Geometric Function Theory: Analysis, Geometry and Topology

This project explores fundamental problems in Geometric Function Theory, focusing on mappings and functions with limited differentiability, such as convex functions, Sobolev functions, and Lipschitz and Hölder continuous mappings. Geometric Function Theory has its roots in classical complex analysis and quasiconformal mappings, but over the past decades it has evolved into a rich and modern field with deep connections to other areas, including convex analysis, analysis on metric spaces and Heisenberg groups, contact and symplectic geometry, and geometric measure theory. This broadening has been driven by the need to understand nonlinear phenomena and low-regularity structures in both pure mathematics and applied sciences. These types of maps are essential in modeling irregular behavior, where classical smooth tools fail. By studying their analytic, geometric, and topological behavior, the project seeks to uncover new mathematical principles that improve our understanding of irregular structures. The research is structured around 21 well-defined objectives, most of which are formulated as precise mathematical conjectures with definitive yes-or-no answers. These investigations aim to generate new directions in geometric analysis and topology while contributing to the broader mathematical community. The research is expected to support the national interest by advancing mathematical knowledge, training students, and providing tools applicable to areas that depend on the analysis of non-smooth structures. The investigator studies several interconnected areas of research. These include: (1) Lusin approximation and rectifiability questions in convex analysis; (2) regularity of homeomorphisms in Euclidean spaces, including the study of the sign of the Jacobian; (3) Guth's conjecture about the homotopy theory of continuously differentiable maps whose derivatives have low rank; (4) Gromov's conjecture about Hölder continuous embeddings into the Heisenberg group and related questions about the topology of Lipschitz and Hölder continuous maps in the Heisenberg group; (5) Sobolev extension domains; (6) analysis on metric spaces and the geometric measure theory of Lipschitz mappings into metric spaces. The project combines analytic, geometric, and topological methods to address both longstanding open problems and newly formulated questions. The anticipated outcomes include theoretical advances, publication of results in leading journals, and training of graduate students in cutting-edge mathematics. 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.