Multi-scale geometry of bi-Lipschitz and quasiconformal maps

距离的概念是人类感知周围物理空间的基础，也是几何空间理想化的基础。度量空间的概念源于距离最显著的性质的升华，它广泛应用于计算机科学、机器学习、数学遗传学、统计学和其他对物体的定量相异性感兴趣的学科。理解度量空间的一个正确方法是考虑它们的变换，这些变换将空间上的距离收缩或拉伸一个有界的量。这些被称为双Lipschitz变换，它们是拟议调查的重点。作为这种变换的应用的一个例子，可以提到的是，在欧几里得空间内精确表示给定的，可能非常不规则的度量空间是数据可视化和分析的一个重要方面。赋范线性空间提供了一个方便的模型，可以与不太规则的度量空间进行比较。因此，我们很想知道一个给定的度量空间什么时候是双李普希兹等价于赋范空间，或者可以嵌入赋范空间。这是该提案所要解决的主要问题之一。即使对于二维空间（度量平面），我们对双李普希茨等价的理解也远未完成。解决这个问题的一种方法是引入和研究度量空间的数值不变量，例如维度类型的概念，它捕获空间内连通性的大小或程度。另一种不同但相关的方法涉及空间上支持的函数的分析性质：庞加莱不等式和各种扩展性质（Sobolev，Lipschitz等）。该提案涉及这两种方法。特别是，双Lipschitz扩展问题的定量方面进行了研究。

The notion of distance is fundamental both to human perception of the physical space around us, and to the idealization of this space in geometry. The distillation of the most salient properties of distance leads to the concept of a metric space.Applications of this concept abound in computer science, machine learning, mathematical genetics, statistics and other disciplines where quantitative dissimilarity of objects is of interest. A proper way to understand metric spaces is to consider their transformations that shrink or stretch the distance on the space by a bounded amount. These are called bi-Lipschitz transformations, and they are at the focus of the proposed investigation. As an example of an application of such transformations, one can mention that accurate representations of a given, possibly quite irregular, metric space within a Euclidean space is an essential aspect of data visualization and analysis.Normed linear spaces provide a convenient model to which less regular metric spaces can be compared. Thus, it is of interest to know when a given metric space is bi-Lipschitz equivalent to a normed space, or can be embedded into one. This is one of the principal problems addresses in the proposal. Even for two-dimensional spaces (metric planes) our understanding of bi-Lipschitz equivalence is far from complete. One approach to this problem is to introduce and study numeric invariants of metric spaces, such as dimension-type concepts which capture the size or degree of connectivity within the space. A different, but related approach involves analytic properties of functions supported on the space: Poincare inequality and various extension properties (Sobolev, Lipschitz, etc). The proposal involves both of these approaches. In particular, the quantitative aspects of the bi-Lipschitz extension problem will be investigated.