Mathematical Sciences: Studies in Commutative Algebra and its Applications

该奖项侧重于交换代数的理论方面，并试图利用它们来打破计算代数中几个问题的复杂性的僵局。首席研究员将：(a)推广了Koszul同调在Cohen-Macaulay奇点结构理论中的作用，(b)研究了线性代数的结构，(c)研究了各种类型的Rees代数方程，(d)研究了李代数交换变体的不变量和奇点及其表示。(e)开发了一些将图论数据转换为理想论结构的新方法。同时，他将研究最近发现的计算自由基的封闭公式，等维分解和计算无一般投影的归一化的方法，以及伴随的算法，在代数/数值方程求解器的发展中的作用。这个项目是在交换代数和这个领域的计算机辅助计算方面的一般领域。人们对使用计算机来回答代数中的理论问题越来越感兴趣，相反，代数对算法的发展非常有用。本项目认真运用符号计算的理论和实践来解决交换代数中的实际问题。

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.