The Collaborative Research project of the University of Massachusetts and Staten Island College (CSI) of the City University of New York, currently supported by NSF under Grants PHY-0426881 and PHY-0426814|is studying collective phenomena in trapped ultra-cold atomic Fermi gases, multi-component condensates in optical lattices, the supersolid state of 4He and the theory of deconfined criticality by the state-of-the-art Monte Carlo (MC) approaches. The numeric schemes will be based on Worm algorithm (WA) and Diagrammatic Monte Carlo (DMC)|generic high-performance techniques introduced by the research team for simulations of systems with topologically complex configuration spaces and large numbers of continuous variables.