Multigraded commutative algebra

多级交换代数

基本信息

批准号：
2409776
负责人：
Daniel Erman
金额：
$ 26.5万
依托单位：
University of Hawaii
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-11-15 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2409776&HistoricalAwards=false
关键词：
Multigraded commutative algebra

项目摘要

Polynomial equations are some of the most fundamental objects in mathematics and science, and they arise in a huge array of examples: Fermat’s Last Theorem, physical models of motion, the ideal gas law, and much more. Understanding the solutions of polynomial equations is thus a fundamental challenge that cuts across all quantitative fields. The PI’s work aims to develop techniques—both theoretical and algorithmic—for better understanding the solutions of polynomial equations that are endowed with extra symmetries. As such equations often arise in mathematical or scientific applications, the work has the potential for wide impact.The PI aims to expand the literature on multigraded polynomials, building on his prior work of using virtual resolutions as a foundation for geometric applications of multigraded syzygies. One project would provide novel multigraded analogues of the Hilbert Syzygy Theorem and of Beilinson's resolution of the diagonal, thus establishing two foundational results. A second project on a multigraded version of Green's Linear Syzygy Theorem would have more direct geometric applications. And a third project would have computational applications, yielding a new and potentially much faster algorithm for computing sheaf cohomology on a toric variety. The project will also have broader impacts through the PI’s mentoring of PhD students.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.

多项式方程是数学和科学中最基本的对象，它们出现在大量的例子中：费马大定理、运动物理模型、理想气体定律等等，因此理解多项式方程的解是很重要的。 PI 的工作旨在开发理论和算法技术，以更好地定量理解具有额外对称性的多项式方程的解，因为此类方程经常出现在数学中。 PI 的目标是扩大有关多级多项式的文献，其基础是使用虚拟分辨率作为多级 syzygies 几何应用的基础。一个项目将提供新颖的多级类似物。希尔伯特 Syzygy 定理和 Beilinson 的对角线分辨率，从而建立了两个基础结果关于格林线性 Syzygy 的多级版本的第二个项目。定理将有更直接的几何应用，第三个项目将有计算应用，产生一种新的、可能更快的算法，用于计算环面簇上的层上同调。该项目还将通过 PI 对博士生的指导产生更广泛的影响。授予 NSF 的法定使命，并通过评估反映使用基金会的智力优点和更广泛的影响审查标准，被认为值得支持。