Spectral geometry with sub-Laplacians

In this project we consider the Heisenberg group, a classical group built from Heisenberg’s commutation relation LaTeX: XY-YX=Z $XY-YX=Z$ . The aim of the project is to study the sub-Laplacian operator LaTeX: X^2+Y^2 $X^2+Y^2$ acting on functions in three dimensions, the exotic spectral properties it displays and how it captures the geometry of the Heisenberg group.