Mathematical Sciences: A Topological Invariant for Surface Singularities

该奖项支持拓扑不变量的初步研究，-P.P，与任何复杂的正常表面奇异的胚芽。 我们的目标是开发技术，既找到一个体积形式的链接的表面奇点，并制定封闭的表达式-P. P.的主要研究人员将首先确定的行为-P.P下拼接和有限分歧的地图。 这是代数几何领域的研究，它是现代数学中最古老的部分之一，但在过去的四分之一世纪里，它已经有了革命性的发展。 在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。 如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，并且反过来在这些领域以及理论计算机科学和机器人学中找到应用。

This award supports preliminary research on a topological invariant, -P.P, associated to the germ of any complex normal surface singularity. The goal is to develop techniques for both finding a volume form on the link of a surface singularity, and for formulating closed expressions for -P.P. The principal investigator will initially work on determining the behavior of -P.P under splicing and finite ramified maps. This is research 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.