Topology of Nonisolated Singularities and the Geometry of Functions

非孤立奇点拓扑和函数几何

• 批准号: 9803467
• 负责人: James Damon
• 金额: $ 7.93万
• 依托单位: University of North Carolina at Chapel Hill
• 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 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803467&HistoricalAwards=false
• 关键词: Topology; Nonisolated; Singularities; Geometry; Functions

9803467 Damon Professor Damon's research will apply infinitesimal methods from singularity theory (originally introduced by Mather) to investigate the topological structure of a general class of highly nonisolated singularities arising as nonlinear sections of varieties. This class includes discriminants of mappings, bifurcation sets, nonlinear arrangements of hypersurfaces, etc. He will use analogues of Morse singularities for sections of varieties to derive general algebraic formulae for certain fundamental topological invariants. He will also apply these ideas involving sections of varieties to understand the geometry of functions using the notion of "relative critical sets," which extend the notion of ridges and "cores" used in computer imaging. He will combine these geometric techniques with other singularity-theoretic methods developed for determining generic properties of solutions to partial differential equations. Together they will be applied to "pixel intensity functions" representing medical images. In conjunction with Gaussian blurring or nonlinear blurring techniques, they will associate certain geometric structures that can be used for addressing various questions in medical imaging. The first part of Professor Damon's research project will concern qualitative properties of spaces defined as the set of solutions of specific types of systems of nonlinear equations. The specific form of the equations will allow the investigator to use methods from the field of mathematics called singularity theory. Specifically, this allows the analysis of the set of solutions by perturbing the equations and reducing to the analysis of certain basic cases. The total contributions from the basic cases can be determined algebraically, yielding a qualitative understanding of the structure of the set of solutions. Second, such systems of equations will be specifically applied to analyze medical images. They allow one to associate to a medical image a g eometric structure, which captures essential features of the image. The geometric structure is defined using systems of equations of the above type, and it even allows one first to remove "noise" from the image by applying appropriate filters. The basic properties of the geometric structure obtained from the first part will allow it to be used to address several questions in medical imaging. ***

小行星9803467 达蒙教授的研究将应用奇点理论（最初由马瑟介绍）的无穷小方法来研究一类高度非孤立奇点的拓扑结构，这些奇点是由各种非线性部分产生的。 这一类包括判别式的映射，分歧集，非线性安排的超曲面等，他将使用类似物的莫尔斯奇异性的部分品种，以获得一般代数公式的某些基本拓扑不变。 他还将应用这些想法涉及部分品种理解几何的功能使用的概念“相对临界集”，这扩展了概念的脊和“核心”在计算机成像中使用。 他将结合联合收割机这些几何技术与其他奇异理论的方法，确定一般性质的解决方案，偏微分方程。 它们将一起应用于表示医学图像的“像素强度函数”。 结合高斯模糊或非线性模糊技术，它们将关联某些几何结构，可用于解决医学成像中的各种问题。 达蒙教授的研究项目的第一部分将关注空间的定性性质，定义为特定类型的非线性方程组的解的集合。 方程的特定形式将允许研究人员使用数学领域的方法，称为奇点理论。 具体来说，这允许通过扰动方程和减少到某些基本情况下的分析的解决方案的集合的分析。 从基本情况下的总贡献可以确定代数，产生一个定性的理解的结构的解决方案。 第二，这样的方程组将被具体地应用于分析医学图像。 它们允许人们将医学图像与几何结构相关联，该几何结构捕获图像的基本特征。 几何结构是使用上述类型的方程组定义的，并且它甚至允许一个 首先通过应用适当的滤波器从图像中去除“噪声”。 从第一部分获得的几何结构的基本属性将允许它被用来解决医学成像中的几个问题。 ***