The study of invariant geometries is a topic of fundamental importance in mathematics. These geometries arise naturally in many areas, including several complex variables (SCV), partial differential equations (PDE), and algebraic, complex, and differential geometry. This research project by the Principal Investigator is centered around a particular geometry - CR geometry - that arises in the study of SCV and complex geometry. It has deep and profound connections to central topics in mathematical and theoretical physics, including quantum field theory, general relativity, and string theory, as well as applications in systems engineering and control theory. The study of obstruction flatness, e.g., which has a prominent role in this project, has a direct link, via the Lorentzian Fefferman space, to the equations of motion in conformal gravity. The ideas and techniques that are needed for the investigations in this project come from a broad range of mathematical areas: complex analysis/geometry, PDE, and differential geometry; and, at the same time, the techniques and tools developed in this project will benefit these areas as well. The project will also provide interesting research topics and learning experiences for graduate students and postdocs. The seminar activity that will result from the project should be inspiring and stimulating for both students and other researchers. The goal of this mathematics research project is to study geometric and analytic aspects of invari