Geometric Structures on 3-manifolds

I will talk about bilipschitz geometry of complex algebraic sets, focusing on the local geometry in dimension 2 (complex surface singularities), where the topological classification has long been understood in terms of 3-manifolds, while the analytic classification, even in this low dimension, is still very much out of reach. Bilipschitz geometry lies between the extremes of topological and analytic type. Although it leads to a "tame" classification (as first conjectured by Siebenmann and Sullivan in the 70's and proved by Mostowski in the mid-eighties), it is only in the last decade that the richness of the bilipschitz classification has become evident. I will describe work of Lev Birbrair, Anne Pichon and myself, on bilipschitz classification of complex surface germs in terms of refined JSJ decomposition of their 3-manifold links plus associated rational invariants, and if time permits, also the relationship of the classification with Zariski equisingularity, which was Zariski's approach to a tame classification.