The objects of study in directed algebraic topology are topological spaces with a selected subset of the continuous paths, the dipaths. A prominent example is Euclidean space where the dipaths are paths increasing in all coordinates. Other examples are products of directed graphs.

Questions asked (and to a certain extent answered) are: What is the space of dipaths? How many dipaths are there up to directed homotopy equivalence in a given space? When do two spaces have equivalent spaces of dipaths? What are good combinatorial models of spaces and of spaces of dipaths? Are there combinatorial equivalences of directed spaces directed collapsing and what should they preserve? The tools used are built to fit based on well-known ones coming from algebraic topology, combinatorial topology, category theory.

Directed algebraic topology emerged around 1996 as a possible answer to questions in concurrent computing (L.Fajstrup, E. Goubault, M. Raussen and independently M. Grandis). When several programs are executed concurrently/in parallel, the model for a solo execution does not suffice, since interaction between the executions should be taken into account. One model of concurrency, the PV-model, focuses on shared resources and the way these are accessed by the executions: who gets them first and how many can own them at the same time. A geometric model of the situation goes back to E.W. Dijkstra. In this model, an execution is a continuous directed path and executions are equivalent if the corresponding paths can be deformed into each other through directed paths they are directed homotopic. Classification of executions, verification of programs, equivalence of programs and many other questions now have a directed topology analogue and, in good cases, an answer. The lectures will give an introduction to the methods and problems of directed algebraic topology and the questions in concurrent computing which are behind.