Arnol'd's Chord Conjecture for Conormal Legendrian Lifts

The chord conjecture, due initially to Arnol'd in the case of the standard contact three-sphere, asserts the existence of a Reeb chord with boundary on every closed Legendrian submanifold of a closed contact manifold for every contact form. This conjecture was established in various settings by Cieliebak, Mohnke, Hutchings and Taubes, and others. In this talk, I will sketch a proof of the chord conjecture for conormal bundles of closed submanifolds of any closed manifold seen as Legendrians in the co-sphere bundle. This generalizes a result of Grove in Riemannian geometry regarding the existence of geodesics normal to the submanifold. The method of proof involves wrapped Floer cohomology with local coefficients. This talk is based on joint work with Dylan Cant and Egor Shelukhin.