Abstract

The isoperimetric problem asks which sets minimize perimeter among all sets with a fixed volume. This is one of the most ancient questions in mathematics, dating back to the legend of Queen Dido. Despite the ancient origins, a turning point came with the work of De Giorgi only around the 1950s. He introduced a general definition of perimeter in the n-dimensional Euclidean space and this allowed for a more rigorous formulation of the isoperimetric problem. Later the problem has been generalized by other mathematicians to different frameworks, like Riemannian manifolds and metric spaces.

In this talk, after an historical introduction, we will at first recall the well-known isoperimetric inequality in the euclidean case, where the solution is the ball, examining in detail the notion of perimeter on which it is based. We will then introduce the Heisenberg group as the simplest and most studied example of a sub-Riemannian manifold and we will discuss the isoperimetric problem in this setting, highlighting Pansu’s conjecture which provides the candidate isoperimetric sets, known as Pansu spheres.

The video of the seminar will appear shortly afterwards in this Mediaspace channel.

Seminario Dottorato