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

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。所考虑的度量空间通常不是平滑的，即它们在所有尺度和位置上看起来都是崎岖不平的，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。文献中分形一词的另一种用法是指具有一定自相似性的空间，即它的一部分表现为整个空间。在文献和本项目中，通常会考虑这两种意义上的分形空间。拟对称形成了一类重要的度量变形，其范围足够广，可以“拉直”一些分形，但仍适合分析方法。该项目研究的空间的主要例子是 Ahlfors 规则表面和 Sierpinski 地毯，其度量不一定来自欧几里得或球形几何。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，即坎农猜想和卡波维奇-克莱纳猜想。虽然坎农猜想和卡波维奇-克莱纳猜想最初是由瑟斯顿的几何化计划推动的，但它们并不遵循佩雷尔曼对庞加莱猜想和几何化猜想的解，并且在几何群论和三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源于复杂的分析。从一开始，复分析就为解决自然科学、工程和其他数学领域的问题提供了手段。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。复杂分析的许多方面已经发展到产生各种离散对应物和对一般度量空间的分析。该项目中考虑的分形空间在分析中作为分数维集出现，在动力学中作为朱莉娅集，在克莱因群理论中作为极限集，在几何中作为格罗莫夫双曲群无穷大的边界，仅举几个例子。研究人员希望该项目能够为度量空间的几何分析领域，特别是拟对称参数化问题，添加新的工具和想法，并为坎农猜想和卡波维奇-克莱纳猜想提供线索。他还希望引起更广泛的数学界对该项目中讨论的想法和结果的兴趣，并吸引学生进入该领域。