Mathematical Sciences: Topics in Commutative Algebra and Algebraic Geometry

该奖项支持交换代数和代数几何方面的研究。Roger Wiegand和Sylvia Wiegand将研究Cohen-Macaulay模的直接和分解，特别是在二维的noether环上。他们还将研究二维环的基本理想结构。Brian Harbourne将研究投影平面上处于良好位置的点，其目标是理解完整线性系统在平面上n个点爆炸时整体截面空间维度的行为。Thomas Marley将研究局部环（R,m）的m-初等理想的Rees代数和共轭梯度环的深度性质。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 research in commutative algebra and algebraic geometry. Roger Wiegand and Sylvia Wiegand will study direct-sum decompostions of Cohen-Macaulay modules, particularly over Noetherian rings of dimension two. They will also study the prime ideal structure of two-dimensional rings. Brian Harbourne will investigate points in good position in the projective plane, with a goal of understanding the behavior of the dimension of the space of global sections of complete linear systems on blowings up of the plane at n points. Thomas Marley will study the depth properties of Rees algebras and associative graded rings of m-primary ideals of a local ring (R,m). 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.