Geometric Models of Concurrent Computations

Since the 90s, geometric models have been introduced for concurrent programs. In those, a point corresponds to a state, a path to an execution and a deformation of a path to an equivalence between executions. They are useful to analyze programs because they provide a convenient representation of their state space, on which one can use some of the well-developed tools and invariants from geometry (curvature, homology, etc.). Conversely, the study of the spaces arising as models brings new problems of purely geometric nature: most importantly, they are naturally equipped with a direction (of time), which requires adapting most usual notions. In this habilitation thesis, we present such models that we have developed and studied, as well as general techniques to do so and the results they have allowed us to obtain. Those have been applied to various notion of "concurrent programs": programs in an imperative language extended with a parallel construction and resources, but also distributed protocols, version control systems, or rewriting systems. The “geometric models” we have studied for those are also of various nature: precubical sets, directed topological spaces, generalized metric spaces, or polygraphs