Topology of Nonisolated Singularities and Scale-based Geometry

Damon教授的研究将应用奇点理论中的无穷小和分层方法来研究一类一般高度非孤立奇异空间的拓扑和变形特性。这些空间是某些自然泛变的非线性截面。例如，映射在各种等价下的几何性质和变形性质可以通过这样的泛变来捕获。切向量场的结构将用于推导某些基本拓扑不变量的代数公式。这些将在一定的自然代数和几何多样性的条件下给出，测量相对于相关几何结构（如叶状）的映射的奇异行为。其次，他将通过在这种高度奇异的空间中开发与图像中的物体和特征相关的几何结构来完善这些思想，以解决计算机成像中的问题。图像中存在的离散性、噪声和失真需要“基于比例的几何”，它适用于不可微函数、度量甚至分布。这种几何结构将适用于给定类型中的“几乎所有”对象，并将在比例空间中产生稳定的几何结构。这将允许使用函数和措施来区分图像中的各种特征对图像进行几何分析。Damon教授研究的第一部分将确定特定类型的非线性方程组，解集的定性性质。这些可以从某些通用方程组中得到。他提出利用方程的某些无穷小对称性来推导解集的代数不变量的性质，这些代数不变量既反映了普适方程的性质，又反映了特定方程与普适方程的关系。其次，本研究将应用于计算机成像问题。对于图像中的物体和特征，人们可以将几何结构与捕获其属性联系起来，用于各种成像目的。这样的结构是用上述方程组来定义的。图像中存在的离散性、噪声和失真会干扰对几何特征的识别。该研究将通过“基于比例”的版本来改进上述方法，该版本引入了克服这些困难的坚固几何结构，然后可用于各种计算机成像问题。

DMS-0103862James N. DamonProfessor Damon's research will apply infinitesimal and stratification methods from singularity theory to investigatethe topology and deformation properties of a general class of highly nonisolated singular spaces. These spaces arise as nonlinear sections of certain natural universal varieties. For instance, geometric and deformation properties of mappings under various equivalences can be captured by such universal varieties. The structure of the tangent vector fields will be used to deduce algebraic formulae for certain fundamental topological invariants. These will be given in terms of certain natural algebraic and geometric multiplicities, measuring the singular behavior of mappings relative to associated geometric structures such as foliations Second, he will refine these ideas for questions in computer imaging by developing geometric structures associated to objects and features in images in terms of such highly singular spaces. The presence of discreteness, noise, and distortions in images require a "scale-based geometry" which is applicable to nondifferentiable functions, measures, and even distributions. Such a geometry will apply to "almost all" objects in a given type, and will yield stable geometric structures in scale space. This will allow the geometric analysis of images using functions and measures discriminating various features in images.The first part of Professor Damon's research will determine for specific types of systems of nonlinear equations, the qualitative properties of the set of solutions. These can be obtained from certain universal systems of equations. He proposes to use certain infinitesimal symmetries of the equations to deduce properties of the set of solutions in terms of algebraic invariants which reflect both properties of the universal equations and how the specific equations relate to the universal ones. Second, the research will be applied to problems in computer imaging. To objects and features in images, one may associate geometric structures capturing their properties for various imaging purposes. Such structures are defined using systems of equations as above. The presence of discreteness, noise and distortions in images interferes with identifying geometric features. The research will refine the methods described above via "scale-based" versions which introduce robust geometric structures overcoming these difficulties, which can then beused for a variety of computer imaging problems.