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The notion of a homotopy flow on a directed space was introduced in [19] as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all parameter directed maps preserve the homotopy type of path spaces, such a flow (and these parameter maps) are called inessential.For a directed space, one may consider various categories whose objects are pairs of reachable points and whose morphisms may be induced by these inessential d-maps. Localization with respect to subcategories with these inessential d-maps as morphisms can be combined with a path space functor into the homotopy category; the quotient pair component category has as objects pair components along which the homotopy typeis invariant – for a coherent and transparent reason.This paper follows up [7, 14, 19] and removes some of the restrictions for their applicability. If one uses homology equivalence instead of homotopy equivalence as the basic relation, it yields an alternative to computable versions of “natural homology” introduced in [4] and elaborated in [3]. It refines, for good and for evil, the stable components introduced and investigated in [22]

Andrew J. Wiles is the recipient of the 2016 Abel Prize of the Norwegian Academy of Science and Letters. The interview was conducted by Martin Raussen And Christian Skau in Oslo on May 23, 2016, in conjunction with the Abel Prize celebration.

Louis Nirenberg is one of the recipients of the 2015 Abel Prize of the Norwegian Academy of Science and Letters.

Higher Dimensional Automata (HDA) are topological models for the studyof concurrency phenomena. The state space for an HDA is given as a pre-cubical complex in which a set of directed paths (d-paths) is singled out. The aim of this paper is to describe a general method that determines the space of directed paths with given end points in a pre-cubical complex as the nerve of a particular category.The paper generalizes the results from Raussen [19, 18] in which we had to assume that the HDA in question arises from a semaphore model. In particular, important for applications, it allows for models in which directed loops occur in the processes involved. Higher Dimensional Automata (HDA) are topological models for the studyof concurrency phenomena. The state space for an HDA is given as a pre-cubical complex in which a set of directed paths (d-paths) is singled out. The aim of this paper is to describe a general method that determines the space of directed paths with given end points in a pre-cubical complex as the nerve of a particular category.The paper generalizes the results from Raussen [19, 18] in which we had to assume that the HDA in question arises from a semaphore model. In particular, important for applications, it allows for models in which directed loops occur in the processes involved.

Higher Dimensional Automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek [26]. For a topologist, they are attractive since they can be modeled as cubical complexes - with an inbuilt restriction for directions´of allowable (d-)paths. In Raussen [25], we developed a new method describing, for a certain subclass of HDA, the homotopy type of the space of execution paths (d-paths) as a finite simplicial complex. Several restrictions that were made to ease the presentation in that latter paper will be removed in this article in order to make the results applicable in greater generality. Furthermore, we take a close look at semaphore models with semaphores all of arity one. It turns out that execution spaces for these are always homotopy discrete with components representing sets of “compatible” permutations. Finally, we describe a model for the complement of the execution space seen as a subspace of a product of spheres – with the aim to make the calculation of topological invariants easier and faster.

Directed Algebraic Topology studies topological spaces in which certain directed paths (d-paths) - in general irreversible - are singled out. The main interest concerns the spaces of directed paths between given end points - and how those vary under variation of the end points. The original motivation stems from certain models for concurrentcomputation. So far, spaces of d-paths and their topological invariants have only been determined in cases that were elementary to overlook.In this paper, we develop a systematic approach describing spaces of directed paths - up to homotopy equivalence - as prodsimplicial complexes (with products of simplices as building blocks). This method makes use of certain poset categories of binary matrices and applies to a class of directed spaces that arise from a class of models of computation; still restricted but with a fair amount of generality. In the final section, we outline a generalization to model spaces known as Higher Dimensional Automata.In particular, we describe algorithms that allow to determine not only the fundamental category of such amodel space, but all homological invariants of spaces of directed paths within it. The prodsimplical complexes and their associated chain complexes are finite, but they will, in general, have a huge number of generators. Directed Algebraic Topology studies topological spaces in which certain directed paths (d-paths) - in general irreversible - are singled out. The main interest concerns the spaces of directed paths between given end points - and how those vary under variation of the end points. The original motivation stems from certain models for concurrentcomputation. So far, spaces of d-paths and their topological invariants have only been determined in cases that were elementary to overlook.In this paper, we develop a systematic approach describing spaces of directed paths - up to homotopy equivalence - as prodsimplicial complexes (with products of simplices as building blocks). This method makes use of certain poset categories of binary matrices and applies to a class of directed spaces that arise from a class of models of computation; still restricted but with a fair amount of generality. In the final section, we outline a generalization to model spaces known as Higher Dimensional Automata.In particular, we describe algorithms that allow to determine not only the fundamental category of such amodel space, but all homological invariants of spaces of directed paths within it. The prodsimplical complexes and their associated chain complexes are finite, but they will, in general, have a huge number of generators.

Book Review: Invitaton to Topological Robotics by Michael Farber. Zurich Lectures in Advanced Mathematics, European Mathematical Society (2008), ISBN 978-3-03719-054-8 Udgivelsesdato: June Book Review: Invitaton to Topological Robotics by Michael Farber. Zurich Lectures in Advanced Mathematics, European Mathematical Society (2008), ISBN 978-3-03719-054-8

Udgivelsesdato: June

In directed algebraic topology, (spaces of) directed irreversible (d)-paths are studied from a topological and from a categorical point of view. Motivated by models for concurrent computation, we study in this paper spaces of d-paths in a pre-cubical complex. Such paths are equipped with a natural arc length which moreover is shown to be invariant under directed homotopies. D-paths up to reparametrization (called traces) can thus be represented by arc length parametrized d-paths. Under weak additional conditions,it is shown that trace spaces in a pre-cubical complex are separable metric spaces which are locally contractible and locally compact. Moreover, they have the homotopy type of a CW-complex. In directed algebraic topology, (spaces of) directed irreversible (d)-paths are studied from a topological and from a categorical point of view. Motivated by models for concurrent computation, we study in this paper spaces of d-paths in a pre-cubical complex. Such paths are equipped with a natural arc length which moreover is shown to be invariant under directed homotopies. D-paths up to reparametrization (called traces) can thus be represented by arc length parametrized d-paths. Under weak additional conditions,it is shown that trace spaces in a pre-cubical complex are separable metric spaces which are locally contractible and locally compact. Moreover, they have the homotopy type of a CW-complex.

Udgivelsesdato: March

This interview has appeared also in EMS Newsletter no. 53 The interview took place in Oslo on the 24th of May 2004 prior to the Abel prize celebrations. This interview has appeared also in EMS Newsletter no. 53 The interview took place in Oslo on the 24th of May 2004 prior to the Abel prize celebrations.

Udgivelsesdato: FEB