Quasisymmetric Maps-Parametrization, Extension and Factorization

该项目以解决拟共形分析中长期存在的问题为特色。拟共形映射在经典函数理论的发展中起着举足轻重的作用。拟对称映射近年来在几何群论、流形结构和分形分析中得到了重要的应用。然而，仍然存在大量的基本问题：欧几里德空间对度量空间的准对称参数化；拟对称映射在环境空间中的扩展拟共形映射分解成小扩张映射。当膨胀较小时，扩展和平滑更简单；带有小膨胀拟共形结构的流形是光滑的；如果一个拟共形映射可以被分解成小扩张的映射，那么这些因子就可以被扩展，然后被平滑。PI建议研究这一系列问题。经典几何拓扑学中有许多空间的例子，这些空间都是通过极大的巧思才被证明与欧氏空间同胚的，例如同调三球的双悬和某些分解空间。本课题研究此类空间的准对称参数化，这些空间不仅在拓扑上，而且在几何上和测量理论上都与欧几里德空间相似。这些发现将有助于更好地理解一般理论。准共形和准对称映射因其数学之美以及在物理和生物学中自然发生的缺乏光滑结构的物体的潜在科学应用而被研究。最近，在应用拟共形映射来研究大脑皮层表面的图像，确定身体的电导率，以及研究渗透和晶体生长方面有了令人兴奋的发现。本文从几何函数理论和经典几何拓扑学的角度讨论了这些映射的内在性质。除了理论上的进步，这些发现还将揭示一些更实际的问题。拓扑学中丰富的例子将扩大研究生对几何分析研究活动的参与。

The project features new approaches to long-standing problems in quasiconformal analysis. Quasiconformal maps have played a pivotal role in the development of classical function theory. Quasisymmetric maps have recently found important applications in geometric group theory, structure of manifolds and analysis on fractals. However a large number of fundamental questions remain: quasisymmetric parametrization of metric spaces by Euclidean spaces; extension of quasisymmetric maps to an ambient space; factorization of quasiconformal maps into maps of small dilatation. Extension and smoothing are simpler when the dilatation is small; a manifold carrying a quasiconformal structure of small dilatation is smoothable; if a quasiconformal map can be factored into maps of small dilatation, then the factors can be extended,then smoothed. The PI proposes to study this circle of problems. Classical geometric topology is rich with examples of spaces which were proved homeomorphic to Euclidean spaces only with great ingenuity, e.g., the double suspension of homology 3-spheres and certain decomposition spaces. This project deals with quasisymmetric parametrization of such spaces, which resemble the Euclidean spaces not only topologically, but also geometrically and measure-theoretically. The findings will lead to a better understanding of the general theory.Quasiconformal and quasisymmetric maps have been studied for their mathematical beauty as well as potential scientific applications to objects lacking a smooth structure that occur naturally in physics and biology. Recently, there have been exciting discoveries in applying quasiconformal mappings to study images of brain cortical surfaces, to determine the conductivity of a body, and to study percolation and crystal growth. This proposal deals with intrinsic properties of these mappings at the interface of geometric function theory and classical geometric topology. In addition to theoretical advances, the findings will shed light on some of these more practical problems. The richness of the examples from topology will broaden the participation of graduate students in research activities in geometric analysis.