Quasisymmetric deformations of topologically planar fractal spaces

这个项目的目的是研究各种拓扑平面度量空间的变形特性，特别是在准对称变形下。所考虑的度量空间通常不是光滑的，也就是说，它们在所有尺度和位置上看起来都很粗糙，就像著名的冯·科赫雪花一样。在这种情况下，我们说这样的度量空间是分形的。在文献中，分形一词的另一种用法是指具有某种自相似性质的空间，即它的某些部分表现为整个空间。通常在文献和这个项目中，空间在这两种意义上都是分形的。准对称形成了度量变形的一个重要类别，它的范围足够广泛，可以“拉直”一些分形，但又适合于分析方法。该项目研究的主要空间例子是Ahlfors规则表面和Sierpinski地毯，其度量不一定来自欧几里得或球面几何。这种空间的出现与一般参数化问题有关，特别是与几何群论中的两个主要猜想有关，即Cannon猜想和kapoovich - kleiner猜想。虽然最初是由瑟斯顿的几何化程序激发的，但坎农和卡波维奇-克莱纳猜想并不遵循佩雷尔曼对庞加莱猜想和几何化猜想的解决方案，并且在几何群论和3流形拓扑中仍然是重要的开放问题。用于解决项目中问题的工具起源于复杂分析。从一开始，复杂分析就为解决来自自然科学、工程和其他数学领域的问题提供了方法。最近的例子包括统计物理中二维晶格模型连续体极限的共形不变性的研究以及在量子引力中的应用。复分析的许多方面已经发展到各种离散对应物和对一般度量空间的分析。项目中考虑的分形空间在分析中作为分数维的集合出现，在动力学中作为Julia集合出现，在Kleinian群理论中作为极限集出现，在几何中作为Gromov双曲群的无穷边界出现，举几个例子。研究者希望该项目将为度量空间的几何分析领域，特别是准对称参数化问题，增加新的工具和思想，并将阐明坎农和卡波维奇-克莱纳猜想。他还希望在更广泛的数学界对该项目中讨论的思想和结果产生兴趣，并吸引学生进入该领域。

The aim of this project is to investigate deformation properties of various topologically planar metric spaces, specifically under quasisymmetric deformations. The metric spaces under consideration usually are not smooth, i.e., they look rugged on all scales and locations like the famous von Koch snowflake. In this case we say that such metric spaces are fractal. Another use of the term fractal in the literature is to refer to a space that has a certain self-similarity property, i.e., parts of it appear as the whole space. Often in the literature and in this project spaces that are fractal in both of these senses are considered. Quasisymmetries form an important class of metric deformations that is broad enough to "straighten out" some of the fractals and yet amenable to methods of analysis. The primary examples of spaces studied in the project are Ahlfors regular surfaces and Sierpinski carpets with metrics that do not necessarily come from the Euclidean or the spherical geometries. Such spaces arise in relation to the general parametrization problem and in particular to two major conjectures in geometric group theory, namely Cannon's and the Kapovich-Kleiner conjectures. While originally motivated by Thurston's geometrization program, Cannon's and the Kapovich-Kleiner conjectures do not follow from Perelman's solution of the Poincare and Geometrization conjectures and remain important open problems both in geometric group theory and in 3-manifold topology. The tools used to attack the questions in the project originate in complex analysis. From the onset, complex analysis has provided means for solving problems that come from the natural sciences, engineering and other fields of mathematics. Recent examples include the investigations of the conformal invariance of continuum limits of two-dimensional lattice models in statistical physics and applications to quantum gravity. Many aspects of complex analysis have evolved to lead to various discrete counterparts and analysis on general metric spaces. Fractal spaces considered in the project arise in analysis as sets of fractional dimension, in dynamics as Julia sets, in the theory of Kleinian groups as limit sets, and in geometry as boundaries at infinity of Gromov hyperbolic groups, to mention a few examples. It is the hope of the investigator that the project would add new tools and ideas to the field of geometric analysis on metric spaces, in particular to the quasisymmetric parametrization problem, and would shed light on Cannon's and the Kapovich-Kleiner conjectures. He also hopes to generate interest to the ideas and results discussed in the project in the broader mathematical community and attract students to the field.