Metric Geometry of Algebraic Singular Spaces

We consider algebraic sets (real or complex) as metric spaces with a natural inner metric, obtained from the embeddings of these sets into the affine (or projective) space. We are interested in the cases when the sets have singularities (isolated or nonisolated) and study the properties of these metrics near singular points. In particular, we consider the question of Lipschitz classification of singularities of curves and surfaces. For real surfaces we present a complete answer, but the case of complex surfaces is more difficult.