Research objectives and content
The purpose of the project is to prove that the Floer homology of the cotangent bundle of a Riemannian manifold M is naturally isomorphic to the homology of the loop space. The main step of the proof is to obtain the gradient flow of the classical action functional on the loop space of M as an adiabatic limit of the Floer gradient flow of the symplectic action on the loop space of T*M. The limit is one where the metric on the momentum coordinate converges to zero. There is a natural correspondence between the critical points in both theories (perturbed geodesics) and the limit argument relates the heat flow of the classical action to perturbed J-holomorphic curves in the cotangent bundle.
We intend to investigate implications of our result to - existence of Lagrangian submanifolds - spectral geometry
Training content (objective, benefit and expected impact)
Carrying out this project in collaboration with one of the leading experts in the field will give me detailed knowledge of analyzing nonlinear partial differential equations - a topic of fundamental interest in pure mathematics as well as in theoretical physics. The one-year symposium on symplectic geometry at Warwick university provides direct contact and access to researchers as well as research in symplectic geometry.