Mathematical Sciences: Geometry of Singularities

点击翻译按钮获取中文摘要

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.) ***