One of the most important questions in modern Mathematical Physics deals with understanding the space of all possible quantum theories. Unraveling such "theory space" is bound to have significant impact in both Mathematics (with the emergence of new mathematical structures, as the space of all quantum field theories itself is expected to form a new, or at least very subtle, mathematical structure) and Physics (with the description of all possible quantum phenomena). However, this task is a difficult one. Heuristically, the space of all quantum field theories is expected to be infinite-dimensional, with special fixed points (known as conformal field theories (CFT)) linked by paths corresponding to renormalisation-group flows. Understanding this space, encompassing all its geometrical and algebraic substructures, thus begins with understanding some of its special subsets --- in the sense that these CFTs are precisely the building blocks of all possible quantum theories (including theories of quantum gravity and black hole singularities). Our goal in this project is to pursue this long-term goal of understanding the space of all quantum theories, by first understanding special subsets of this space. Furthermore, we shall use complementary approaches, based on the different expertise of the members of this collaboration, associated to Resurgent Analysis, Bootstrap Techniques, and Localisation, to address these special subsets. These approaches allow for computing the spectrum and correlation functions of quantum theories described by random matrix models, of CFTs in diverse dimensions, and of quantum theories with correlation functions which localize to random matrices. Note that the very same theory may be approachable using these different techniques, and they might lead to complementary information. By solving different special sets of quantum theories, we expect to describe local patches, or local subsets, of the full space of quantum theories. In particular we hope to shed light on geometrical and algebraic structures within these (smaller) spaces. We also expect to have different spin-offs into other areas of Mathematics beyond Mathematical Physics, including, e.g., nonlinear ODEs, Lie theory and functional integration, vertex operator algebras, among others.