Minimal Free Resolutions and Syzygies

最小的自由分辨率和 Syzygies

• 批准号: 2001064
• 负责人: Irena Peeva
• 金额: $ 27.23万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001064&HistoricalAwards=false
• 关键词: Minimal; Free; Resolutions; Syzygies

A core goal in the mathematical areas Algebraic Geometry and Commutative Algebra 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. The solution set forms a geometric object. The main idea is to study the rich and beautiful interplay between its geometric and algebraic properties. Closely related to this is the concept of a free resolution, which was introduced by the famous mathematician David Hilbert in two papers in 1890 and 1893. Constructing a free resolution amounts to repeatedly solving systems of polynomial equations. 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 and string theory. Recent computational methods have made it possible to compute some free resolutions by computer. The main research goal in this project is to make significant progress in understanding the structure of free resolutions and their numerical invariants. The main research topics are:(1) The structure of minimal free resolutions of binomial edge ideals with a quadratic initial ideal in a polynomial ring. The project is inspired by a conjecture of Ene, Herzog, and Hibi, that the graded Betti numbers of such an ideal coincide with the graded Betti numbers of its quadratic initial ideal (with respect to the lex order). (2) The structure of infinite minimal free resolutions of lexicographic ideals in Clements-Lindstrom quotient rings, which will be studied using the methods recently introduced by Eisenbud and Peeva in their research monograph "Minimal free resolutions over complete intersections”.(3) The PI plans to explore a new area of research: minimal free resolutions of binomial edge ideals over exterior algebras. The PI plans to train graduate students in her research field.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.

代数几何和交换代数数学领域的核心目标是理解多项式方程系统的解，可能涉及大量变量和大量方程。解集形成一个几何对象。其主要思想是研究其几何和代数性质之间丰富而美丽的相互作用。与此密切相关的是自由分辨率的概念，它是由著名数学家大卫·希尔伯特在1890年和1893年的两篇论文中提出的。构造一个自由分辨率相当于反复求解多项式方程组。对这些物体的研究在20世纪下半叶蓬勃发展，最近取得了惊人的进展。该领域非常广泛，与其他数学领域和弦理论有很强的联系和应用。最近的计算方法使得用计算机计算一些自由分辨率成为可能。本课题的主要研究目标是在理解自由分辨率的结构及其数值不变量方面取得重大进展。主要研究内容有：(1)多项式环中具有二次初始理想的二项式边理想的最小自由分辨率结构。该项目的灵感来自Ene， Herzog和Hibi的一个猜想，即这种理想的分级Betti数与其二次初始理想的分级Betti数重合（相对于lex阶）。(2) Clements-Lindstrom商环中字典理想的无限最小自由分辨率的结构，将使用Eisenbud和Peeva在其研究专著《完全相交上的最小自由分辨率》中最近介绍的方法进行研究。(3) PI计划探索一个新的研究领域：对外部代数的二项式边理想的最小自由分辨率。PI计划在她的研究领域培养研究生。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。