Lipschitz geometry of complex surfaces

Lipschitz geometry is a branch of singularity theory that studies a complex analytic germ (X,0) in (C^n,0) by equipping it with either one of two metrics: its outer metric, induced by the euclidean metric of the ambient space, and its inner metric, given by measuring the length of arcs on (X,0). Whenever those two metrics are equivalent up to a bi-Lipschitz homeomorphism, the germ is said to be Lipschitz normally embedded (LNE).  I will give an overview of several results obtained together with  André Belotto, András Némethi, Walter Neumann, Helge Pedersen, Anne Pichon, and Bernd Schober on the Lipschitz geometry of surfaces, and more precisely on their inner metric structure, properties of LNE surfaces, criteria to prove that a germ is LNE, and the so-called problem of polar exploration, which is the quest of determining the generic polar curves of a complex surface from its topology.