Algebraic topology, which started out as an approach to problems in physics, then developed as a curiosity driven area for almost a century. The last 15-20 years, a number of new applications have had the benefit of this very well developed mathematical area. The focus in this talk is on new models for concurrency, and how this has sparked new mathematical problems, in fact a new area, directed topology, while also giving rise to algorithms and theoretical insight on the computer science side. There will be an introduction to the model and the mathematics and examples of applications. In particular symmetric problems, where a thread runs in parallel with itself - so called cut-off problems.