Title: Recent Advances in Directed Algebraic Topology
Speaker: Eric Goubault & Jeremy Dubut
Abstract: In directed algebraic topology, the object of study is topological spaces with a « preferred direction », modulo a form of homotopy equivalence that should respect this direction, i.e. not revert it. The idea originally appeared in computer science, in the study of concurrent and distributed processes during the early nineties, but the field of directed topology is now really about more general space-time phenomena.
Some notions are not too difficult to translate from the undirected to the directed world, such a the equivalent of the fundamental groupoid, into the fundamental category. Still, some notions have long eluded us, and for some, still elude us even if we have some candidates. Among these, we will mention directed homology theories, directed homotopy equivalences, and some elements of classification of directed spaces.
In this talk, we begin by introducing the basic notions of directed topology, and show that in the simple case of non-positively curved spaces, they are not too different from the classical undirected case. For more general spaces, classification is more intricate, and for that matter, we introduce a directed homology theory, which in spirit looks like both non-abelian homology (in that we use natural systems of abelian groups), and persistent homology (in that these abelian groups are seen as evolving along some base category). We will end up by some conjectures on a "directed homotopy hypothesis", if time permits.
Seminar URL: http://www.tgda.osu.edu/tgda-seminar.html