‘Geometry and Concurrency’ is not (yet) a well-established domain of research, but consists rather of a collection of seemingly related techniques, algorithms and formalizations, coming from different application areas, and accumulated over a long period of time. There is currently a certain amount of effort being made to unify these, following the workshop ‘New Connections between Computer Science and Mathematics’ held at the Newton Institute in Cambridge, England in November 1995 (and sponsored by HP/BRIMS). More recently, the first workshop on the same subject was held in Aalborg, Denmark (see http://www.math.auc.dk/˜raussen/admin/workshop/workshop.html) where the articles of this issue, among others, were first sketched.

As a starter, and to put in context the papers by Lisbeth Fajstrup, Philippe Gaucher, Maurice Herlihy, Vaughan Pratt, Sergio Rajsbaum and Martin Raussen that appear later in this issue, I have written a brief historical note, or survey, which follows and naturally completes this foreword.

Last and not least, I must express my gratitude to Giuseppe Longo, who gave me the opportunity to act as a guest editor for this special issue, and who very kindly helped me through the many different steps that editing necessitates. My very warm thanks also go to the authors who did a very nice job, in a relatively short time. Finally, I wish to thank the referees of this issue, who found ‘bugs’ that I would not have found, and who suggested very useful improvements to the authors. The referees were Nir Shavit, Stefan Sokolowski, Christophe Tabacznyj, Jean Goubault-Larrecq, Arnaud Venet, Ian Mackie (who helped me with the survey article), Ronnie Brown and Tim Porter, as well as some of the authors who cross-checked other authors' articles, Philippe Gaucher, Martin Raussen and Lisbeth Fajstrup.

Geometrical methods in concurrency theory (and in distributed systems theory) have appeared recently for modelling and analyzing the behaviour of systems and also for solving computability and complexity issues. We identify some of the main directions of research, and survey some of the major ideas on which all this is based (some of which are now more than thirty years old).

This paper focusses on the determination of the dihomotopy classes of dipaths in cubical complexes, representing the essentially different computations in a given concurrent program due to different schedules. Several new notions have to be developed, for example, the domains of dependence (Definition 2.4), which are borrowed from relativity theory (Penrose 1972). However, it turns out that the algorithm determining deadlocks and unsafe regions described in Fajstrup et al. (1998a; 1998b) can be modified and applied to do the essential calculational work.

In Fajstrup et al. (1998a) the authors proposed a fast algorithm for deadlock detection, which was based on a geometric model of concurrency. We propose here an extension of this approach to deal with recursive processes (loops) and branchings.

The strict globular ω-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ω-category [Cscr ] three homology theories. The first one is called the globular homology. It contains the oriented loops of [Cscr ]. The two other ones are called the negative (respectively, positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (respectively, the merging areas of execution paths or positive corners) of [Cscr ]. Two natural linear maps called the negative (respectively, the positive) Hurewicz morphism from the globular homology to the negative (respectively, positive) corner homology are constructed. We explain the reason why these constructions allow the reinterpretation of some geometric problems coming from computer science.

The dual of a true concurrency schedule appears to be a false concurrency automaton, a paradox we resolved in a previous paper by extending the latter to higher dimensions. This extension may be formalized via such discrete geometries as n-categories, simplicial complexes, cubical complexes and Chu spaces. We advocate the last as having a clear notion of event, a well-defined process algebra uniformly extending that for event structures, and ease of extension beyond the basic before-during-after analysis.

Topological methods have yielded a variety of lower bounds and impossibility results for distributed computing. In this paper, we introduce a new tool for proving impossibility results, which is based on a core theorem of algebraic topology, the acyclic carrier theorem, and unifies, generalizes and extends earlier results.