Elastic deformations, which describe how materials stretch, compress, fold, and tear, are central to many applied sciences, including materials science, and engineering. Understanding these processes requires not only physical modeling but also sophisticated mathematical theory. This project develops new tools and refines existing methods to address longstanding challenges in the field. Specifically, the project analyzes the mathematical foundations of elastic deformations through the framework of Sobolev homeomorphisms. These mappings serve as natural models for nonlinear elasticity, though energy minimizers often fall outside this class. The research addresses this challenge by examining the limits of energy-minimizing homeomorphisms and focusing on the inner-variational equation. Key objectives include understanding the weak and strong closures of Sobolev homeomorphisms and developing Sobolev analogues of topological results such as the Jordan–Schönflies theorem. The project also advances a new theory of quasiregular values, providing a pointwise characterization of quasiregularity and uncovering new links to classical results. New topological arguments grow into decisive components of Geometric Function Theory and Nonlinear Elasticity. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.