Quasisymmetric deformations of topologically planar fractal spaces

拓扑平面分形空间的拟对称变形

• 批准号: 1001144
• 负责人: Sergiy Merenkov
• 金额: $ 12.6万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001144&HistoricalAwards=false
• 关键词: Quasisymmetric; deformations; topologically; planar; fractal; Quasisymmetric; deformations; topologically; planar; fractal

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.

该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。所考虑的公制空间通常不光滑，即它们在著名的冯·科赫雪花等所有尺度和位置上看起来都很坚固。在这种情况下，我们说这样的度量空间是分形的。文献中分形术语的另一种用途是指具有具有一定自相似属性的空间，即部分属性显示为整个空间。在文献和这个项目空间中，通常都会考虑到这两种感觉。准对称形成了一类重要的度量变形类，足以“弄清”某些分形，但可以接受分析方法。该项目中研究的空间的主要例子是AHLFORS常规表面和Sierpinski地毯，其指标不一定来自欧几里得或球形几何形状。这种空间是与一般参数化问题有关的，尤其是几何群体理论中的两个主要猜想，即坎农和卡波维奇 - 克莱纳的猜想。尽管最初是由瑟斯顿（Thurston）的几何化计划激发的，但坎农（Cannon's）和卡波维奇·克莱纳（Kapovich-Kleiner）的猜想并不遵循佩雷尔曼（Perelman）对庞加罗（Poincare）和几何化猜想的解决方案，并且在几何群体理论和三个manifold拓扑学中仍然是重要的开放问题。用于攻击项目中问题的工具起源于复杂分析。从一开始，复杂的分析就可以解决来自自然科学，工程和其他数学领域的问题。最近的例子包括研究统计物理学中二维晶格模型的连续限制的保形不变性，以及用于量子重力的应用。复杂分析的许多方面已经演变为导致各种离散的对应物以及对一般度量空间的分析。该项目中考虑的分形空间是在分析中作为分数维度的集合，在朱莉娅集合中，克莱尼亚群体作为极限集的理论，而在Gromov高音群体无穷大的边界中，以几何形状为限制。调查员希望该项目将在公制空间（尤其是准对称参数化问题）的几何分析领域中添加新的工具和想法，并将阐明Cannon's和Kapovich-Kleiner的猜想。他还希望对在更广泛的数学社区中讨论的项目中讨论的想法和结果产生兴趣，并吸引学生进入该领域。