Mathematical Sciences: Geometry of Singularities

小行星9403708 加夫尼研究解析集和映射的局部几何。 这一领域的目标是找到具有几何内容的数字，这些数字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，其在一个对象族中的恒定性意味着该族的几何形状以某种明确的方式是恒定的。 具体来说，加夫尼期待不变量的恒定性是必要的，充分的惠特尼equisingularity家庭的集或地图正在考虑。 这个条件意味着族的嵌入拓扑是常数。 对于集合，加夫尼提出使用与非有限字长模的积分闭包相关的不变量来控制家族的惠特尼等奇异性。 这些不变量将扮演的角色，多重性没有理想和模的有限色长。 Gaffney也将寻找来自映射芽的A轨道的切空间的类似不变量来描述映射族的等奇异性。 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个集合族--每一个形状对应一个时刻。 一般来说，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， 在此之后，集合将再次看起来相似。 给定描述形状的方程，这个项目将寻找可以从这些方程中计算出来的数字，这些方程将告诉我们什么时候是例外时刻。 一些可能的应用是机器人导航，以及丰富的物理问题，可以研究通过寻找平衡点的家庭微分方程。 (In在机器人的情况下，形状由机器人或其成员之一不可接近的点的集合组成;当这个形状的性质改变时，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.) ***