Tools of Applied Algebraic Topology

Lead Research Organisation: Durham University
Department Name: Mathematical Sciences

Abstract

The project develops tools of algebraic topology which are relevant to various applications. We will study topological aspects of the problem of building motion planning algorithms and homotopy invariants of topological spaces which reflect the complexity of these algorithms. These invariants were computed previously for some interesting examples, and in this research we plan to resolve several remaining important challenges and also adopt the results for a number of applications. We plan to express the topological complexity TC(X) for aspherical spaces X in terms of the fundamental group of the space X hoping to find new connections with homological algebra and geometric group theory. We plan to explore further the connection between the motion planning problem and the classical problems of geometric topology (embeddings and immersions of manifolds) in the case of lens spaces hoping to find generalizations of recent results concerning real projective spaces. We also plan to study a modification of the theory of motion planning algorithms and their topological complexity relevant to the theory of concurrent computation. In this research we will also study new problems of a mixed topological - probabilistic character dealing with topological spaces depending on a large number of random parameters. The main motivation for studying such spaces comes from engineering applications involving large mechanical systems; their configuration spaces are major examples of such random topological spaces . In our previous work we studied in detail configuration spaces of mechanical linkages with large number of links and with random bar lengths. In this research we plan to investigate other mechanisms producing random topological spaces, such as configuration spaces of particles of random size, and various models based on random graphs.

Publications

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Cohen D (2011) Topology of Random 2-Complexes in Discrete & Computational Geometry

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COSTA A (2011) TOPOLOGY OF RANDOM RIGHT ANGLED ARTIN GROUPS in Journal of Topology and Analysis

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COSTA A (2011) MOTION PLANNING IN SPACES WITH SMALL FUNDAMENTAL GROUPS in Communications in Contemporary Mathematics

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FARBER M (2011) The Walker conjecture for chains in R d in Mathematical Proceedings of the Cambridge Philosophical Society

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FARBER M (2011) TELESCOPIC LINKAGES AND A TOPOLOGICAL APPROACH TO PHASE TRANSITIONS in Journal of the Australian Mathematical Society

 
Description In this research we proved a version of the Walker conjecture for configuration spaces of certain mechanisms in Euclidean spaces. This means that one may recover the metric parameters of a mechanism (up to an equivalence relation) by examining the topology of the mechanism's configuration space. This is an example of an inverse problem, similar to the question "Can you hear the shape of the drum" of Mark Kac.
Exploitation Route Our findings and the method we used to prove our results can be applied in some other interesting situations which appear in engineering and computer science.
Sectors Aerospace, Defence and Marine,Energy,Manufacturing, including Industrial Biotechology