[Winetalk.com]

PD, you don't get it because you have not
studied singular semialgebraic, subanalytic and complex analytic sets from a metric viewpont. Each connected set of these types can be considered as a metric space with different natural metrics. The most interesting metrics are induced (euclidian) metric and intrinsic metric. A set is called normally embedded if these two metrics are bi-Lipschitz equivalent. The main result of the talk is the following : Every closed semialgebraic set is intrinsically bi-Lipschitz equivalent to some normally embedded semialgebraic set. This result does not admit a complex analog. We present an example of a complex analytic set which is not intrinsically bi-Lipschitz equivalent to any normally embedded complex analytic set.

Do you see how simple this chart really is???