03. Juli 2025

19.06.2025: Kolloquium

Vortragender: Dr. Stephan Mescher (Universität Halle)

"On topological and geodesic complexities of manifolds“

Beginn: 16:15 Uhr

Ort: HS 1.04, VSP 1

Abstract:

In this talk, I want to outline the contents of my habilitation proposal to the institute, which I would to submit to the faculty in the near future. I will give an overview of my research projects from the previous years. Topological complexity (TC), as introduced by M. Farber in 2003, is an integer-valued homotopy invariant of topological spaces that is motivated by the motion planning problem from robotics.  After a short introduction to topological complexity (TC), I will first present lower bounds on topological complexities of closed manifolds. As consequences of these bounds,  we obtain a characterization of manifolds dominated by products in terms of TC as well as a computation of the TC of certain symplectic manifolds. I will also discuss the TC of frame bundles and explain how it can be seen as an abstraction of oriented motion planning. Afterwards, I will talk about geodesic complexity, an abstraction of motion planning along paths of minimal length. Several results on the geodesic complexity of homogeneous Riemannian manifolds, obtained by M. Stegemeyer and myself, will be discussed.

Moreover, I will present spherical complexities, which are invariants for loop and sphere spaces that resemble topological complexity and which can be used to obtain topological lower bounds for the number of critical orbits of  functionals on loop and sphere spaces which are invariant under orthogonal reparametrizations. I will also show how to apply such a bound to derive several new existence results for closed geodesics in spheres and projective spaces under curvature pinching conditions. If time permits, I may also speak about results on algebraic characterizations of the TC of aspherical spaces in terms of their fundamental groups.