Mathematical Sciences: Commutative Algebra

主要研究人员将继续研究分次环和局部环及其奇点上的交换代数。主题包括希尔伯特函数，希尔伯特多项式，双数扩张和形式纤维。这项工作在代数几何中有具体的应用，特别是在奇点分解问题上。这是对代数几何领域的研究，代数几何是现代数学中最古老的部分之一，但在过去的25年里，它取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，这个领域不仅使用代数的方法，而且使用分析和拓扑学的方法，相反，在这些领域以及理论计算机科学和机器人学中找到了应用。

The principal investigator will pursue research in commutative algebra on graded rings and local rings and their singularities. Topics include Hilbert functions, Hilbert polynomials, birational extensions and formal fibers. The work has specific applications to algebraic geometry, particularly to the problem of resolution of singularities. 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.