Geodesics and Invariants of Riemannian Manifolds

• Applicant: Maximilian Stegemeyer
• Subject Area: Mathematics
• Term: from 2023 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 518920559

The aim of this project is to study connections between the global behavior of geodesic curves and invariants of Riemannian manifolds. In particular, problems in string topology and geodesic complexity shall be investigated. String topology is the study of algebraic structure in the homology and cohomology of the free loop space of a compact orientable manifold. Morse theory gives a closed connection between the closed geodesics in a Riemannian manifold and the homology of the free loop space of this manifold. One of the central ideas of this project is to use geometric properties of geodesics to gain information about string topology operations and conversely, to use properties of the string topology of the manifold to study the behavior of geodesics. In particular, the string topology coproduct will be studied. Geodesic complexity is a geometric version of Michael Farber's topological complexity. The topological complexity of a path-connected topological space is a numerical homotopy invariant and can be seen as an abstraction of the robot motion planning problem. If the topology of the space is induced by a metric it is natural to ask for the motion planners to only use length-minimizing curves. This leads to the definition of geodesic complexity of a Riemannian manifold. Geodesic complexity strongly depends on the global behavior of geodesics. The main goals of the proposed project are to study the dependence of geodesic complexity on the metric and to understand the precise relation between geodesic and topological complexity.

• DFG Programme: WBP Fellowship
• International Connection: Denmark
• Host: Professorin Nathalie Wahl, Ph.D.