The Heisenberg group with its sub-Riemannian metric comes up in multiple contexts involving asymptotic geometry and exhibits essential non-Euclidean and fractal behavior that stands in the way of generalizing standard methods from Euclidean geometry and analysis. On the other hand, as a 3-dimensional space with a large isometry group, its geometry retains enough analogs of Euclidean concepts for the Heisenberg group serve as a gateway to understanding the geometry of broader classes of metric spaces.

In this talk, I will describe the Heisenberg group, provide some motivation for its study, and then provide some examples of the above principle of using the Heisenberg group as a gateway to metric geometry. In particular, I will provide results from the well-developed field of analysis on metric spaces and some more recent developments in discrete geometry and number theory on the Heisenberg group.

Note: this is a practice job talk. Everyone is welcome, and feedback would be appreciated. Speaker(s): Anton Lukyanenko (UM)