  We present a new approach to Morse and Novikov theories, based on the deRham
Federer theory of currents, using the finite volume flow technique of Harvey
and Lawson. In the Morse case, we construct a noncompact analogue of the Morse
complex, relating a Morse function to the cohomology with compact forward
supports of the manifold. This complex is then used in Novikov theory, to
obtain a geometric realization of the Novikov Complex as a complex of currents
and a new characterization of Novikov Homology as cohomology with compact
forward supports. Two natural ``backward-forward'' dualities are also
established: a Lambda duality over the Novikov Ring and a Topological Vector
Space duality over the reals.
