Topology of Nonisolated Singularities and the Geometry of Functions

Damon教授的研究将应用奇点理论（最初由Mather引入）的无穷小方法来研究一类高度非孤立奇点的拓扑结构，这些奇点是作为非线性部分产生的。这类包括映射的判别式、分岔集、超曲面的非线性排列等。他将使用摩尔斯奇点的类似物来推导某些基本拓扑不变量的一般代数公式。他还将运用这些涉及到变化部分的思想，使用“相对临界集”的概念来理解函数的几何形状，这扩展了计算机成像中使用的脊和核的概念。他将把这些几何技术与其他奇点理论方法结合起来，这些方法是用来确定偏微分方程解的一般性质的。它们将一起应用于代表医学图像的“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。Damon教授研究项目的第一部分将关注空间的定性性质，它被定义为特定类型非线性方程组的解的集合。方程的具体形式将允许研究者使用称为奇点理论的数学领域的方法。具体地说，这允许通过扰动方程来分析一组解，并简化为对某些基本情况的分析。基本情况的总贡献可以用代数方法确定，从而产生对解集结构的定性理解。其次，这样的方程组将专门用于分析医学图像。它们允许人们将医学图像与捕获图像基本特征的几何结构联系起来。几何结构是用上述类型的方程组定义的，它甚至允许人们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. ***