Sobolev spaces in analysis and geometry

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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.