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.

该研究项目涉及交换代数，该代数为广泛的数学奠定了基础，包括代数几何和代数数理论。交换代数在许多科学和工程领域都发现了应用，包括计算机科学，密码学，编码理论，机器人技术，模式识别和理论物理学。 该项目的一部分工作旨在将Kakeya针问题连接起来，这是一个经典的分析问题，讲述了在整个圆圈中旋转针的空间量，并具有来自通勤代数的现代思想。该项目的另一部分将使用交换代数来设计新算法，用于执行有关具有非凡对称性的一类形状的几何计算，以及其他应用。这项研究将在交换代数和其他领域之间建立新的边界。第一个项目通过谐波分析将交换代数连接起来。 Kakeya的猜想是谐波分析中的一个核心问题，它在有限领域催生了平行文献。该项目旨在扩展到P-ADIC整数的平行，从而与原始分析问题产生更紧密的联系。第二个项目为新的几何设置开发了同源代数方法。对于嵌入在投影空间以外的其他物质中，免费分辨率通常无法与几何形状提供尖锐的连接；该项目将开发出更适合感谢您的几何形状的同源机械。这个多方面的项目提供了一系列潜在的应用程序：在曲线品种上划分的矢量束，Hurwitz空间的合理性证明以及新的捆式同学算法。