"The research field of this proposal is situated in the domains of computational physics, mostly quantum Monte Carlo simulations, field theoretical approaches to condensed matter systems and dilute ultra-cold atoms. We want to systematically develop a new diagrammatic Monte Carlo method, which will be applied to cold gases and strongly interacting systems in solid state physics. The method has the attractive feature that it can be formulated for totally different systems ranging from ultracold atoms to particle physics. It has shown some successes for benchmark systems, but its full potential for the hardest models remains unknown. This would require a systematic study for which the time has come. Monte Carlo simulations of cold atomic systems have in recent years shown a remarkable one-to-one agreement with experiments without free fitting parameters, a field to which the PI has greatly contributed. Now that such experiments are maturing, novel numerical methods are needed to describe the parameter regimes which experiments are beginning to reach. The methods we want to explore have the full capability of doing that. If successful, the message for the numerical methods is that they have been validated and can be applied to other systems that are otherwise intractable. Learning about a hard model by simulating it with another quantum system is known as ""quantum simulation"" and we wish to extend this paradigm to the future. The PI has strong experience in numerical methods (especially Monte Carlo simulations) and ultracold atom physics, and has acquired international recognition with more than 50 publications and several prizes. With the close connection to the experimental cold atom group by I. Bloch in Munich, and the computer infrastructure present in Munich, no better combination of expertise and infrastructure can be found to bring this project to a good end, determine the ultimate fate of this method, and potentially push the frontiers of theoretical physics."