# Friday Febuary 28 , 1:00 pm, Carnegie 122

The Heisenberg groups are metric spaces with a lot of explicit structure, but which exhibit fractal-like qualities making them highly non-Euclidean. A calculus may be developed in this setting and as such they form excellent test cases for the more general program of analysis on metric spaces. I will introduce the first Heisenberg group, its Lie algebra structure, and the way this structure manifests as a 'horizontal' distribution to which we associate a Carnot-Caratheodory metric and various objects of the calculus. Next I will introduce the Pansu derivative, and time permitting, explain Semmes' observation that Pansu's Rademacher type theorem shows us just how non-Euclidean these spaces are.