Mathematics Colloquium, Fall 2007

String Topology

What physicists mean by a string (at least in the most naive sense) is a map from the circle to a manifold M. This makes it interesting to study the space LM of all continuous maps from the circle to M. About 8 years ago Moira Chas and Dennis Sullivan discovered that the homology of LM has previously unsuspected structure: a product and a Lie bracket. Their work has led to many further developments. In particular, it has been shown that certain, perhaps all, Gromov-Witten invariants of the cotangent bundle of M can be identified with Chas-Sullivan operations on the homology of LM. The talk will give a survey of this subject and an elementary construction of the Chas-Sullivan product.