Joint Colloquium: Le Donne -- Mathematical appearances of sub-Riemannian geometries

Sub-Riemannian geometries are a generalization of Riemannian geometries.  Roughly
speaking, in order to measure distances in a sub-Riemannian manifold, one is allowed to
only measure distances along curves that are tangent to some subspace of the tangent
space.  These geometries arise in many areas of pure and applied mathematics (such as
algebra, geometry, analysis, mechanics, control theory, mathematical physics,
theoretical computer science), as well as in applications (e.g., robotics, vision).
This talk introduces sub-Riemannian geometry from the metric viewpoint and focus on a
few classical situations in pure mathematics where sub-Riemannian geometries appear.
For example, we shall discuss boundaries of rank-one symmetric spaces and asymptotic
cones of nilpotent groups.  The goal is to present several metric characterizations of
sub-Riemannian geometries so to give an explanation of their natural manifestation.  We
first give a characterization of Carnot groups, which are very special sub-Riemannian
geometries.  We extend the result to self-similar metric Lie groups (in collaboration
with Cowling, Kivioja, Nicolussi Golo, and Ottazzi).  We then present some recent
results characterizing boundaries of rank-one symmetric spaces (in collaboration with
Freeman).