Quasisymmetric Maps-Parametrization, Extension and Factorization

拟对称映射-参数化、扩展和因式分解

• 批准号: 1001669
• 负责人: Jang-Mei Wu
• 金额: $ 18.35万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001669&HistoricalAwards=false
• 关键词: Quasisymmetric; Maps; Parametrization; Extension; Factorization

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.

该项目采用新的方法来解决长期存在的问题，在准共形分析。拟共形映射在经典函数论的发展中起着举足轻重的作用。近年来，拟对称映射在几何群论、流形结构和分形分析中有着重要的应用。然而，大量的基本问题仍然存在：拟对称参数化的度量空间的欧氏空间;延长拟对称映射的环境空间;因式分解的拟共形映射成地图的小膨胀。扩张和光滑化在扩张量小的时候比较简单;带有小扩张量的拟共形结构的流形是可光滑的;如果拟共形映射可以分解为小扩张量的映射，那么这些因子可以被扩张，然后被光滑化。PI建议研究这一系列问题。经典几何拓扑学中有很多空间的例子，它们被证明是与欧几里得空间同胚的，只是有很大的独创性，例如，同调3-球面的双悬置和某些分解空间。这个项目涉及准对称参数化的空间，这类似于欧几里德空间不仅拓扑，但也几何和测量理论。这些发现将使人们更好地理解一般理论。拟共形和拟对称映射因其数学之美以及对物理学和生物学中自然存在的缺乏光滑结构的物体的潜在科学应用而受到研究。最近，有令人兴奋的发现，在应用拟共形映射研究大脑皮层表面的图像，以确定身体的导电性，并研究渗流和晶体生长。 这个建议涉及这些映射的内在性质的接口几何函数理论和经典几何拓扑。除了理论上的进步，这些发现将揭示一些更实际的问题。丰富的拓扑学的例子将扩大研究生在几何分析的研究活动的参与。