The significance of the research is that in multiscale systems we need to extract an appropriate summary of the microscale dynamics to form a closure of macroscale dynamics. This empowers us to simulate the emergent behaviour on the large scale with a much simpler model, one that only resolves the macroscale quantities of interest. The whole process of making simplifying approximations, so cen- tral to quantitative science, is vastly improved by my development of these new systematic and accurate techniques. Most physical situations of interest in the world around us have an enormous number of fine details which are of little concern in many situations. For example, the microscopic detail of molecular motion is almost completely irrelevant on our macroscopic scale; consequently we work with the much less informative, but nonetheless much more tractable, continuum approximation. Similarly, when engineers consider a beam they are not overly interested in the detailed distribution of elastic stresses, instead they are primarily concerned with how the beam simply bends in response to a given load—for this, beam theory was developed. In these, and many situations, the equations which scientists deal with are simplifications of the ‘true’ but intractable or overly-complicated equations that describe all the details. Traditionally these simple models of actual physical complexity emerge via specific heuristic arguments which are very limited in scope. The trick has always been to argue cogently that neglected effects are indeed small and to be careful not to ‘throw out the baby with the bath-water’; the ‘baby’ being the important dynamics of interest on the large scale, while the ‘bathwater’ is the uninteresting microscopic fine detail. It is this process of creating simple macroscale approximations for otherwise intractable complex multiscale dynamical systems which my research has addressed