"The goal of my mathematical research is to force a breakthrough in solving and understanding a number of long-standing open problems that are rooted in physics and chemistry. My objects of study are systems consisting of a large number of random components that interact locally but exhibit a global dependence. Typically, the components of these systems are subject to a simple microscopic dynamics. The challenge lies in understanding the complex macroscopic phenomena that may arise from this dynamics. Core to my proposal are macroscopic phenomena that are very hard to grasp with heuristic or numerical methods: pinning, localisation, collapse, porosity, nature vs. nurture, metastability, condensation, ageing, catalysis, intermittency and trapping. My main line of attack is to combine large deviation theory, which is a well-established technically demanding yet flexible instrument, with a number of new variational techniques that I have recently developed with my international collaborators, which are based on space-time coarse-graining. My goal is to apply this powerful combination to a number of complex systems that are at the very heart of the research area, in order to arrive at a complete mathematical description. The idea is to use the coarse-graining techniques to compute the probability of the possible trajectories of the microscopic dynamics, and to identify the most likely trajectory by maximising this probability in terms of a variational formula. The solution of this variational formual is what describes the macroscopic behaviour of the system, including the emergence of phase transitions. My proposal focuses on five highly intriguing classes of random interacting systems that are among the most challenging to date: (1) polymer chains; (2) porous domains; (3) flipping magnetic spins; (4) lattice gases; (5) evolving random media. The unique reward of the variational approach is that it leads to a full insight into why these systems behave the way they do."