Homogeneous model theory as a framework for Algebraic Stability Theory

In this thesis we aim to persuade the readers to reconsider homogeneous model theory as a framework in which to study Algebraic Stability Theory. We argue that the tight link between homogeneous model theory and categorical amalgamation constructions gives it a flexibility beyond what the ordinary first-order context can offer while simultaneously it allows the development of a stability-theoretic machinery very close to classical first-order stability. Thus we first introduce homogeneous model theory and categorical amalgamation theory separately (Chapters 1 and 2), laying out our own contributions alongside the known theory, before we utilise their interactions to prove related results pertinent to several different questions in algebraic stability theory. In Chapter 3 we give a very general result on omitting (possibly isolated) types from homogeneous models, generalising an analogous theorem by Hrushovski and Itai [18] regarding differential fields. Our method makes it possible to omit any family of types of trivial geometry from a superstable (and simple) homogeneous model without impacting the dividing structure and thus the stability class of the model. In Chapter 4 we apply homogeneous model theory to the study of Hrushovski constructions, enabling us to give a general treatment of the fundamental stabilitytheoretic properties beyond the combinatorial case to which such general accounts are usually confined. We obtain a general characterisation of dividing and by extension of U-Rank in the amalgamation limits of those constructions, and since we are not limited to finitary cases we can use our results for studying pseudo-exponentiation in the next two chapters. In Chapters 5 and 6 we thus narrow our focus even more and apply the tools developed above to the single class of exponential fields, illustrating the power of our general results through application to a topic which has been under close investigation ever since Boris Zilber proposed pseudo-exponentiation as a model-theoretic approach to complex exponentiation in the 2000s [42]. In passing we will also construct a non-almost-exponentially-algebraically closed quasiminimal exponential field, answering a question asked by Bays and Kirby in their recent [7].