  Presented in this thesis are a set of projects which lie at the intersection
between strong correlations and topological phases of matter. The first of
these projects is a treatment of an infinite dimensional generalization of the
SSH model with local Coulomb interactions which is solved exactly using
DMFT-NRG. Observed in the solution is power-law augmentation of the
non-interacting density of states, as well as a Mott transition. This
calculation represents an exact solution to an interacting topological
insulator in the strongly correlated regime at zero temperature. The second set
of projects involves the development of methods for formulating non-interacting
auxiliary models for strongly correlated systems. These auxiliary models are
able to capture the full dynamics of the original strongly correlated model,
but with only completely non-interacting degrees of freedom, defined in an
enlarged Hilbert space. We motivate the discussion by performing the mapping
analytically for simple interacting systems using non-linear canonical
transformations via a Majorana decomposition. For the nontrivial class of
interacting quantum impurity models, the auxiliary mapping is established
numerically exactly for finite-size systems using exact diagonalization, and
for impurity models in the thermodynamic limit using NRG, both at zero and
finite temperature. We find that the auxiliary systems take the form of
generalized SSH models, which inherit the topological characteristics of those
models. These generalized SSH models are also formalized and investigated in
their own right as novel systems. Finally, we apply this methodology to study
the Mott transition in the Hubbard model. In terms of the auxiliary system, we
find that the Mott transition can be understood as a topological phase
transition, which manifests as the formation and dissociation of topological
domain walls.
