Sobolev spaces in analysis and geometry

索波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解的存在性和正则性以及变分问题的工具而创建的，但它的应用范围远远不止于此。PI将专注于理论在几何分析中以自然方式出现的问题的各种应用。更具体地说，PI和他的研究生将学习凸函数和次调和函数的逼近，保形映射的刘维尔定理，非规则区域中的Sobolev嵌入定理，度量空间上的Sobolev空间，Sobolev映射到度量空间的逼近，Heisenberg群的Lipschitz同伦群，以及关于Heisenberg群的Holder连续同胚的Gromov猜想。然而，该项目不会局限于这些主题。作为调查的结果，许多新的问题将会出现，一些问题将不得不修改。拟议的研究处于当代数学的许多领域的边缘。如果成功，它将在不同的数学领域之间建立桥梁，如分析、变分、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。国际和平研究所的广泛合作将加强美国和海外研究机构的科学合作伙伴关系。该项目的成果将以预印本的形式免费提供，并将在期刊上发表。此外，还将在国际听众的会议和讲座中介绍成果。

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.