Recent 'Geometric (Analyst's) Traveling Salesman' Theorems.

Given a set $K$ in a metric space $M$ one may ask when is $K$ contained in
$\Gamma$, a connected set of finite 1-dimensional Hausdorff length, and
for estimates on the minimal length of such a $\Gamma$.  This was first
answered for $M$ = the Euclidean plane by P. Jones and extended to $M=R^d$
by K. Okikiolu.  Recently, there have been several new relevant results
(by several people) for $M$ being: a Hilbert space, the Heisenberg group,
and a general metric space.  for some of these one restricts the
discussion to $K$ and $\Gamma$ in specific categories. In some of these
categories which we will discuss the anwswer is in IFF form, whereas in
others it is not. The answer to this question usually comes together with
a multiresolutional analysis of the set $K$ and a construciton of a
$\Gamma$ containing $K$ which is not 'too long'. Essentially no prior
knowledge in analysis is needed to understand this talk.