论中出现的多样性概念对模块的积分依赖性的数值表征。这种定理与方程式理论有关，实际上它们导致了纤维的数值条件，使一个分析空间家族成为惠特尼式式，因此在拓扑上是微不足道的。最后一个目标是研究定义图形图的隐式方程和投射空间之间理性图的形象。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来提供支持。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Multiplicity sequence and integral dependence

多重序列和积分依赖性

DOI：
10.1007/s00208-020-02059-5
发表时间：
2020
期刊：
Mathematische Annalen
影响因子：
1.4
作者：
Polini, Claudia;Trung, Ngo Viet;Ulrich, Bernd;Validashti, Javid
通讯作者：
Validashti, Javid

Degree bounds for local cohomology

局部上同调的度界

DOI：
10.1112/plms.12364
发表时间：
2020
期刊：
Proceedings of the London Mathematical Society
影响因子：
1.8
作者：
Kustin, Andrew R.;Polini, Claudia;Ulrich, Bernd
通讯作者：
Ulrich, Bernd

Relations between the 2 × 2 minors of a generic matrix

泛型矩阵的 2××2 个次数之间的关系

DOI：
10.1016/j.aim.2021.107807
发表时间：
2021
期刊：
Advances in Mathematics
影响因子：
1.7
作者：
Huang, Hang;Perlman, Michael;Polini, Claudia;Raicu, Claudiu;Sammartano, Alessio
通讯作者：
Sammartano, Alessio

Simple D -module components of local cohomology modules

局部上同调模的简单 D 模分量