Graded Syzygies: Geometry and Computation

分级 Syzygies：几何和计算

• 批准号: 1900792
• 负责人: Jason McCullough
• 金额: $ 15万
• 依托单位: Iowa State University
• 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: 2019-07-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900792&HistoricalAwards=false
• 关键词: Graded; Syzygies; Geometry; Computation

This research focuses on several problems in computational commutative algebra and algebraic geometry. This project is concerned with the study of systems of polynomial equations in several variables. Polynomial equations model many real world phenomena, ranging from robotics motion planning to physics to conservation programs. To each system of polynomial equations, we associate a set of points, called a variety, which correspond to the set of solutions for all the equations. There is a growing body of literature that shows that the computational complexity of the system of polynomial equations has deep connections to the geometry of the associated variety. In nice cases, when the associated variety is smooth, there are well understood bounds limiting the computational complexity. In the worst possible cases, complexity is doubly exponential. Recent work of the PI shows there is a middle ground which is still poorly understood. The goal of this research is to better understand this connection.The research aims to attack several open questions concerning projective bounds on syzygies. One goal is to study Rees-Like Algebras, which were essential in the PI's construction of counterexamples to the Eisenbud-Goto Conjecture (joint with I. Peeva), and to relate their properties to the more well-studied Rees Algebras. A second goal is to provide effective bounds on invariants, such as the Castelnuovo-Mumford regularity or the degrees of individual syzygies, in terms of other invariants. All of the projects have a computational flavor and include writing code for Macaulay2, an NSF-sponsored computer algebra system maintained by Grayson and Stillman. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

这项研究着重于计算交换代数和代数几何的几个问题。该项目涉及多个变量中多项式方程系统的研究。多项式方程模拟了许多现实世界现象，从机器人运动计划到物理学到保护计划。对于每个多项式方程组的每个系统，我们将一组称为一个变体的点相关联，该点对应于所有方程的解决方案集。越来越多的文献表明，多项式方程系统的计算复杂性与相关品种的几何形状有着深厚的联系。在不错的情况下，当相关的品种平滑时，有充分理解的界限限制了计算复杂性。在最坏的情况下，复杂性是双重指数的。 PI的最新工作表明，有一个中间立场仍然知之甚少。这项研究的目的是更好地理解这种联系。该研究旨在攻击有关Syzygies投影范围的几个开放问题。一个目标是研究类似REES的代数，这对于PI对Eisenbud-Goto猜想（与I. Peeva的关节）的构建至关重要，并将其特性与更精心养成的REES代数相关联。第二个目标是根据其他不变式等不变式（例如Castelnuovo-Mumford的规律性或单个Syzygies的程度）提供有效的界限。所有项目都有计算味，包括为Macaulay2编写代码，这是由Grayson和Stillman维护的NSF赞助的计算机代数系统。该项目由代数和数理论计划和启发竞争性研究的既定计划共同资助（EPSCOR）。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的审查标准的评估来获得支持的。

项目成果

The Regularity Conjecture for prime ideals in polynomial rings

• DOI: 10.4171/emss/38
• 发表时间: 2021-01
• 期刊: EMS Surveys in Mathematical Sciences
• 影响因子: 2.3
• 作者: J. McCullough;I. Peeva

On the maximal graded shifts of ideals and modules

论理想与模的最大分级位移

• DOI: 10.1016/j.jalgebra.2018.09.037
• 发表时间: 2021
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: McCullough, Jason

Canonical Modules and Class Groups of Rees-Like Algebras

类Rees代数的规范模和类群

• DOI: 10.1307/mmj/20205974
• 发表时间: 2023
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Mantero, Paolo;McCullough, Jason;Miller, Lance Edward
• 通讯作者: Miller, Lance Edward

G-quadratic, LG-quadratic, and Koszul quotients of exterior algebras

外代数的 G 二次商、LG 二次商和 Koszul 商

• DOI: 10.1080/00927872.2022.2029875
• 发表时间: 2022
• 期刊: Communications in Algebra
• 影响因子: 0.7
• 作者: McCullough, Jason;Mere, Zachary
• 通讯作者: Mere, Zachary

Singularities of Rees-like algebras

类里斯代数的奇点

• DOI: 10.1007/s00209-020-02524-6
• 发表时间: 2020
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Mantero, Paolo;Miller, Lance Edward;McCullough, Jason
• 通讯作者: McCullough, Jason