Dual complex of resolution of singularities

Dual complex encodes the combinatorics of intersections of the components of exceptional divisor in a resolution of singularities and generalizes to higher dimensions the notion of dual graph of a resolution of a surface singularity. We shall survey some recent results on the dual complex. We shall try to cover the following topics: the proof of homotopy invariance of the dual complex based on the factorization theorem for birational maps and on the theory of Berkovich spaces; calculation of the dual complex of hypersurface singularities with a help of embedded toric resolution; restrictions on the dual complex coming from Hodge theory (dual complex of a complement to an affine set, results of Arapura–Bakhtary–Wlodarzcyk); Payne's example of a rational singularity with the dual complex homeomorphic to $\mathbb RP^2$; Kollar's results on the realizability of any simplicial complex as the dual complex of an isolated singularity.