Mathematical Sciences: Studies in Commutative Algebra and its Applications

该奖项侧重于理论方面的交换 代数，并试图利用它们来打破僵局， 计算代数的几个问题的复杂性。 的 首席研究员将：（a）扩大科苏尔的作用 Cohen-Macaulay奇点结构理论中的同调， (b)研究线性型代数的结构，（c） 检查各种Rees代数的方程，（d） 研究交换变量的不变量和奇异性 李代数及其表示，（e）利用一些 一种将图论数据转换为 理想理论结构 与此同时，他将调查 最近发现的计算封闭公式的作用 根式、等维分解和计算方法 规范化，不含通用投影， 伴随的算法，在开发 代数/数值方程求解器。 这个项目是在交换代数的一般领域 以及这个领域的计算机辅助计算方面。 人们对使用计算机回答问题的兴趣越来越大 代数中的理论问题，反过来说，代数是相当 这对算法的开发很有用。 该项目使 认真运用象征主义的理论和实践 计算来解决交换代数中的真实的问题。

This award focuses on theoretical aspects of commutative algebra and seeks to make use of them to break the logjam of complexity in several problems of computational algebra. The principal investigator will: (a) extend the role of Koszul homology in the structure theory of Cohen-Macaulay singularities, (b) investigate the structure of algebras of linear type, (c) examine the equations of Rees algebras of various kinds, (d) study the invariants and singularities of commuting varieties of Lie algebras and their representations, and (e) exploit some novel means of converting graph-theoretic data into ideal-theoretic structures. Concomitantly, he will investigate roles for the recent discovery of closed formulas for computing radicals, equi-dimensional decompositions and a method to compute normalization that is free of generic projections, with accompanying algorithms, in the development of algebraic/numerical equation solvers. This project is in the general area of commutative algebra and the computer-aided computational aspects of this field. There is a growing interest in using computers to answer theoretical questions in algebra and conversely, algebra is quite useful for the development of algorithms. This project makes serious use of both the theory and practice of symbolic calculations to solve real problems in commutative algebra.