Asymptotic Commutative Algebra and Multigraded Syzygies

渐近交换代数和多级 Syzygies

• 批准号: 1601619
• 负责人: Daniel Erman
• 金额: $ 20.46万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601619&HistoricalAwards=false
• 关键词: Asymptotic; Commutative; Algebra; Multigraded; Syzygies

This research project concerns commutative algebra, which provides the foundation for a broad range of mathematics, including algebraic geometry and algebraic number theory. Commutative algebra finds application in many fields of science and engineering, including computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. Part of the work in this project aims to connect the Kakeya needle problem, a classical analysis problem about the amount of space needed to turn a needle in a full circle, with modern ideas from commutative algebra. Another part of the project will use commutative algebra to design new algorithms for performing geometric computations about a class of shapes with extraordinary symmetries, among other applications. This research will build new frontiers between commutative algebra and other fields. The first project connects commutative algebra with harmonic analysis. The Kakeya conjecture is a central problem in harmonic analysis that has spawned a parallel literature over finite fields. The project aims to expand this parallel to the p-adic integers, yielding closer connections with the original analytic questions. The second project develops homological algebra methods for new geometric settings. For a variety embedded in something other than projective space, free resolutions often fail to provide sharp connections with geometry; this project will develop homological machinery better suited to toric geometry. This multifaceted project offers an array of potential applications: splitting theorems for vector bundles on toric varieties, rationality proofs for Hurwitz spaces, and new sheaf cohomology algorithms.

这个研究项目涉及交换代数，它为广泛的数学，包括代数几何和代数数论提供了基础。交换代数在许多科学和工程领域都有应用，包括计算机科学、密码学、编码理论、机器人、模式识别和理论物理。该项目的部分工作旨在将Kakeya针问题（一个经典的分析问题，关于将针转一圈所需的空间量）与交换代数的现代思想联系起来。该项目的另一部分将使用交换代数来设计新的算法，用于对一类具有非凡对称性的形状进行几何计算，以及其他应用。这项研究将为交换代数与其他领域的研究开辟新的前沿。第一个项目将交换代数与调和分析联系起来。Kakeya猜想是谐波分析中的一个中心问题，在有限域上产生了平行的文献。该项目旨在将这种并行扩展到p进整数，从而与原始分析问题产生更紧密的联系。第二个项目开发新的几何设置的同调代数方法。对于嵌入到射影空间之外的事物，自由分辨率通常无法提供与几何的清晰联系；该项目将开发更适合于环形几何的同构机械。这个多方面的项目提供了一系列潜在的应用：环上矢量束的分裂定理，Hurwitz空间的合理性证明，以及新的束上同调算法。