Stability Theory Modulo a Predicate

In this thesis we develop the analysis of the structure of a model, modulo the structure induced by a part of the model interpreting a predicate, P. We develop the 'Morley Rank Modulo a Predicate', PMR, and define an independence relation based on this rank. We analyze this relation in a nice setting (where every formula has PMR) in terms of the eight axioms of stability theory. We prove a dichotomy theorem classifying PMR-Minimal structures and a two-cardinal result. Finally, we give a classification of the norms one can place on a finite dimensional vector space over the reals (up to model-theoretic equivalence).