Mathematical Sciences: Geometry of Singularities

数学科学：奇点几何

• 批准号: 9403708
• 负责人: Terence Gaffney
• 金额: --
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403708&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Singularities

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.) ***

9403708 Gaffney Gaffney研究解析集和映射的局部几何。这个领域的目标是找到具有几何内容的数字，它将描述被研究对象的几何形状。第一步是找出这种类型的数字，其在对象族中的恒定意味着该族的几何以某种明确定义的方式是恒定的。具体地说，Gaffney寻找的不变量的不变性对于所考虑的集合或映射族的惠特尼等价性是必要的和充分的。此条件意味着该族的嵌入拓扑是恒定的。对于集合，Gaffney建议使用与非有限长度模的积分闭包相关的不变量来控制族的Whitney等奇性。这些不变量将扮演重数对有限长度的理想和模所起的作用。Gaffney还将寻找来自映射芽的切线空间到A轨道的类似不变量来描述映射族的均衡性。集合族和映射族(它们只是集合之间的可能关系)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--每个时刻对应一个形状。一般说来，形状看起来基本上是一样的，除了发生剧烈变化的特殊时刻，在那之后，布景看起来又会相似。给出描述形状的方程，这个项目将寻找可以从这些方程中计算出来的数字，这些数字将告诉我们什么时候是异常时刻。一些可能的应用是机器人导航，以及大量的物理问题，这些问题可以通过查看微分方程组的平衡点来研究。(在机器人的情况下，形状由机器人或其成员之一无法到达的一组点组成；当该形状的性质改变时，通往目标的路径可能会改变。)*