Mathematical Sciences: Topological Properties of Singularities and Solutions of Nonlinear Equations

9400930 Damon教授一直在开发Mather介绍的无穷小方法的类似方法，用于回答有关非线性映射的结构和特殊但重要的高度非孤立奇点的拓扑结构的问题，例如出现在映射的判别式中。在完全不同的情况下，他还发现了如何修改这些无穷小的方法，以理解计算机视觉中使用的高斯模糊下“像素强度函数”的一般行为。他打算在他早期结果的基础上，推导出高度奇异空间的某些基本拓扑不变量的一般代数公式。其次，他将进一步发展确定偏微分方程解的一般性质的方法，特别是将它们应用于最近的非线性版本的高斯模糊。第三，他将应用关于奇异空间拓扑的结果来理解如何从发生的奇点推导出解的定性行为。本研究项目将研究非线性方程组解的某些定性性质。这些性质将应用于偏微分方程解的一般形式。达蒙教授一直在开发确定这种定性性质的方法。此外，他已经开始应用它们来理解应用于计算机视觉中分析图像的“像素强度函数”的高斯模糊的效果。他打算进一步发展这些方法，使其适用于一般偏微分方程式。这将包括研究解的某些不变量与它们的定性性质之间的关系。一个特别的应用将是正在开发的日益复杂的非线性模糊方案，该方案用于识别诸如边缘、脊线等图像的基本特征和属性。

9400930 Damon Professor Damon has been developing analogues of infinitesimal methods introduced by Mather for answering questions about the structure of nonlinear mappings and the topological structure of special but important classes of highly nonisolated singularities, such as occur in discriminants of mappings. In quite different circumstances, he has also discovered how to modify these infinitesimal methods for understanding the generic behavior of "pixel intensity functions" under Gaussian blurring used in computer vision. He intends to build on his earlier results and derive general algebraic formulae for certain fundamental topological invariants for highly singular spaces. Second, he will further develop methods for determining the generic properties of solutions to partial differential equations and, in particular, apply them to recent nonlinear versions of Gaussian blurring. Third, he will apply the results on the topology of singular spaces to understand how the qualitative behavior of the solutions can be deduced from the singularities which occur. This research project will investigate certain qualitative properties of the set of solutions of systems of nonlinear equations. Such properties will be applied to generally appearing solutions of partial differential equations. Professor Damon has been developing methods for determining such qualitative properties. Moreover, he has begun applying them to understand the effects of Gaussian blurring applied to "pixel intensity functions" for analyzing images in computer vision. He intends to develop these methods further so that they apply to general partial differential equations. This will include the investigation of relations between certain invariants of the solutions and their qualitative properties. One particularapplication will be to the increasingly sophisticated nonlinear blurring schemes which are being developed for identifying essential features and properties of images suc h as edges, ridges, etc. ***