Mathematical Sciences: Geometry of Singularities

Gaffney研究了解析集和映射的局部几何。该领域的目标是找到具有几何内容的数字，这些内容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。具体地说，Gaffney寻找不变量，其常数对于所考虑的集合或映射族的Whitney等奇性是充分必要的。这个条件意味着家族的嵌入式拓扑结构是恒定的。对于集合，Gaffney提出使用与非有限长度模的整闭包相关的不变量来控制族的Whitney等奇异性。这些不变量将发挥多重性对有限长度的理想和模块所起的作用。Gaffney还将寻找来自映射源的A轨道的切空间的类似不变量，以描述映射族的等奇性。集合族和映射族（它们只是集合之间可能的关系）出现在许多不同的情况下。空间中任何随时间变化的形状都可以被认为是一组集合——每一个瞬间都有一个形状。一般来说，这些形状看起来基本上是一样的，除了一些特殊的时刻，当剧烈的变化似乎发生时，之后这些集合又会看起来相似。给定描述形状的方程，该项目将寻找可以从这些方程中计算出的数字，这些数字将告诉我们特殊时刻的时间。一些可能的应用是机器人导航，以及可以通过观察微分方程族的平衡点来研究的大量物理问题。（在机器人技术中，形状由机器人或其成员之一无法到达的点组成；当该形状的性质发生变化时，到达目标的路径可能会改变。）＊＊＊

9403708 Gaffney Gaffney studies the local geometry of analytic sets and mappings. The goal in this area is to find numbers with a geometric content which 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. Specifically, Gaffney looks for invariants whose constancy is necessary and sufficient for the Whitney equisingularity of the family of sets or maps being considered. This condition implies that the embedded topology of the family is constant. For sets, Gaffney proposes to use invariants associated with the integral closure of a module of non-finite colength to control the Whitney equisingularity of the family. These invariants will play the role that the multiplicity does for ideals and modules of finite colength. Gaffney will also look for similar invariants coming from the tangent space to the A orbit of a map-germ to describe the equisingularity of a family of mappings. Families of sets and families of mappings (which are just the possible relationships between sets) occur in many different situations. Any shape in space which changes with time can be thought of as a family of sets--one shape for each instant of time. Generally speaking, the shapes will seem essentially the same, except for exceptional moments when drastic change seems to occur, after which the sets will appear to be similar again. Given the equations that describe a shape, this project will look for numbers which can be calculated from these equations which will tell us when the exceptional moments are. Some possible applications are to robot navigation, and to the wealth of physical problems which can be studied by looking at equilibrium points of families of differential equations. (In the case of robotics, the shape consists of the set of inaccessible points to a robot or one of its members; when the nature of this shape changes, the path to a goal may change.) ***