Mathematical Sciences: A Topological Invariant for Surface Singularities

该奖项支持对拓扑不变量p的初步研究。P，与任何复杂法向曲面奇点的胚芽有关。目标是开发在表面奇点链接上寻找体积形式的技术，以及制定- p.p.的封闭表达式。首席研究员将首先确定-P的行为。P在拼接和有限分支映射下。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，但在过去的四分之一个世纪里，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 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.