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

几何函数论是一个广泛的数学领域，它植根于经典的单复变量解析函数理论。从一开始，这个领域就与位势理论、偏微分方程、变分和几何拓扑学有联系。二十世纪下半叶出现了拟共形和拟正则映射等新领域，它们与非线性偏微分方程组和调和分析有联系。该研究小组正计划通过利用我们的不同优势来解决这个被广泛解释的领域中的一些最重要的公开问题。这些问题的例子包括了解共形映射导数的可积性，找到双Lipschitz或拟共形等价的度量空间的识别准则，进一步发展全纯曲线理论及其拟正则推广，以及研究与能量泛函的拟凸性有关的代数条件。我们活动的智力价值在于加深了对几何函数论基本问题的理解，增加了与其他数学领域的联系，以及更广泛的可能应用范围。核心数学不断地出现在自己的领域之外，取得了戏剧性的成功和后果。最近的例子从宇宙学(关于所提出的宇宙的新维度出现了深层次的拓扑问题)到材料科学(通过变分演算方法研究弹性物体的变形)到工程学(其中函数论方法导致了控制理论的进步)。后两个例子与我们研究组的工作直接相关，与三维空间几何有关的拓扑问题也是如此。另一个可能产生深远影响的新特征是使用函数论方法研究经典意义上不光滑的空间；当黎曼结构退化并形成奇点时，这种空间自然出现。我们集团的主要优势是成员有共同的根基，但利益多样，从而在不同领域之间取得进展和联系。我们活动的更广泛的影响将是教育新的学者，他们了解该领域的方法和技术，并知道如何将他们的知识应用于数学和科学的其他部分。我们将非常重视把重要的问题传递给年轻一代，让他们能够独立研究。

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.