Invasion fronts, or the moving boundaries between unstable and stable states caused by disturbances such as invasive species or a novel disease, play a key role in the self-organized development of coherent structures in many scientific fields, including epidemiology, developmental biology, fluid dynamics, materials science, and more. Historically, these invasion processes have been poorly understood, with clear results available only for special systems of limited interest and are difficult to study computationally. This project involves the development of a broad, model-independent framework for studying complex invasion fronts both theoretically and computationally, which can be used to make efficient predictions of invasion speeds and resulting spatial structures in broad classes of physical systems. The development of computational tools will focus on making these theoretical advances available to be put into practice by a broad range of scientists. Furthering the understanding of the selection of spatial structures in the wake of invasion fronts has particular promise for semiconductor manufacturing technologies which aim to harness this self-organized pattern formation to efficiently create nanomaterials with desirable electrical and optical properties. The project will involve both undergraduate and graduate students in the research, helping to develop the next generation of applied mathematicians. Prior mathematical results on invasion fronts have been limited to