This proposal deals with a new and rapidly growing subject: superfluidity in ultracold fermionic atomic systems. The capability (via Feshbach resonances) of tuning the attractive interaction strength is unprecedented in condensed matter systems. This knob in the atomic system makes it possible to study a realization of the BCS-Bose Einstein condensation (BEC) scenario. The work builds on our past contributions which reproduce the correct BCS ground state and can address finite temperature effects as well. The work will focus on (i) vortices and vortex lattices over the entire range of crossover (ii) the three dimensional superfluid phase diagram (iii) the nature of T non 0 collective modes and (iv) BEC-BEC crossover on optical lattices. In addition (v) we will expand our research to consider a more general ground state which is posited here. The ultimate goal is to work with the theoretical community to help formulate the appropriate fermionic analogue of the Gross-Pitaevskii theory which has been so useful for bosonic superfluids.
