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.

这个项目的目的是研究各种拓扑平面度量空间的变形性质，特别是在准对称变形下。所考虑的度量空间通常是不光滑的，即，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。在这种情况下，我们说这样的度量空间是分形的。在文献中，术语分形的另一种用法是指具有某种自相似性的空间，即，它的一部分看起来像整个空间。在文献和本项目中，经常考虑在这两种意义上都是分形的空间。准对称性形成了一类重要的度量变形，它的范围足以“理顺”某些分形，而且还适用于分析方法。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面几何。这种空间的出现与一般的参数化问题有关，特别是与几何群论中的两个主要命题有关，即坎农命题和卡波维奇-克莱纳命题。虽然最初的动机瑟斯顿的几何化计划，坎农的和Kapovich-Kleiner拓扑不遵循佩雷尔曼的解决方案的庞加莱和几何化拓扑，仍然是重要的开放问题，无论是在几何群论和3-流形拓扑。用于解决项目中问题的工具来自复杂分析。从一开始，复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手段。最近的例子包括统计物理学中二维格点模型连续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导致各种离散的对应物和一般度量空间上的分析。在该项目中考虑的分形空间出现在分析作为集的分数维，在动力学作为朱莉娅集，在理论的克莱因集团作为极限集，并在几何边界在无穷大的格罗莫夫双曲群，举几个例子。这是希望的调查，该项目将增加新的工具和思想领域的几何分析度量空间，特别是准对称参数化问题，并将阐明坎农的和Kapovich-Kleiner的结构。他还希望在更广泛的数学界对该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。