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I have added some more material and some slides at persistent homology.
Joao Pita Costa: Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
Seminar za algebro, Ljubljana
Sreda (Wednesday), 8. oktobra 2014, ob 10. uri v Plemljevem seminarju, Jadranska 19/III, Ljubljana
João Pita Costa, Mikael Vejdemo Johansson, Primož Škraba, Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology http://arxiv.org/abs/1409.8613
A topos theoretic generalisation of the category of sets permits ideas as for sets varying according to time intervals. In general it provides tools for unification of techniques for mathematics having had a great importance in the recent developments of Quantum Theory. Persistent homology is a central tool in topological data analysis, which examines the structure of data through topological structure. The basic technique is extended in many different directions, permuting the encoding of topological features by barcodes and correspondent persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between correspondent persistence bars through the algebraic properties of its underlying lattice structure. In this paper we shall look at the topos of sheaves over such algebra, discuss its construction and potential for a generalised simplicial homology over it. In particular we are interested in es tablishing a topos theoretic unifying theory for the various flavours of persistent homology that have emerged so far, providing a global perspective over the algebraic foundations of applied and computational algebraic topology.
added pointer to this work on “persistent Cohomotopy” (kindly pointed out by David C.):
Peter Franek, Marek Krčál, Persistence of Zero Sets, (arXiv:1507.04310);
accompanying talk: Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, (slides)
added pointer to this followup on persistent Cohomotopy:
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