The project is aimed at developing new methods to create ultracold gases with unexplored many-body properties, and we construct the theory to realize the proposed opportunities. We intend to develop new ideas to induce resonant, long-range, and many-body interaction between particles. This includes novel near-zero-field Feshbach resonances in gases tightly confined to 1 or 2 dimensions that will enable the exploration of the physics of spinor gases in ultralow magnetic fields (<1mG). The idea is that the resonating state of the closed channel is a field-tunable confinement-induced weakly bound state. By the low field one avoids field-induced accumulation of particles in a given Zeeman state and encounters a variety of novel many-body states with interaction-broken spin rotation symmetry. The new resonances for polar molecules in 2 dimensions (layers) are provided by their coupling to the interlayer 2-molecule bound state. This allows one to reduce the short-range 2-body interaction making a 3-body repulsion important for bosons, so that the resulting many-body states can be various supersolids. It is further proposed to work on intriguing open problems. The creation of an itinerant ferromagnet of 2-component fermions is blocked in 3D by the formation of weakly bound dimers at the strong intercomponent repulsion required by the Stoner mechanism. In 1D this state is impossible for contact interactions. Our idea is to include an antisymmetric interaction (p-wave in 3D), which can practically make the ground state ferromagnetic. We then focus on non-conventional transport of rotational excitations of polar molecules randomly distributed in a deep optical lattice. The amplitude of hopping of an excitation from an excited to a ground state molecule decays as a cubic power of the distance between them. This is a long-range behavior which may lead to Levy flights, antilocalization, algebraic localization of the excitations, and we develop a theory of all these regimes.
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