Geometry, Topology and Dynamics Seminar
Marshall Williams
UIC
Metric currents in Heisenberg groups
Abstract: Recently, Ambrosio and Kirchheim defined currents in metric spaces. This generalized Federer and Fleming's theory of Normal and Integral currents, used to formulate and study variational problems in Euclidean space. In particular, Ambrosio and Kirchheim proved that metric currents satisfy the closure and compactness theorems of Federer and Fleming, and that in Euclidean spaces, the metric and classical theories are equivalent.
I will discuss some results concerning metric currents in Heisenberg groups (and briefly, other Carnot Groups), equipped with their Carnot-Carathéodory metrics. These spaces, which are highly fractal, provide a fertile testing ground for generalizations of classical theories to more general settings. It will turn out that in low dimensions, we obtain a reasonable theory, whereas in higher dimensions, the theory becomes vacuous.
Monday October 19, 2009 at 3:00 PM in SEO 612