In the first half of the talk I will give a very compressed introduction into parts of Theoretical Distributed Computing from the point of view of mathematician. I will describe how to construct simplicial models whose combinatorics contains important information about computability and complexity of standard distributed tasks. In the second part, I will outline our recent progress on estimating the complexity of the so-called Weak Symmetry Breaking task, where we are able to derive some quite surprising results. At the technical core of our argument we need to construct complete matchings on specific graphs associated to the so-called iterated standard chromatic subdivision of a simplex. The talk is based on our monograph "Distributed Computing through Combinatorial Topology" (with Herlihy and Rajsbaum), as well as on a recent series of preprints.