Collaborative Research: FRG: Geometric Function Theory: From Complex Functions to Quasiconformal Geometry and Nonlinear Analysis

合作研究：FRG：几何函数理论：从复杂函数到拟共形几何和非线性分析

• 批准号: 0244408
• 负责人: Steffen Rohde
• 金额: $ 18.84万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244408&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Geometric; Function

FRGGeometric Function Theory is a broad area of mathematics that has its roots in the classical theory of analytic functions of one complex variable. From the very beginning this field has had connections to potential theory, partial differential equations, the calculus of variations, and geometric topology. The second half of the twentieth century brought about new areas like quasiconformal and quasiregular mappings, with links to nonlinear PDEs and harmonic analysis. The research group is planning to tackle some of the most important open problems in this broadly construed field by using our diverse strengths. Examples of the problems include understanding the integrability properties of derivatives of conformal mappings, finding criteria for recognizing metric spaces up to bi-Lipschitz or quasiconformal equivalence, further developing the theory of holomorphic curves and its quasiregular generalizations, and investigating algebraic conditions related to quasiconvexity of energy functionals. The intellectual merit of our activity will be found in a deepened understanding of fundamental questions in Geometric Function Theory, in an increase of the links to other fields of mathematics, and in a broader scope of possible applications.Core mathematics keeps reappearing outside its own realm with dramatic success and consequences. Recent examples range from cosmology (where deep topological issues arise regarding the proposed new dimensions for the universe) to material science (where deformation of elastic bodies are studied by methods of the Calculus of Variations) to engineering (where function theoretic methods have led to advances in control theory). The latter two examples are directly connected with the work of our research group, as are topological issues pertaining to the geometry of three dimensional spaces. Another new feature with possible far reaching reverberations is to use function theoretic methods in studying spaces that are not smooth in the classical sense; such spaces naturally occur when Riemannian structures degenerate and form singularities. The main strength of our group is that its members have common roots, but multifarious interests, so as to make advancement in and connections between separate fields. The broader impact of our activity will be the education of new scholars who understand the methods and techniques in the field, and who know how to find applications of their knowledge to other parts of mathematics and sciences. We will put great weight on passing on the important questions to the younger generation and on enabling them to perform independent research.

手术函数理论是数学的广泛领域，它源于一个复杂变量的分析函数的经典理论。 从一开始，这个领域就可以与潜在理论，部分微分方程，变化的计算和几何拓扑结构联系在一起。 二十世纪下半叶带来了与非线性PDES和谐波分析的链接，构成了诸如准文献和准磁映射等新领域。 研究小组计划通过使用我们的多样化的优势来解决这个广泛解释的领域中一些最重要的开放问题。 问题的示例包括理解共形映射的衍生物的可合转性，找到识别为Bi-lipschitz或准形式等效性的度量空间的标准，进一步发展了全体形态曲线及其准词的理论，并研究了与Quasiconve x quasiconve tiquasiconve tiquasiconve titersites of quasiconve coldienconsitys cormine vilitys的概括。 我们活动的智力优点将在对几何功能理论中的基本问题，增加与其他数学领域的联系以及更广泛的可能应用范围内的链接中加深理解，从而达到了巨大的成功和后果。最近的示例范围从宇宙学（有关宇宙的新维度出现的深层拓扑问题）到材料科学（通过变异的计算方法的方法研究弹性物体的变形）到工程（其中功能理论方法导致了控制理论的进步）。后两个示例与我们的研究小组的工作直接相关，与三维空间的几何形状有关的拓扑问题也是如此。另一个新功能具有遥远的回响，是在研究经典意义上不光滑的空间中使用功能理论方法。当Riemannian结构退化并形成奇异性时，这种空间自然就会发生。我们小组的主要优势在于，其成员具有共同的根源，但有多种利益，以便在单独的领域之间发展和联系。我们活动的更广泛的影响将是对了解该领域方法和技术的新学者的教育，并且知道如何在数学和科学的其他部分中找到知识的应用。我们将在将重要问题传递给年轻一代，并使他们能够进行独立研究方面放大重视。