Sobolev spaces in analysis and geometry

Sobolev空间理论在当代数学的许多领域中起着基础性的作用。虽然它是作为研究偏微分方程和变分问题解的存在性和正则性的工具而创建的，但应用范围远远超出了这一点。PI将专注于该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和他的研究生是包括在该项目将研究逼近凸和次调和函数，刘维定理的共形映射，索伯列夫嵌入定理在不规则域，索伯列夫空间的度量空间，逼近索伯列夫映射到度量空间，Lipschitz同伦群的海森堡群和格罗莫夫猜想的保持器连续同胚的海森堡群。然而，该项目将不限于这些主题。许多新的问题将会出现，有些问题将不得不作为调查的结果进行修改。所提出的研究处于当代数学许多领域的边缘。当成功时，它将在不同的数学领域之间建立桥梁，如分析，变分法，拓扑学，亚黎曼几何，非线性弹性甚至数值分析。该项目的一个重要部分是培训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，还将在会议和讲座上向国际听众介绍成果。

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.