Metric geometry and functions of bounded variation

自然界中的物体很少是光滑的。从分形物体到多孔介质，控制热量和压力等量的耗散的方程都有非光滑分量。这种非平滑对象也作为某些平滑对象的限制出现。为了理解和利用这些物体的行为，我们需要从黎曼几何中移除光滑性假设。对于这样的非光滑对象，我们需要研究非光滑能量极小化器的行为及其与几何的联系，即底层空间或对象的结构属性。这个项目的度量几何和有界变差的功能有助于这一目标，通过探索之间的互连几何的基础度量空间配备了一个措施，和属性集的最小表面积。这里研究的对象是度量空间，它配备了一个加倍的测度（也就是说，一个球的测度相当于一个半径为两倍的同心球的测度），并且使得一个球上的Lipschitz连续函数的方差可以根据它在该球上的能量（使用其局部振荡计算）的平均值来控制。在这种情况下，这个项目的目的是研究属性，如可矫正性，孔隙率，形状和自然尺寸的局部最小表面积集。 分析度量措施空间产生了许多来源;研究复杂的解析函数，微分方程管理骨折的混合材料和相关映射的有限变形，控制理论在工程和相关的卡诺-Caratheodory空间，研究著名的庞加莱猜想，是一些根源，这一领域的研究。然而，不同于在欧几里得的情况下，极小曲面（具有最小表面能的曲面）的结构性质被很好地理解，在自然界和物理学和工程学的控制理论问题中出现的非光滑设置中，极小曲面的结构知之甚少。这个项目旨在探索这种结构，并扩大我们在非光滑环境中的最小表面的知识。这项研究的结果将有助于进一步了解由于热和压力等自然效应而改变的物体的行为变化。除了提高我们对非光滑物体的认识外，本项目进行的研究还将有助于度量空间分析理论的不断发展;这种发展中的理论的一个很大的好处是，由于在经典欧几里德分析中可用的许多结构工具在非光滑设置中不可用，因此必须开发新的工具和方法，使理论更容易为更广泛的受众所接受。

Objects occurring in nature rarely are smooth in appearance. From fractal objects to porous media, the equations that govern dissipation of quantities such as heat and pressure have non-smooth components. Such non-smooth objects also occur as limits of certain smooth objects. To understand and exploit the behavior of such objects, we need to remove the smoothness assumptions from Riemannian geometry. For such non-smooth objects we need to study behaviors of non-smooth energy minimizers and their connection to the geometry, that is, the structural properties, of the underlying space or object. This project on metric geometry and functions on bounded variation contributes to this goal by exploring interconnections between the geometry of the underlying metric space equipped with a measure, and the properties of sets of minimal surface areas. The study of objects here are metric spaces equipped with a measure that is doubling (that is, measure of a ball is comparable to the measure of a concentric ball of double the radius) and such that the variance of a Lipschitz continuous function on a ball can be controlled in terms of the average value of its energy (computed using its local oscillation) on that ball. In this setting, this project aims to study properties such as rectifiability, porosity, shape, and natural dimension of sets of locally minimal surface areas. Analysis on metric measure spaces arose from many sources; the study of complex analytic functions, differential equations governing fractures in mixed material and associated mappings of finite distortion, control theory in engineering and associated Carnot-Caratheodory spaces, the study of the famous Poincare conjecture, are some of the roots of this field of study. However, unlike in the Euclidean situation where structural properties of minimal surfaces (surfaces with smallest surface energy) are well understood, in the non-smooth setting that arise in nature and in control theory problems of physics and engineering, the structure of minimal surfaces is poorly understood. This project seeks to explore such structures and expand our knowledge of minimal surfaces in a non-smooth setting. The results of this study will be useful in further understanding the change in the behavior of objects that are transformed due to natural effects such as heat and pressure. In addition to advancing our knowledge of non-smooth objects, the study conducted in this project will also contribute to the ongoing development of the theory of analysis on metric spaces; a great benefit of this developing theory is that since much of the structural tools available in classical Euclidean analysis are not available in the non-smooth setting, new tools and methods have necessarily to be developed, making the theory more accessible to a wider audience.