Mathematical Sciences: Geometry of Singular Spaces

代数簇（集合）的拓扑性质 一些多项式方程的解决方案）将在两个研究 方法，使用对他们来说特别的结构。 一是 是关于物体光滑部分的大小和距离的概念， 代数簇，继承自其嵌入空间。 使用 这定义了平方可和函数，更一般地， 在品种的光滑部分上的微分形式。 第一 要研究的问题是，这些周期在多大程度上 函数或形式由拓扑性质决定 的品种，特别是由拓扑不变，称为 “交集同调群。“第二个问题是找到 如何将量化措施的交叉点， 循环的品种与其局部拓扑性质。 这些几何调查将不限于 非奇异的情况下，使他们更困难，但更多 适用于理论上产生的空间 物理学和其他分析理论。

The topological properties of an algebraic variety (the set of solutions of some polynomial equations) will be studied in two ways, using structures which are special to them. First, there are notions of size and distance on the smooth part of an algebraic variety, inherited from its imbedding in space. Using this, one defines square-summable functions and, more generally, differential forms on the smooth part of the variety. The first problem to be studied is to what extent the periods of these functions or forms are determined by the topological properties of the variety, in particular, by a topological invariant called "the intersection homology group." The second problem is to find ways to relate quantitative measures of the intersection of cycles on the variety with its local topological properties. These geometrical investigations will not be confined to the non-singular case, rendering them more difficult, yet more suitable of application to spaces which arise in theoretical physics and in other analytic theories.