On Wednesday, March 6, 2019, Dr. Nancy Hingston will be giving a talk, “Loop products, closed geodesics and self-intersections,” as part of the Intitute for Advanced Studies (IAS) Workshop on Geometric Functionals: Analysis and Applications. Per the IAS website, the workshop covers “current developments in several different geometric variational problems and their applications. Topics include harmonic maps, Willmore surfaces, Yamabe-type problems, geometric measure theory, systolic geometry, Einstein metrics and others.”
Let M be a compact Riemannian manifold. Morse theory for the energy function on the free loopspace LM of M gives a link between geometry and topology, between the growth of the index of the iterates of closed geodesics on M, and the algebraic structure given by the Chas-Sullivan product on the homology of LM. I will discuss this link, and a new geometric property of the loop coproduct: the nonvanishing of the kth iterate of the coproduct on a homology class ensures the existence of a loop with a (k+1)-fold intersection in every representative of the class. No knowledge of loop products will be assumed. Joint work with Nathalie Wahl.