Mathematical Conversations

Geometers since Morse are interested in Morse Theory on the free loop space $LM$ of a Riemannian manifold $M$, because the critical points of the energy function on $LM$ are the closed geodesics on $M$. I will discuss an observed symmetry of the Morse theory that looks a lot like Poincaré duality. So far there are many manifestations of this symmetry principle, but no clear statement.