# Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups

Main funder

Funder's project number: 352649

Funds granted by main funder (€)

- 159 190,00

Funding program

Project timetable

Project start date: 01/09/2022

Project end date: 10/02/2025

Summary

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.