Mathematical Sciences: Studies in Commutative Algebra and its Applications

该奖项专注于交换代数的理论方面，并试图利用它们来打破计算代数几个问题中复杂性的僵局。主要的研究人员将：(A)推广Koszul同调在Cohen-Macaulay奇点结构理论中的作用，(B)研究线性类型代数的结构，(C)研究各种类型的Rees代数的方程，(D)研究交换李代数的不变量和奇点及其表示，以及(E)开发一些将图论数据转换为理想理论结构的新方法。同时，他将调查最近发现的用于计算根的封闭公式、等维分解和一种不需要通用投影的归一化计算方法以及伴随的算法在开发代数/数值方程解算器中所起的作用。这个项目是在交换代数和该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。这个项目认真地利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.