"Proposed research is concerned with theoretical study of phase transitions in many-fermion systems in a framework of the exact renormalisation group (ERG) approach. This method allows for a nonperturbative treatment of the evolution of the physical systems with some momentum/length scale. I intend to develop the ERG-based theoretical description of phase diagrams in a variety of many-fermion systems, ranging from cold atoms to quark matter. I particular, I plan to explore BEC-BCS transition in cold fermionic atoms both at zero and finite temperature and apply the developed theory to describe recent experiments in ultra-cold atomic Fermi gases. The formal ERG-based formal methods will also be used to study nuclear/neutron matter - nucleon gas transition, meson condensation phenomena, phase transitions of quark matter at finite density /temperature and crossover quark to hadron matter. Apart from specific projects proposed during the course of the fellowship, the experience in some neighbouring areas such as lattice simulations,finite temperature field theory and modern many-body theory will be obtained. This is one of the objectives of the outgoing phase of the whole project, which is supposed to be spent in the Institute of Theoretical and Experimental Physics, Moscow, Russia. The knowledge and skills gained will then be used at full strength to make a further progress in the project related subjects during the return phase at the University of Manchester."