  Consider the infinite dimensional hyperbolic dynamical system provided by the
(forward) heat semi-flow on the loop space of a closed Riemannian manifold M.
We use the recently discovered backward {\lambda}-Lemma and elements of Conley
theory to construct a Morse filtration of the loop space whose cellular
filtration complex represents the Morse complex associated to the downward
L2-gradient of the classical action functional. This paper is a survey. Details
and proofs will be given in [6].
