Title: Directed topological complexity
Speaker: Eric Goubault (Ecole Polytechnique, France)
Abstract: In this talk, I will introduce a notion of « directed » topological complexity (based on the work on « classical » topological complexity of M. Farber) that would help characterize the complexity of planification and control, in robotics for instance. The « classical » notion of topological complexity is the Schwartz genus of the fibration which, to any continuous path of a topological space X, associates its pair of start and end points. Said differently, this is the minimal number of chunks Fi inside XxX on which there exists a continuous selection function of one continuous path between x and y, for any (x,y) in Fi. This describes the complexity of a path-planning algorithm that would make you move from any point x to any point y in X. It is also an interesting homotopy invariant and has deep links with sheaf cohomology.
In practice, physical systems can only be controlled using parameters that would vary within some bounds. This means that the controlled dynamics can only be solutions to differential inclusions, i.e. would need to respect a future cone, at each point, within the corresponding tangent space. This is where topological complexity meets directed topology. I will introduce the corresponding notion, show some examples and differences with respect to the classical case, and hints at links with a notion of directed homotopy invariance, and of a directed homology theory.
Seminar URL: https://research.math.osu.edu/tgda/tgda-seminar.html