This project investigates a metric space that behaves very differently from the familiar Euclidean space: in the sub-Riemannian Heisenberg group H, a line segment can have infinite length, and translations do not commute. The resulting geometry is well-suited to model constrained motion and it has intriguing connections to the theory of subelliptic partial differential equations (PDE). The objective of the project is twofold. The first part aims to promote a particular branch of mathematical analysis, namely a theory of quantitative rectifiability, in the setting of H. New tools will be developed to study the regularity of surface-like sets in H. The second goal is to apply these tools to gain information about boundaries of sets (i) on which a certain PDE can be solved with rough boundary data, or (ii) which arise as perimeter minimizers in an isoperimetric problem on H. The project involves international collaboration with researchers at the Universities of Connecticut and Padova. |