Employing Symmetry in Commutative Algebra

在交换代数中运用对称性

• 批准号: 1600765
• 负责人: Claudiu Raicu
• 金额: $ 15.9万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600765&HistoricalAwards=false
• 关键词: Employing; Symmetry; Commutative; Algebra

The project pertains primarily to the field of commutative algebra and is concerned with the study of systems of polynomial equations: these equations should be thought of as the algebraic relationships between the set of parameters in some model. Oftentimes mathematical models of natural phenomena come equipped with a large group of symmetries, and the main goal of the current project is to develop techniques that employ the symmetry in order to reveal both qualitative and quantitative features of the models. The questions and the methods are strongly influenced by classical algebraic geometry, which analyzes the models from a geometric perspective, by representation theory, which is an algebraic study of symmetry, and by D-module theory, which is an algebraic perspective on the infinitesimal study of the models. Computer experimentation is important for the success of the project. The investigator, together with students and collaborators, will develop and disseminate the necessary computer algebra software and will make available the results of experimentation that are of general interest.This research project investigates a number of open questions concerning fundamental invariants associated to algebraic varieties endowed with a large group of symmetries, by analyzing the implications of the symmetries to the structure of the invariants. The main emphasis will be on the study of syzygies and local cohomology groups, with a view towards understanding further invariants such as regularity and projective dimension, cohomological dimension and arithmetic rank, or Lyubeznik numbers and Bernstein-Sato polynomials. The project will combine methods from commutative algebra and classical algebraic geometry, as well as techniques from representation theory and the theory of D-modules. For instance, the investigator will explore the D-module structure of the local cohomology groups associated to natural stratifications of the spaces of binary forms. On a separate note, the investigator will research the connections between syzygies of ideals in rings of polynomial functions on spaces of matrices, and representations of general linear Lie superalgebras.

该项目主要与交换代数有关，并且与多项式方程系统的研究有关：这些方程应被视为某些模型中一组参数之间的代数关系。通常，自然现象的数学模型配备了大量的对称性，当前项目的主要目标是开发采用对称性的技术，以揭示模型的定性和定量特征。这些问题和方法受经典代数几何形状的强烈影响，该几何形状从几何学的角度分析了模型，它是对对称性的代数研究和D模块理论的代数研究，该理论是对模型无限研究的代数观点。计算机实验对于项目的成功很重要。研究人员与学生和合作者一起将开发和传播必要的计算机代数软件，并将提供一般感兴趣的实验结果。本研究项目调查了许多与代数相关的基本不变性的开放问题，通过与一系列Symmeties相关的基本变种，通过分析组成型的组合构成了该组件的构成，从而构成了符号的组合。主要的重点将是对Syzygies和本地共同体学组的研究，以期了解进一步的不变性，例如规律性和投影维度，共同体学维度和算术等级，或Lyubeznik数字和Bernstein-Sato-Sato多项式。该项目将结合换向代数和经典代数几何形状的方法，以及代表理论和D模块理论的技术。例如，研究者将探索与二进制形式空间自然分层相关的局部共同体学组的D模块结构。另外，研究者将研究矩阵空间多项式函数环中理想环的共同关系之间的连接，以及一般线性谎言超级级别的表示。