Wednesday 4 May 2016 h. 14:30, Room 2BC30

Valentina Franceschi (Dip. Mat.)

"Isoperimetric inequalities in Carnot-Caratheodory spaces"

Abstract

One of the most ancient mathematical problems is Dido's problem, appearing in Virgil's Aeneid: what is the shape to give to a rope in order to enclose a maximal region of land? The expected solution is of course the circle. Despite the ancient origins, a rigorous mathematical formulation and solution is quite recent, dating back to the 1950s when Caccioppoli and De Giorgi introduced the notion of perimeter in the n-dimensional Euclidean space. The latter notion led to the study of isoperimetric inequalities and to the solution of Dido's problem generalized to n dimensions. Mathematicians then generalized isoperimetric inequalities to different frameworks, such as riemannian manifolds and metric spaces. After an overview of the classical definitions, in this talk, we present isoperimetric inequalities in a class of metric spaces arising from the study of hypoelliptic differential operators, called Carnot-Caratheodory spaces. We conclude presenting the main conjecture in this framework (Pansu's conjecture) ad some related results.