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

More documents

Recommendations

Info

Computational Topology, an Introduction,by H. Edelsbrunner & J.L Harer, AmericanMathematical Society, 2010).The development underpins the needfor fast algorithms to compute homologyfor data sets of a few million elements ormore, which arise in diverse areas suchas image analysis, dynamical systems,robotics, electromagnetic engineering,and material science. Similar methods areapplied in the classification of shapes andtheir retrieval from databases. An essentialingredient in this undertaking is a measureof dissimilarity between shapes. If weenhance shapes by functions on them, wecan use the pseudo-distance that can beapproximated by persistent homology. Wecontinue by listing a few of the questions tobe investigated by members of this project:• Well modules have recently beenintroduced to measure the robustness ofintersections and to prove the stability ofcontours of mappings. They can also beused to formulate a notion of robustness forfixed points of mappings. Can these ideasbe further developed to obtain a notion ofpersistence for dynamical systems? On arelated note, we need a notion of persistenthomology for maps, and fast algorithms.• Image processing raises a number ofchallenges to our understanding of ourtopological tools. How do we use persistenthomology under partial information, suchas partially occluded shapes?• Filtrations are generated by real-valuedfunctions on the space. Replacing thefunctions by vector-valued mappingsgives multi-parameter filtrations. A deeperunderstanding of the rank invariant of thesemore general filtrations is important forapplications in shape analysis and retrieval.• Homology algorithms can be extendedto computing cohomology, which isparticularly important in electromagneticengineering. The running timecharacteristics of the cohomology variantsare likely to be different from the homologyalgorithms, and this difference needs to beexplored.• Matrix reduction algorithms lendthemselves to implementations on parallelcomputer architectures, which promise afurther increase in efficiency. We need tounderstand which reduction algorithms arebest suited for this effort.Topological RoboticsThis mathematical discipline studiestopological problems relevant to practicalapplications in modern robotics,engineering and computer science. Withany mechanical system, one associatesthe configuration space, which encodes alladmissible configurations of the system.Many important engineering questionsabout the system reduce to geometricquestions about the configuration space.For instance, the connectivity of theconfiguration space means that themechanical system is fully controllable.In other words, we can bring the systemfrom any initial state to any desiredstate by a continuous motion. Curiously,the interaction between topology andmechanical engineering is bi-directionalbecause any smooth manifold can berealised as configuration space of amechanical system. We continue byoutlining a few broad topics within the area.• Configuration spaces of simple linkagesrepresent an interesting class of closedsmooth manifolds. These remarkablespaces are also known as polygon spacesbecause they parameterise the spaceof all n-sided polygons with given sidelengths. In the last few years significantprogress has been made in classification ofconfiguration spaces of linkages leading toa solution of the Walker conjecture, whichis a question about the invertibility of themapping from a linking to its configurationspace.4 • ACAT
• 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
Page 1 and 2: Research Networking ProgrammeApplie
Page 3: Scientific BackgroundFigure 1.Two i
Page 7 and 8: Fundingthese are most often not inv

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

Create successful ePaper yourself

Delete template?

Save as template?