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

FRG几何函数论是一个广泛的数学领域，其根源在于一个复变量的解析函数的经典理论。 从一开始这个领域已经连接到潜在的理论，偏微分方程，变分法，和几何拓扑。 世纪的后半叶带来了新的领域，如拟共形和拟正则映射，与非线性偏微分方程和调和分析。 该研究小组正计划通过利用我们的各种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。 这些问题的例子包括了解共形映射的导数的可积性，寻找识别度量空间到双Lipschitz或拟共形等价的准则，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。 我们的活动的智力价值将被发现在几何函数理论的基本问题的加深理解，在增加的联系到数学的其他领域，并在更广泛的范围内可能的应用.核心数学不断出现在自己的领域外与戏剧性的成功和后果.最近的例子从宇宙学（其中出现了关于宇宙的新维度的深层拓扑问题）到材料科学（其中弹性体的变形通过变分法进行研究）到工程学（其中函数论方法导致了控制理论的进步）。后两个例子与我们的研究小组的工作直接相关，就像与三维空间几何有关的拓扑问题一样。另一个新的功能与可能深远的反响是使用函数理论的方法在研究空间是不光滑的经典意义上，这样的空间自然发生时，黎曼结构退化，形成奇点。本集团的主要优势在于其成员拥有共同的根源，但兴趣多样，以便在不同的领域中取得进步并建立联系。我们的活动的更广泛的影响将是谁了解该领域的方法和技术的新学者的教育，谁知道如何找到他们的知识应用到数学和科学的其他部分。我们将非常重视将重要问题传递给年轻一代，并使他们能够进行独立的研究。

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.