Given a variety X, a closed subvariety Y, and a resolution of the pair (X,Y), one may construct a simplicial complex from the combinatorics of the exceptional divisors of the resolution; this is called the dual complex of the resolution. The dual complex (and other related constructions) provide useful tools to study the singularities of the subvariety Y; however, it depends on the choice of resolution. In 2007, Thuillier proved that the homotopy type of the dual complex is independent of the choice of the resolution (provided we work over a perfect field), generalizing a theorem of Stepanov. The proof relies crucially on the theory of Berkovich spaces over a trivially-valued field. Our goal is to introduce the dual complex of a resolution, and to explain some of the ideas in Thuillier's proof. Speaker(s): Matt Stevenson (UM)