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，它为包括代数几何和代数数论在内的广泛数学奠定了基础。交换代数在科学和工程的许多领域都有应用，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。 该项目的部分工作旨在将挂谷针问题（一个有关将针转动一整圈所需空间量的经典分析问题）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设计新算法，用于对一类具有非凡对称性的形状执行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前沿。第一个项目将交换代数与调和分析联系起来。挂谷猜想是调和分析中的一个核心问题，它催生了有限域上的平行文献。该项目旨在将这种平行扩展到 p 进数整数，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同调代数方法。对于嵌入在射影空间以外的事物中的变量，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复曲面几何的同调机制。这个多方面的项目提供了一系列潜在的应用：环面簇上向量束的分裂定理、Hurwitz 空间的有理性证明以及新的层上同调算法。