The aim of this seminar is to present some results about minimal bubble clusters in some sub-Riemannian spaces. This amounts to find the best configuration of $min mathbb N$ regions in a manifold enclosing given volumes, in order to minimize their total perimeter. In a $n$-dimensional sub-Riemannian manifold, the perimeter is a non-isotropic $(n-1)$-dimensional measure that is defined according to the geometry. After an introduction to the subject, we will present some results concerning the cases $m=1$ (isoperimetric problem) and $m=2$ (double bubble problem), in a class of sub-Riemannian structures connected to the Heisenberg geometry. This is the framework of an open problem about the shape of isoperimetric sets, known as Pansu’s conjecture. We start by presenting the isoperimetric problem in Grushin spaces and Heisenberg type groups, under a symmetry assumption that depends on the dimension (based on joint work with R. Monti, University of Padova). We conclude by showing some recent results in collaboration with Giorgio Stefani (SNS, Pisa) concerning the double bubble problem in the Grushin plane.

Jan Derezinsky

(Université de Varsovie)

Abstract: First I will describe a certain natural holomorphic family of closed operators with interesting spectral properties. These operators can be fully analyzed using just trigonometric functions. Then I will discuss one- dimensional Schrödinger operators with inverse square potential and general boundary conditions, which I studied recently with S.Richard. Even though their description involves Bessel and Gamma functions, they turn out to be equivalent to the previous family.

Some operators that I will describe are homogeneous – they get multiplied by a constant after a change of the scale. In general, their homogeneity is weakly broken-scaling and induces a simple but nontrivial ow in the parameter space. One can say (with some exaggeration) that they can be viewed as « toy models of the renormalization group ».

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