  We explore the universal properties of interacting fermionic lattice systems,
mostly focusing on the development of pairing correlations from attractive
interactions. Using renormalization group we identify a large number of fixed
points and show that they correspond to resonant scattering in multiple
channels. Pairing resonances in finite-density band insulators occur between
quasiparticles and quasiholes living at different symmetry-related wavevectors
in the Brillouin zone. This allows a BCS-BEC crossover interpretation of both
Cooper and particle-hole pairing. We show that in two dimensions the run-away
flows of relevant attractive interactions lead to charged-boson-dominated low
energy dynamics in the insulating states, and superfluid transitions in bosonic
mean-field or XY universality classes. Analogous phenomena in higher dimensions
are restricted to the strong coupling limit, while at weak couplings the
transition is in the pair-breaking BCS class. The models discussed here can be
realized with ultra-cold gases of alkali atoms tuned to a broad Feshbach
resonance in an optical lattice, enabling experimental studies of pairing
correlations in insulators, especially in their universal regimes. In turn,
these simple and tractable models capture the emergence of fluctuation-driven
superconducting transitions in fermionic systems, which is of interest in the
context of high temperature superconductors.
