Potential Theory of Functions of Bounded Variation and Quasiconformal Maps

当我们观看物体的图像时，我们看到的是物体的不同位置发出不同强度的光。我们可以认为物体的图像是由不同形状的表面组成的，每个表面都有自己均匀的亮度。同样，在研究各种数学对象时，例如测量不同位置温度的函数，测量电磁强度的函数，以及测量流体中不同位置流体粒子速度的函数，都可以根据函数的水平集的形状来理解。（水平集是函数采用给定常数值的集合。）这个度量空间分析的项目探讨了在水平集方面的势理论和拟共形映射研究中出现的函数的行为。该研究项目的应用包括图像处理和边缘检测。该项目的许多组成部分都适合研究生的论文材料，因此将有助于培养STEM劳动力的未来成员。本课题研究了由有限周长集合给出的度量空间的几何与非线性势理论和拟共形映射之间的联系。所考虑的空间配备了支持1-庞加莱不等式的加倍测度。在项目的第一部分，PI将探索有限周长集合集合与拟共形映射之间的相互作用，以及有限周长集合集合与非线性势理论之间的相互作用。项目的第二部分将探索拟极小边界曲面集的“切空间”规则性。项目的第三部分是发展有界变分函数的潜在理论。项目的最后一部分是用有限周长集合族的模来刻画某些庞加莱不等式。该研究的应用包括进一步理解与有限周长集相关的度量几何与某些非线性偏微分方程解之间的联系。这样的集合出现在图像处理和边缘检测的研究中，而抽象度量空间出现在黎曼流形理论中，考虑到Cheeger、Gromov和Perelman的作品中发现的Gromov- hausdorff极限空间。此外，这些项目将扩展现有的几何知识和Carnot-Caratheodory空间中有限周长集的理论。

When we view images of objects, what we see are various locations of the object emitting different intensities of light. We can think of the image of the object as formed by a collection of surfaces of different shapes, each with its own uniform brightness. Similarly, in the study of various mathematical objects such as functions that measure temperature at different locations, functions that measure electro-magnetic intensities, and functions that measure the velocity of fluid particles at various places in a fluid, can be understood in terms of the shape of the level sets of the function. (A level set is a set where the function takes on a given constant value.) This project in metric space analysis explores the behavior of functions that arise in the study of potential theory and quasiconformal mappings in terms of level sets. Applications of the the research project include image processing and edge detection. Many components of this project are suitable dissertation material for graduate students, and therefore will contribute to the training of future members of the STEM workforce.This project is concerned with links between geometry of a metric space given in terms of its sets of finite perimeter on the one hand, and nonlinear potential theory and quasiconformal mappings on the other hand. The spaces considered are equipped with a doubling measure supporting a 1-Poincare inequality. In the first part of the project the PI will explore interactions between collections of sets of finite perimeter and quasiconformal mappings, and between collections of sets of finite perimeter and nonlinear potential theory. The second part of the project will explore "tangent space" regularity of sets of quasiminimal boundary surfaces. The third part of the project is to develop a potential theory for functions of bounded variation. The last part of the project is to obtain a characterization of certain Poincare inequalities in terms of modulus of families of sets of finite perimeter. Applications of the research include further understanding of connections between metric geometry related to sets of finite perimeter and solutions to certain nonlinear partial differential equations. Such sets arise in the study of image processing and edge detection, while abstract metric spaces arise in Riemannian manifolds theory when considering Gromov-Hausdorff limit spaces as found in the works of Cheeger, Gromov, and Perelman. Furthermore, these projects will expand the current knowledge about geometry and the theory of sets of finite perimeter in Carnot-Caratheodory spaces.