Mathematical Sciences: "Uniform Bounds in Noetherian Rings, The Theory of Tight Closure, and Big Cohen-Macaulay Algebras"

数学科学：“诺特环的一致界、紧闭理论和大科恩-麦考利代数”

• 批准号: 9301053
• 负责人: Craig Huneke
• 金额: $ 34.06万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9301053&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Uniform; Bounds; Noetherian

This award supports work on several projects in commutative algebra especially concentrating on two topics: the theory of tight closure and big Cohen-Macaulay algebras, and the existence of uniform bounds in noetherian rings. The first topic involves a continuing joint effort with Melvin Hochster. The principal investigator and Hochster have been developing the theory of tight closure over the last six years and plan on continuing this work. The latter topic is centered around issues arising in the proof of the uniform Artin-Rees property. The principal investigator would also like to continue work on several other topics, notably local cohomology, Rees algebras, computer algebra, and general hyperplane sections of algebraic varieties. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.

该奖项支持在交换代数的几个项目中的工作，尤其是专注于两个主题：紧密封闭和大型科恩·马库拉伊代数的理论，以及在诺埃里戒指中存在统一界限的理论。 第一个主题涉及与梅尔文·霍奇斯特（Melvin Hochster）的持续共同努力。 在过去的六年中，首席研究员和Hochster一直在发展紧密关闭的理论，并计划继续这项工作。 后一个主题围绕着统一的Artin-Rees财产的证明而引起的问题。 首席研究人员还希望继续研究其他几个主题，特别是本地同居，REES代数，计算机代数和代数品种的一般超平面段。 这是在代数几何学领域的重新搜索，这是现代数学的最古老部分之一，但在过去的四分之一世纪中，它具有革命性的开花。 在其起源中，它处理了可以通过最简单的方程（即多项式）在平面中定义的数字。 如今，该领域不仅利用了来自代数的方法，还利用了分析和拓扑的方法，相反，在这些领域以及理论计算机科学和机器人技术中找到了应用。