Isoperimetric inequalities for subriemannian analogues of the Gaussian measure

We will study isoperimetric inequalities for a family of exponential power type probability measures defined on a class of stratified Lie groups. We begin by introducing the Heisenberg group, the simplest example of such a group. Depending on time, we discuss various objects of interest, such as vector fields, gradients, laplacians, and norms, and also provide some tools, such as functional inequalities, emphasising the similarities and differences between these objects and their euclidean counterparts. In the second part, we will study three probability measures, namely the heat kernel measure, a Gaussian measure where the Euclidean norm is replaced by the subriemannian metric, and lastly when the Euclidean norm is replaced by a smooth metric.