Isoperimetric Problem and Groups in Geometric Analysis

In this presentation I will focus on the isoperimetric structure of spaces with curvature bounded from below. 

The isoperimetric profile $I$ is the function that associates to each volume the infimum of the perimeter of sets with that volume. I will describe the background, the challenges, and the main consequences of the following result: \textit{Let $n\geq 2$. On a nonnegatively Ricci curved $n$-dimensional smooth complete Riemannian manifold, $I^{\frac{n}{n-1}}$ is concave.}

In the compact case the proof uses the existence of isoperimetric regions, and the first, and second variation of the surface area. In the noncompact case isoperimetric regions might fail to exist for some volumes and might escape at infinity in isoperimetric sets in possibly nonsmooth spaces. Here, nonsmooth geometry naturally enters into play and tools from Analysis in metric spaces are needed. Finally, I will mention open questions and possible future applications of the discussed results.

Moreover, time allowing, I will survey a second line of research concerning rectifiability in the metric setting, especially on Carnot groups. First, I will briefly introduce Carnot groups. Then I will discuss an extension of Preiss' Theorem in the first Heisenberg group, and some questions concerning the problem of characterizing metric spaces with unique tangents almost everywhere.