Geometry of Singularities

Gaffney研究了解析集和映射的局部几何。他的目标是从定义映射的集合或组成部分的方程中找到可计算的数字，这些映射将描述所研究对象的几何形状。第一步是找到这种类型的数，其在一组对象中的常数意味着该族的几何形状以某种明确定义的方式是常数。加夫尼认为最重要的条件之一是惠特尼等奇异性。这个条件意味着家族的嵌入式拓扑结构是恒定的。模的积分闭包理论为研究这一条件提供了一个强有力的框架。通过与Kleiman的合作，Gaffney将解决具有非孤立奇点的完全交叉口族的这个问题。通过适当的修改，将Gaffney在具有非孤立奇点的超曲面中使用的思想转移到与Kleiman一起开发的完全相交的框架中来实现这一点。Gaffney还将研究超曲面奇点的几何形状与与这些奇点相关的例外除数中出现的分量之间的关系。要考虑的一个具体情况是有限确定的映射芽和靠近好维边界的拓扑稳定的映射芽的判别式。这项研究将是朝着惠特尼等奇异地图的家庭标准迈出的一步。这个项目是几个世纪以来探索映射几何和奇异形状的努力的一部分。（映射是形状之间的关系。）圆锥是具有奇点的形状的一个简单例子，而圆锥的图像是映射的一个简单例子。这幅图在圆锥体的点和平面的点之间建立了一种关系，在这种情况下，这幅图的平面。如果我们拍一张即使是光滑形状的照片，图像也会有边缘，而边缘也经常会有奇异点，所以对形状和映射的研究是相关的。方程描述形状。本研究的目标是从一个形状的方程开始，并从中提取数字，这些数字可以描述形状的几何形状，用于确定形状族中的一个成员何时与其他成员不同。例如，如果你把一根绳子穿过它自己，它就会形成一个环。当你把这个环拉紧，它会形成一系列曲线，而这个环会在t时刻消失，我们的直觉告诉我们这一系列的曲线是相似的，直到t时刻，当这个环消失的时候。我们可以写出这个家族成员的方程。用两个尺度来研究这种情况是有帮助的。在宏观层面上，单个曲线弯曲形成环路；在无限小层面上，曲线的切向量和曲线的并集存在。当你沿着一个环移动时，你可以沿着切向量移动。值得注意的是，尽管在宏观层面上，随着时间趋于t，这个循环消失了，但在无穷小的层面上，这个循环却留下了痕迹。这种轨迹出现在不变量中，称为米尔诺数，我们用它来研究平面上的曲线，它可以从曲线方程中计算出来。模的整闭包理论是研究这个无穷小水平上的集合和从集合族中提取检测这个水平上的变化的数的一个强有力的框架。由于在复空间中由解析函数定义的形状在代数和几何之间有很好的联系，这是迄今为止使用这种方法的大多数工作的背景。由n-d方程定义的复n空间的d维子集称为完全交集。Gaffney和Kleiman将结合他们自己和其他人早期的方法来处理更有限的情况，用任意维的奇异集来处理完全交集的族。＊＊＊

9803691 Gaffney Gaffney studies the local geometry of analytic sets and mappings. His goal is to find numbers computable from the equations defining the sets or components of the mappings that will describe the geometry of the object under study. A first step is to find numbers of this type whose constancy in a family of objects implies that the geometry of the family is constant in some well-defined way. One of the most important conditions that Gaffney considers is Whitney equisingularity. This condition implies that the embedded topology of the family is constant. The theory of the integral closure of modules provides a powerful framework for studying this condition. Working with Kleiman, Gaffney will solve this problem for families of complete intersections with non-isolated singularities. This will be done by transporting, with suitable modifications, the ideas Gaffney used in the case of hypersurfaces with non-isolated singularities into the framework of complete intersections developed with Kleiman. Gaffney will also investigate the relation between the geometry of hypersurface singularities and the components appearing in the exceptional divisor associated to these singularities. One specific case to be considered will be discriminants of finitely determined map germs and topologically stable map germs close to the boundary of the nice dimensions. This study will be a step toward criteria for families of such maps to be Whitney equisingular. This project is part of a centuries old effort to fathom the geometry of mappings and singular shapes. (A mapping is a relation between shapes.) A cone is a simple example of a shape with a singularity, and a picture of a cone is a simple example of a mapping. The picture creates a relation between the points of the cone and the points of a plane, the plane of the picture in this case. If we take a picture of even a smooth shape, the image will have edges, and the edges will also often have singula r points, so the study of shapes and mappings are related. Equations describe shapes. The goal of this study is to start with equations of a shape and extract numbers from them that give a description of the geometry of the shape useful for determining when one member of a family of shapes is different from other members. For example, if you lay a piece of string across itself, it forms a loop. As you pull the loop tight, it forms a family of curves, and the loop disappears at some time t. Our intuition says that the curves in this family are similar until we get to time t, when the loop disappears. We can write down equations for the members of this family. It is helpful to study this situation using two scales. There is the macroscopic level, in which the individual curves bend to form loops, and there is also an infinitesimal level at which the tangent vectors to the curves and to the union of the curves exist. As you move around a loop, you can follow the tangent vectors around. Remarkably, although the loop disappears as time goes to t at the macroscopic level, the loop leaves a trace at the infinitesimal level. This trace shows up in the invariant, called the Milnor number, that we use to study curves in the plane, and it can be calculated from equations of the curves. The theory of integral closure of modules is a powerful framework for studying sets at this infinitesimal level and for extracting numbers that detect change at this level in a family of sets. Because there is a good connection between algebra and geometry for shapes defined in complex space by analytic functions, this is the setting for most work to date using this approach. d-dimensional subsets of complex n-space defined by n-d equations are called complete intersections. Gaffney and Kleiman will combine their own and others' earlier approaches to more limited situations to treat families of complete intersections with singular sets of arbitrary dimension. ***