Lipschitz Graphs in Carnot Groups

Submanifolds with intrinsic Lipschitz regularity in Carnot groups (i.e.,
stratified groups endowed with a sub-Riemannian structure) can be
introduced using the theory of intrinsic Lipschitz graphs started years
ago by B. Franchi, R. Serapioni and F. Serra Cassano. One of the main
related questions concerns a Rademacher-type theorem about the almost
everywhere existence of a tangent plane to intrinsic Lipschitz graphs:
after a gentle introduction to the topic, I will discuss a positive
solution to the problem in Heisenberg groups. The proof uses the
language of currents in Heisenberg groups (in particular, a version of
the Constancy Theorem) and a number of complementary results such as
extension and smooth approximation theorems for intrinsic Lipschitz
graphs. I will also show a recent example, joint with A. Julia and S.
Nicolussi Golo, of an intrinsic Lipschitz graph in a Carnot group that
is nowhere intrinsically differentiable. The talk will be kept at an
introductory level.