Sobolev spaces in analysis and geometry

索博列夫空间理论在当代数学的许多领域起着基础性的作用。虽然它是作为研究偏微分方程和变分问题解的存在性和规律性的工具而创建的，但其应用范围远远不止于此。PI将关注几何分析中自然出现的各种问题的理论应用。更具体地说，PI和他的研究生将包括在这个项目中，研究凸函数和次调和函数的逼近，共形映射的Liouville定理，不规则域上的Sobolev嵌入定理，度量空间上的Sobolev空间，度量空间上Sobolev映射的逼近，Heisenberg群的Lipschitz同伦群和Heisenberg群的Holder连续同纯的Gromov猜想。然而，这个项目并不局限于这些主题。许多新的问题将会出现，一些问题将不得不作为调查的结果加以修改。所提出的研究处于当代数学许多领域的边缘。一旦成功，它将在数学的不同领域之间建立桥梁，如分析、变分演算、拓扑学、亚黎曼几何、非线性弹性甚至数值分析。该项目的一个重要组成部分是培养这些领域的研究生和青年学者。PI的广泛合作将加强美国和海外研究机构的科学伙伴关系。该项目的结果将作为预印本免费提供，并将在期刊上发表。此外，研究结果将在会议和讲座中向国际听众介绍。

The theory of Sobolev spaces plays a fundamental role in many areas of contemporary mathematics. Although it was created as a tool to study existence and regularity of solutions to partial differential equations and variational problems, the scope of applications goes far beyond that. The PI will focus on a variety of applications of the theory to questions that arise in a natural way in geometric analysis. More specifically the PI and his graduate students that are included in the project will study approximation of convex and subharmonic functions, the Liouville theorem for conformal mappings, Sobolev embedding theorems in irregular domains, Sobolev spaces on metric spaces, approximation of Sobolev mappings into metric spaces, the Lipschitz homotopy groups of the Heisenberg group and the Gromov conjecture about Holder continuous homeomorphisms of the Heisenberg group. The project, however, will not be restricted to these topics. Many new questions will emerge and some questions will have to be modified as a result of the investigation.The proposed research lies on the borderline of many areas of contemporary mathematics. When successful, it will create bridges between different areas of mathematics such as analysis, calculus of variations, topology, sub-Riemannian geometry, nonlinear elasticity and even numerical analysis. An important part of the project is to train graduate students and young scholars in these areas. The wide collaboration of the PI will strengthen scientific partnerships of research institutions within the U.S. and overseas. Results of the project will be made freely available as preprints and will be published in journals. Moreover the results be presented in conferences and lectures for the international audience.