Geometry of Singularities

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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. ***