Mathematical Sciences: A Topological Invariant for Surface Singularities

该奖项支持对拓扑不变量-P.P的初步研究，该不变量与任何复法向曲面奇点的萌芽有关。我们的目标是开发出在表面奇点的链接上寻找体积形式的技术，以及建立-P.P.的闭合表达式的技术。主要研究人员最初将致力于确定-P.P在剪接和有限分支映射下的行为。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，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及理论计算机科学和机器人学中也找到了应用。

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.