Geometry of complex surface singularities

A complex variety has two intrinsic metric space structures in neighborhood of any point (“inner” and “outer” metric) which are uniquely determined from the complex structure up to bilipschitz change of the metric (changing distances by at most a constant factor). In dimension 1 the inner metric (given by minimal arc­length within the variety) carries no interesting information, and it is only very recently, starting with a 2008 paper [1] of Birbrair and Fernandes, that it has become clear how rich metric information is in higher dimensions. Inner metric in dimension 2 is now very well understood through work of Birbrair, Pichon and the speaker [2]. The talk will give an overview of this work, and, given time, describe some results of the speaker and Pichon about outer metric [3].