Free Resolutions

免费决议

• 批准号: 2401238
• 负责人: Irena Peeva
• 金额: $ 35万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2028-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401238&HistoricalAwards=false
• 关键词: Free; Resolutions

This project concerns research in Commutative Algebra. A core goal in the subject deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. Closely related to this is the concept of a free resolution. Constructing such a resolution amounts to repeatedly solving systems of polynomial equations. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra. The idea of constructing free resolutions was introduced by Hilbert in a famous paper in 1890. The study of these objects flourished in the second half of the twentieth century and has seen spectacular progress recently. The field is very broad, with strong connections and applications to other mathematical areas. The broader impacts of the project include the writing of an expository paper, organization of professional development workshops for undergraduate students, and organization of mathematical conferences.The main research goal in this project is to make significant progress in understanding the structure of minimal free resolutions and their numerical invariants. In particular, the PI will: work jointly with M. Mastroeni and J. McCullough on Koszul Algebras; continue work on minimal free resolutions of binomial edge ideals; study the asymptotic structure of minimal free resolutions over exterior 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.

该项目涉及对代数的研究。该主题的核心目标涉及理解多项式方程系统的解决方案，可能在大量变量和大量方程式中。与此密切相关的是免费分辨率的概念。构建这样的分辨率等于反复求解多项式方程的系统。多年来，最少的自由决议一直是中心对象和富有成果的代数工具。 希尔伯特（Hilbert）在1890年的一篇著名论文中引入了建立自由决议的想法。对这些物体的研究在20世纪下半叶蓬勃发展，最近取得了壮观的进步。该领域非常广泛，与其他数学领域具有牢固的联系和应用。该项目的更广泛影响包括撰写说明性论文，针对本科生的专业发展研讨会的组织以及数学会议的组织。该项目的主要研究目标是在理解最小的免费分辨率及其数字不变的结构方面取得了重大进展。特别是，PI将：与M. Mastroeni和J. McCullough共同合作在Koszul代数上； 继续处理最小的二项式边缘理想的自由分辨率；研究最低限度的自由决议的渐近结构对外部代数。该奖项反映了NSF的法定任务，并且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支持。