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

More documents

Recommendations

Info

• A recent and dynamic branch ofcombinatorial algebraic topology alsobrings probability aspects into play. Therehave been several developments studyingprobability spaces arising naturally intopology, and establishing thresholdsfor non-triviality of various algebraicinvariants. The idea is to incorporateinto the computational model not justthe final data sample, but the samplingprocess itself. These methods giveglobal quality assessment invariants forspecific computations in terms of tools ofprobability theory.• One hopes that it will be possibleto answer some famous outstandingopen problems about two-dimensionalcomplexes, such as the Whitehead orthe Eilenberg-Ganea conjectures usingprobabilistic tools. The methods involvingprobability were successfully used inthe past in other areas of mathematicsto construct objects with curiouscombinations of properties.Combinatorial Algebraic Topologyand Concurrency TheoryThe idea of combinatorial algebraictopology is to form complexes thatrepresent collections of configurations,for example the set of all colourings ofa graph, or the set of all executions of aprotocol. The complexes are typically highdimensionaland have a high degree ofsymmetry. These are expressed via actionsof the symmetric group or of a groupcomposed of subgroups of symmetric orrelated groups. Effective computationsof algebraic invariants exploiting thesesymmetries are still a challenge that willbe pursued within this project. There hasalready been some work exploiting toolsfrom the combinatorial context, such asdiscrete Morse theory, to the calculationof persistent homology, as well as of somerelated invariants. Possibilities for furtherFigure 3.State space in the form of a torus with rectangularholes. © CEA-LIST.connections in this direction are muchgreater than has been explored until now.Equivariant methods have been usefulin connecting combinatorial algebraictopology with applications in computerscience, in particular the feasibilityof distributed systems. This topic isembedded in the larger field of concurrencytheory within theoretical computer science.This field investigates the challengesrepresented by parallel architectures withinan individual computer or within computernetworks; in particular for the assessmentof the correctness and safety of nonsequentialdistributed algorithms.A particular concurrency model, thehigher-dimensional automata, can bedescribed by pre-cubical complexes witha direction reflecting the time flow. For amathematical analysis of these models,one has to incorporate direction into toolsand methods from algebraic topology. Thisis the topic of directed algebraic topology,which uses fast homology algorithms toanalyse the large models arising in practicalapplications. In a directed topologicalspace, a subcategory of the pathcategory is singled out as the (allowable)directed paths. It is important to note that6 • ACAT
Fundingthese are most often not invertible. Forapplications, the topological state spacereflects coordination constraints betweenindividual processes, and directedness isa property of the time flow. The main aim isto understand the properties of (directed)path spaces associated to a well-structureddirected topological space, to performcalculations of standard invariants, and toinvestigate the sensitivity of these invariantswith respect to the chosen end points.Directed paths in the same componentmodel computation schedules that willalways yield the same result. We mention afew particular questions in the area.• For geometric models of computation,abstract homotopy theory tools yieldmodels for associated spaces of directedpaths in a combinatorial form, i.e. assimplicial complexes. Ongoing workaims to develop this theoretical methodinto algorithms for applications allowingmachine calculations of their homologygroups.• It is desirable to decompose a givendirected space into components such thatthe homotopy types of path spaces onlydepend on the components of start andend point. If finitely (or countably) manysuch components suffice, it is possibleto describe coarser models that can beused by a machine. The existing theoryand algorithmic methods apply only toa restricted class of model spaces andshould be extended to more general andrealistic settings. A related question isthe application of persistence to possiblyunderstand the hierarchical structure ofsuch decompositions.• The literature contains a variety ofsuggestions for a directed replacement ofthe notion homotopy equivalence. We willinvestigate their properties and single outwhich of them are most suitable in theoryand in applications.ESF Research Networking Programmesare principally funded by the Foundation’sMember Organisations on an à la cartebasis. ACAT is supported by:• Fonds zur Förderung derwissenschaftlichen Forschungin Österreich (FWF)Austrian Science Fund, Austria• Det Frie Forskningsråd (DFF)The Danish Council for IndependentResearch, Denmark• Aalborg Universitet, Denmark• Deutsche Forschungsgemeinschaft(DFG)German Research Foundation, Germany• Centre National de la RechercheScientifique (CNRS)National Centre for Scientific Research,France• Commissariat à l’Énergie Atomique –Institut LIST (CEA LIST)Atomic Energy Commission –LIST Institute, France• National University of Ireland,Galway, Ireland• University of Bologna – Centrodi Ricerca Avanzato sui SistemiElettronici, Italy• University of Warsaw, Poland• Jagiellonian University, Poland• Wyzsza Szkola Biznesu –National Louis University, Poland• Nicolaus Copernicus University, Poland• Centro de Matematica da Universidadedo Minho, Portugal• Javna agencija za raziskovalnodejavnost Republike Slovenije (ARRS)Slovenian Research Agency, Slovenia• University of Ljubljana, Slovenia• Universidad de Malaga, Spain• Schweizerischer Nationalfonds (SNF)Swiss National Science Foundation,Switzerland• University of Warwick, United KingdomACAT • 7
Page 1 and 2: Research Networking ProgrammeApplie
Page 3 and 4: Scientific BackgroundFigure 1.Two i
Page 5: • The motion planning problem pla

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

Create successful ePaper yourself

Delete template?

Save as template?