Applied and Computational Algebraic Topology (ACAT) - European ...

• The motion planning problem plays aprominent role in modern robotics. Anautonomous mechanical system must beable to select a motion once the currentand the desired states are given; such aselection is made by a motion planningalgorithm. Continuous motion planningalgorithms rarely exist, which explainswhy decisions are often discontinuousas functions of the input data. The notionof topological complexity measuresthese discontinuities numerically. Manyproperties of this notion are known, but itscomputation in general is quite difficult; asituation similar to the related Lusternik-Schnirelmann category.• We plan to apply the theory of motionplanning algorithms in the context ofdirected topological spaces when onlydirected paths between the source and thetarget are allowed. This theory would thenbe applicable to problems of concurrentcomputation, as discussed below. We alsoplan to create appropriate cohomologicaltools for estimating the sectional categoryof fibrations. This will involve strengtheningand generalising the technique of categoryweight of cohomology classes and usingcohomology operations, as suggested byFadell and Husseini in the context of theLusternik-Schnirelmann category.Stochastic TopologyIn applications with large mechanicalsystems, the traditional concept of aconfiguration space is unfortunatelyinadequate. For a mechanical system ofgreat complexity, it is unrealistic to assumethat its configuration space can be fullyknown or completely described. It is morereasonable to assume that the space of allpossible states of such a system can beunderstood only approximately, or that itis described using probabilistic methods.Similar problems arise in modeling oflarge financial, biological and ecologicalsystems. This motivates the study ofrandom manifolds and random simplicialcomplexes as models for large systems.We continue with a number of specialisedtopics in the area:• Recent results about topology of linkageswith random length parameters showthat despite limited information one maypredict the outcome topology, say, the Bettinumbers, with surprising precision. Thishappens in situations when the systemdepends on many independent randomparameters, similar to the classical centrallimit theorem.• We plan to study models that producehigh-dimensional random complexes(generalising the well-developed theoryof random graphs) and investigate theirapplications in engineering and computerscience. This includes the Linial-Meshulammodel which has been studied extensivelyin recent years.Figure 2. Underwater robot with sense of touch.© DFKI BremenACAT • 5