The local topology of isolated complex surface singularites is long understood, as cones on closed 3-manifolds obtained by negative definite plumbing. On the other hand a full understanding of the analytic types is out of reach, motivating Zariski´s efforts into the 1980´s to give a good concept of "equisingularity" for families of singularities.

The significance of Lipschitz geometry as a tool in singularity theory is a recent insight, starting (in complex dimension 2) with examples of Birbrair and Fernandes published in 2008. I will describe work with Birbrair and Pichon on classifying this geometry in terms of discrete data associated with a refined JSJ decomposition of the associated 3-manifold link. Also work with Anne Pichon proving that Zariski equisingularity in this dimension (and lower) is equivalent to constant Lipschitz geometry.