Title:
"Rectifiability in Carnot groups"
Lecturer:
Severine Rigot (Nice)
Abstract: The notion of rectifiable sets is one of the basic concepts that plays a central role in Geometric Measure Theory. In Euclidean spaces, they were introduced in works by Besicovitch in the
plane (in the 1920's) and in general dimensions by Federer (in the late 1940's). To develop a theory of rectifiable sets in more general metric spaces has been the object of much research in the
last thirty years. In these lectures, we will focus on Carnot groups, that is, connected and simply connected stratified Lie groups, equipped with homogeneous distances. In these lectures we will
present (some aspects of) a theory of rectifiable sets based on the notion of intrinsic Lipschitz graphs together with some of its applications. We will aim at explaining the concepts that originate
from works by Franchi, Serapioni and Serra Cassano about finite perimeter sets in the Heisenberg groups (near 2000) together with new points of view inspired by recent works by Naor and Young in the
last two years.
The lectures will be divided into two parts:
PART I : The model case of codimension 1 intrinsic Lipschitz graphs and codimension 1 rectifiable sets in the Heisenberg groups (Lectures 1 and 2).
PART II : Generalization to higher codimensions and to general Carnot groups : some known results and open questions (Lectures 3 and 4).
These lectures are intended to be accessible to an audience without a priori specific knowledge about Heisenberg or Carnot groups. We shall explain the relevant notions in the course of the
lectures.