Loop products and closed geodesics

Loop products and closed geodesics

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Nancy Hingston, The College of New Jersey
Fine Hall 314

The critical points of the energy function on the free loop space $L(M)$ of a compact Riemannian manifold $M$ are the closed geodesics on $M$.  Filtration by the length function gives a link between the geometry of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of $L(M)$.  Geometry reveals the existence of a related product on the cohomology of $L(M)$.  For manifolds such as spheres and projective spaces for which there is a metric with all geodesics closed, the resulting homology and cohomology rings are nontrivial, and closely linked to the geometry. I will not assume any knowledge of the Chas-Sullivan product. Joint work with Mark Goresky.