The emergence of laws, i.e. the appearance of novel physical models at different complexity scales is a ubiquitous, yet poorly understood, fact of nature. From quantum field theories to atomic physics, from quantum mechanics to thermodynamics, from atomic physics to chemistry, science always finds the emergence of new effective theories and models at different scales. Logical consistency dictates that these new emergent laws are a natural consequence of the more fine grained microdynamics. And exactly these transitions are still under-researched and central questions remain open: When does a microscopic theory allow for emergent laws? Why does it seem to happen only at particular sizes/coarse grainings? Can we predict this emergence from the microdynamics? Is it unique or just a consequence of the questions we ask/the measurements we do? This project wants to explore this daunting task via a two-fold approach. In a bottom up approach we want to understand the mathematical structure behind any such transition. Understanding physical laws as algorithmic and operational procedures allows a rigorous mathematical treatment and the development of toy models to understand the ultimate structure behind emergent laws. A top-down approach, analysing data from quantum many-body experiments will provide guidance through a physically motivated and relevant question that provides the ideal test-bed for the general mathematical ideas.