There is a deeply rooted belief in the natural sciences that the more fundamental a theory gets, the less important complexity becomes. In particular, there has traditionally been an almost unquestioned assumption that physics at the smallest scales is intrinsically simple, and that complexity is only relevant in the macroscopic or living world. By now significant theoretical and experimental evidence has accumulated indicating that this is wrong. This evidence suggests that the physics determining the observed parameters of nature, the dynamics of the universe at the largest scales, and the microphysics of black holes is fundamentally complex. Nevertheless, little effort has been made to understand the nature and structure of fundamental complexity, and this has been a major obstruction to progress in some of the biggest and oldest questions in physics. This project seeks to change this in important ways, specifically in three topics: string vacua, eternal inflation and black holes. There are many powerful techniques and concepts available to study complex systems, but as they were developed far from the realm of particle physics, they are unfamiliar to most researchers in our field. In the past year our group has familiarized itself with many of the ideas and techniques developed to understand spin glasses, neural networks and other complex systems. Combined with the more standard tools of geometry, supersymmetry and duality in our field of string theory, we will apply these techniques as a completely new approach to the aforementioned questions. Our proposed research is theoretical and meant to be at the leading innovative forefront in the field. The goal is to come to a breakthrough in our understanding of complexity in fundamental physics, opening up a new research direction with several new opportunities to make a lasting impact on our understanding of the nature of space and time, and the origin and fate of the universe.