What is...?

Given two families of loops on a closed smooth manifold, one can concatenate the loops at the intersections points of these families to obtain a new family of loops. This is the Chas–Sullivan product on the homology of the free loop space of a smooth manifold.  This construction can be vastly generalized to a 2D topological field theory structure consisting of operations defined using chord diagrams/fat graphs that record complicated patterns in which families of loops are cut and concatenated with each other.  These operations have applications to a wide array of questions related to geodesics, smooth topology, and symplectic topology of cotangent bundles. After some historical motivation, I will try to compute some toy examples of these operations, and indicate how they connect to some of the applications. If time permits, I will discuss briefly some exciting recent developments related to the application of the Goresky–Hingston string coproduct to smooth topology.