Mathematical Sciences: "Fundamental Groups Of Quasiprojective Varieties"

该奖项支持对基本组的工作， 复域上的光滑拟投射簇 号码 如果这样一个簇恰当地映射到一个双曲 曲线，其基本群覆盖到非交换自由 组 首席研究员将决定在多大程度上 匡威亦然。 这个问题将受到各种各样的攻击， 方法. 其中包括对Castenuovo-De Franchis引理，L2调和1-形式的构造 普遍覆盖，以及对某些同调定义的 的一维字符组的子集 基本群 这在代数几何领域是合理的， 现代数学中最古老的部分，但其中有一个 在过去的四分之一个世纪里，革命性的开花。 在其 起源，它处理的数字，可以定义在平面上， 最简单的方程，即多项式。 如今，该领域 不仅使用代数方法，而且使用分析方法， 拓扑学，反过来说， 以及理论计算机科学和机器人技术。

This award supports work on the fundamental group of a smooth quasiprojective variety defined over the field of complex numbers. If such a variety maps properly onto a hyperbolic curve, its fundamental group surjects onto a nonabelian free group. The principal investigator will determine to what extent the converse holds. The problem will be attacked by a variety of methods. These include a generalization of the Castenuovo-De Franchis lemma, the construction of L2 harmonic 1-forms on the universal cover, and a study of certain homologically defined subsets of the groups of 1-dimensional characters of the fundamental group. 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.