Moduli spaces of graphs and homology operations on loop spaces of manifolds

In this lecture I will show how spaces of graphs and “gradient graph flows” can be used to define (co)homology operations. In the classical case of Morse theory on a compact manifold, one obtains classical cohomology operations such as Steenrod squares. When one uses “fat” or “ribbon graphs” and Morse theory on the loop space of a manifold, one obtains string topology operations. When one uses ribbon graphs and flows of the symplectic action on the loop space of the cotangent bundle, one obtains a variation of Gromov-Witten invariants on the cotangent bundle. (PDF file)