Geometry of Singularities

奇点几何

• 批准号: 9803691
• 负责人: Terence Gaffney
• 金额: $ 8.01万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-15 至 2003-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803691&HistoricalAwards=false
• 关键词: Geometry; Singularities

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. ***

小行星9803691 加夫尼研究解析集和映射的局部几何。 他的目标是从定义映射的集合或分量的方程中找到可计算的数字，这些映射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。 加夫尼认为最重要的条件之一是惠特尼等奇异性。 这个条件意味着族的嵌入拓扑是常数。 模的积分闭包理论为研究这一条件提供了一个强有力的框架。 与克莱曼，加夫尼将解决这个问题的家庭完整的交叉点与非孤立的奇点。 这将通过适当的修改，将加夫尼在具有非孤立奇点的超曲面的情况下使用的思想转移到与Kleiman开发的完全相交的框架中来完成。 加夫尼还将研究超曲面奇点的几何形状和与这些奇点相关的例外因子中出现的分量之间的关系。 要考虑的一个具体情况是，判别式确定的映射芽和拓扑稳定的映射芽接近的边界的好尺寸。 这项研究将是一个步骤的标准，家庭这样的地图是惠特尼等奇异的。 这个项目是一个世纪的努力，以揣摩映射和奇异形状的几何的一部分。 （映射是形状之间的关系。） 圆锥是一个简单的例子，一个形状的奇异性，和一个图片的圆锥是一个简单的例子映射。 这幅图在圆锥体的点和平面的点之间建立了一种关系，在这种情况下，平面就是这幅图的平面。 如果我们拍摄一张即使是光滑形状的照片，图像也会有边缘，而边缘也往往会有奇异点，因此形状和映射的研究是相关的。 方程描述形状。 本研究的目标是从形状的方程开始，并从中提取数字，这些数字给出了形状的几何形状的描述，用于确定形状家族中的一个成员何时与其他成员不同。 例如，如果你把一根绳子穿过它本身，它就会形成一个环。 当你拉紧这个环时，它会形成一个曲线族，这个环会在某个时间t消失。 我们的直觉告诉我们，在到达时间t之前，这个族中的曲线都是相似的，此时循环消失。 我们可以写出这个家庭成员的方程。 使用两个尺度来研究这种情况是有帮助的。 在宏观层面上，单个曲线弯曲形成环，在无穷小层面上，曲线的切向量和曲线的并集存在。 当你绕着一个环移动时，你可以沿着切向量移动。 值得注意的是，虽然在宏观水平上，当时间到达t时，环消失了，但在无穷小水平上，环留下了痕迹。 这个迹线出现在不变量中，称为米尔诺数，我们用来研究平面上的曲线，它可以从曲线的方程中计算出来。 模的积分闭包理论是一个强大的框架，用于研究在这个无穷小水平上的集合，并提取在这个水平上检测集合族中变化的数字。 因为在代数和几何之间有一个很好的联系，在复杂的空间中定义的形状的解析函数，这是大多数工作的设置日期使用这种方法。 由N-D方程定义的复N空间的D维子集称为完全相交。 加夫尼和克莱曼将结合联合收割机自己和他人的早期方法，以更有限的情况下，处理家庭的完整的交叉与奇异集的任意尺寸。 ***