Studies of cold atom gases have diversified over the past few years. To date, a number of experiments work in the strongly-interacting, highly-correlated regime, in which the effects of atom-atom interactions are non-negligible. Many of these systems require full, non-perturbative quantum mechanical many-body treatments. This proposal discusses the theoretical many-body description of (i) quasi-ID two-component Fermi gases and (ii) doped Bose gases. By tuning an external magnetic field or a laser in the vicinity of confinement-induced resonances (CIRs) in the s- and p-wave channels of a quasi-1D two-component Fermi gas, the relative and absolute strengths between the s- and p-wave interactions can be tuned to essentially any value. Motivated by this experimental progress, a study of trapped quasi-1D two-component Fermi gases with competing s- and p-wave interactions and varying polarization using many body quantum Monte Carlo (MC) techniques will be undertaken. Although a rich zero-temperature phase diagram has been predicted, no quantitative studies that treat both s- and p-wave interactions exist. It is anticipated that the study will shed light on the 1D analog of the BEC-BCS crossover, where bound bosonic molecules on the "BEC side" of the CIR in the s-wave channel undergo a transition to paired fermions, i.e., "Cooper pairs", on the "BCS side" of the resonance, and on p-wave pairing. Benchmark results should allow between different mean-field treatments to be discriminated. The study of doped condensates promises rich physics. Compared to liquid 4He, e.g., atomic gases have the advantage of unprecedented control of the interaction strength by utilizing Feshbach resonances. Initial self consistent mean-field studies of a single neutral impurity immersed in a trapped Bose gas indicate that localized impurity states exist and that the degree of localization can be controlled by varying the atom-impurity scattering length. The existence of localized impurity states may lead to the development of novel quantum computing schemes and single atom devices. Investigations, using MC techniques, should shed light on how one can design molecules consisting of non-interacting neutral impurity atoms with varying "bond lengths". The binding of the impurity atoms is predicted to be induced by the atom background and the atom-impurity interactions, i.e., the binding of the impurity molecule is mediated by the atom-atom and atom impurity scattering lengths
