23 oct 2019 -- 17:00 [open in google calendar]

Sala Seminari (Dipartimento di Matematica di Pisa)

**Abstract.**

At the beginning of the seminar, we introduce the setting of Carnot groups and, in this realm, we discuss the link between different notions of rectifiability: the general one for metric spaces by H.Federer and the ones proposed in the specific case of Carnot groups by Pauls and Franchi-Serapioni-Serra Cassano. We focus mainly on the rectifiability for sets of codimension 1 and present some known results that higlight the relationship between those notions.

In the second part of the seminar we construct a smooth non-characteristic hypersurface in a Carnot group of topological dimension 8 that is purely Pauls' unrectifiable. This shows that even very regular objects of codimension 1 that for sure are rectifiable according to the definition of Franchi-Serapioni-Serra Cassano are not so according to Pauls' definition. Finally we discuss how to prove that such an example does not exist in the Heisenberg group $\mathbb H^n$, with $n\geq 2$. That is to say, every smooth hypersurface in $\mathbb H^n$, with $n\geq 2$, is Pauls' rectifiable, even with bi-Lipschitz maps.

This is a joint work with E. Le Donne.