Add to ORKGOur understanding of particles physics derives, in large part, from our ability to do perturbative calculations by expanding around small coupling. This Feynman diagrammatic approach has been incredibly successful in providing physicists a means to approximate physical observables. However, there are many scenarios where a perturbative approach either does not exist or fails completely. In this thesis I will construct several nonperturbative (or singularly perturbative) methods that have applications in particle physics, quantum field theory, and cosmology. We will also investigate a broad range of topics, including grand unified theories, the hierarchy problem, dark matter phenomenology, the mathematical foundations of quantum mechanics, a...