  The dynamical symmetries of the Fermion Dynamical Symmetry Model are used as
a principle of truncation for the spherical shell model. Utilizing the usual
principle of energy-dictated truncation to select a valence space, and
symmetry-dictated truncation to select a collective subspace of that valence
space, we are able to reduce the full shell model space to one of manageable
dimensions with modern supercomputers, even for the heaviest nuclei. The
resulting shell model then consists of diagonalizing an effective Hamiltonian
within the restricted subspace. This theory is not confined to any symmetry
limits, and represents a full solution of the original shell model if the
appropriate effective interaction of the truncated space can be determined. As
a first step in constructing that interaction, we present an empirical
determination of its matrix elements for the collective subspace with no broken
pairs in a representative set of nuclei with $130\le A \le 250$. We demonstrate
that this effective interaction can be parameterized in terms of a few
quantities varying slowly with particle number, and is capable of describing a
broad range of low-energy observables for these nuclei. Finally we give a brief
discussion of extending these methods to include a single broken collective
pair.
