  Tunability of effective two body interactions near Feshbach resonances is a
powerful experimental tool in systems of ultracold atoms. It has been used to
explore a variety of intriguing phenomena in recent experiments. However not
all of the many-body properties of such systems can be understood in terms of
effective models with contact interaction given by the scattering length of the
two particles in vacuum. For example, when a two component Fermi mixture is
quenched to the BEC side of the Feshbach resonance, a positive scattering
length suggests that interactions are repulsive and thus collective dynamics
should be dominated by the Stoner instability toward a spin polarized
ferromagnetic state. On the other hand, existence of low energy two particle
bound states suggests a competing instability driven by molecule formation.
Compe- tition between spontaneous magnetization and pair formation is
determined by the the interplay of two-particle and many-body phenomena. In
these lecture notes we summarize our recent theoretical results, which analyzed
this competition from the point of view of unstable collective modes. We also
comment on the relevance of this theoretical analysis to recent experiments
reported in Ref. (Jo, Lee, Choi, Christensen, Kim, Thywissen, Pritchard and
Ketterle, 2009).
