The first part studies the emergent spatio-temporal oscillations of the system of Min-proteins in the context of dynamic cell geometry. Models of cell growth via elongation and division are developed, and explored in conjunction with a biophysical reaction-diffusion model of the oscillations. Reaction-diffusion equations are solved numerically with model-specific dynamic boundary conditions. The model for elongation correctly captures the behavior of wild-type cells and predicts the emergence of multi-node oscillations in filamentous mutants. The model for cell division predicts that protein binding to the septum during division is important for preventing asymmetric oscillations in daughter cells.
The second part studies superfluidity in multiple species Fermi systems with imbalanced populations starting from the normal Fermi-liquid. Analytic, model-independent calculations are carried out using many-particle Green's functions and Bogoliubov Equations. Population imbalance if found to lift a degeneracy in both the singlet and triplet excitation spectrum of the superfluid state. Employing a Ginzburg-Landau free energy analysis and the d-vector formalism, the effects of population imbalance on the singlet and triplet ground state structure are calculated. Singlet superfluidity is found to be destroyed in the presence of large population imbalance. Arbitrarily small population imbalances are shown to stabilize the A-phase over the B-phase in triplet superfluids. Effects of anisotropic spin-dependent interactions are also calculated and their consequences are discussed.