Topology of Nonisolated Singularities and the Geometry of Functions

9803467达蒙教授的研究将应用奇点理论(最初由马瑟引入)中的无穷小方法来研究一类作为簇的非线性部分而产生的高度非孤立奇点的一般类型的拓扑结构。这一类包括映射的判别式、分岔集、超曲面的非线性排列等。他将利用簇截面的Morse奇点的类比来推导某些基本拓扑不变量的一般代数公式。他还将应用这些涉及变种的部分的想法，通过“相对临界集”的概念来理解函数的几何，“相对临界集”的概念扩展了计算机成像中使用的脊线和“核”的概念。他将把这些几何技术与为确定偏微分方程解的一般性质而开发的其他奇点理论方法结合起来。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，这些结构可用于解决医学成像中的各种问题。达蒙教授研究项目的第一部分将讨论定义为特定类型的非线性方程组的解的集合的空间的定性性质。这些方程的具体形式将允许研究人员使用数学领域中称为奇点理论的方法。具体地说，这允许通过扰动方程和归结到对某些基本情况的分析来分析解的集合。基本情况的总贡献可以用代数来确定，从而对解的集合的结构产生定性的理解。其次，这种方程组将专门应用于分析医学图像。它们允许人们将捕捉图像基本特征的几何结构与医学图像相关联。几何结构是使用上述类型的方程系统来定义的，它甚至允许人们首先通过应用适当的过滤器来消除图像中的“噪声”。从第一部分获得的几何结构的基本属性将使其能够用于解决医学成像中的几个问题。***

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. ***