Applications of Representation Theory in Commutative Algebra

这个项目涉及对代数几个分支的问题的研究。交换代数研究由多项式方程定义的集合。项目的第一部分和第二部分致力于研究多项式方程定义几何特征的某些集合。这个项目的第三部分将从几何的角度系统地研究某些类型的图。这个项目由几个相互关联的部分组成。第一部分是研究有限自由分辨率下的一般环的结构。首席研究员建议发展他的发现，即长度为3的一般环与t形图相关的Kac-Moody李代数之间的联系。第二部分是局部上同的计算。PI计划发展他对某些交换环的局部上同调的计算。第三部分涉及其他几个关于非交换代数的项目。PI将研究为调制颤振定义的图像群和具有关系的颤振的表示空间的分量的几何。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。

This project involves the study of problems from several branches of algebra. Commutative algebra studies sets defined by polynomial equations. The first and second part of the project are devoted to studying polynomial equations defining certain sets characterized geometrically. The third part of this project will be a systematic study of certain kinds of graphs from a geometric point of view.This project consists of several interrelated parts. The first part is to study the structure of generic rings for finite free resolutions. The principal investigator proposes to develop his discovery of a link between the generic ring for resolutions of length 3 and Kac-Moody Lie algebras related to T-shaped graphs. The second part is related to calculating local cohomology. The PI plans to develop his calculation of local cohomology for certain classes of commutative rings. The third part involves several other projects on noncommutative algebra. The PI will study picture groups defined for modulated quivers and the geometry of the components of representation spaces for quivers with relations.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.