Multi-scale geometry of bi-Lipschitz and quasiconformal maps

距离的概念既是人类感知我们周围的物理空间的基础，也是几何上对这个空间的理想化的基础。距离最显著的性质的提炼导致了度量空间的概念。这一概念在计算机科学、机器学习、数学遗传学、统计学和其他对对象的定量不同感兴趣的学科中有大量的应用。理解度量空间的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有界量。这些被称为双Lipschitz变换，它们是拟议调查的焦点。作为这种变换应用的一个例子，可以提到欧氏空间中给定的、可能非常不规则的度量空间的准确表示是数据可视化和分析的一个基本方面。赋范线性空间提供了一种方便的模型，可以将不太规则的度量空间与之进行比较。因此，知道给定的度量空间何时是赋范数空间的双Lipschitz等价的，或者何时可以嵌入到赋范数空间中是很有趣的。这是提案中涉及的主要问题之一。即使对于二维空间(度量平面)，我们对双Lipschitz等价的理解也远不完全。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，例如捕捉空间内连通度大小或程度的维型概念。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：Poincare不等式和各种延拓性质(Soblev，Lipschitz等)。该提案涉及这两种方法。特别地，将研究双Lipschitz延拓问题的定量方面。

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.