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

该奖项支持对交换代数中几个项目的工作，特别是集中在两个主题上：紧闭和大Cohen-Macaulay代数理论，以及noether环中一致界的存在性。第一个主题涉及与Melvin Hochster的持续共同努力。首席研究员和Hochster在过去的六年里一直在发展紧密闭合理论，并计划继续这项工作。后一个主题是围绕统一的Artin-Rees性质的证明中出现的问题。首席研究员还希望继续研究其他几个课题，特别是局部上同、Rees代数、计算机代数和代数变种的一般超平面截面。这是代数几何领域的研究，是现代数学中最古老的部分之一，但在过去的25年里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，而且还使用分析和拓扑的方法，相反，它在这些领域以及理论计算机科学和机器人技术中也得到了应用。

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.