UC Berkeley

In this thesis we study the Restricted Trichotomy Conjectures for algebraically closed and o-minimal fields. These conjectures predict a classification of all sufficiently complex, that is, non-locally modular, strongly minimal structures which can be interpreted from such fields. Such problems have been historically divided into `lower dimensional' and `higher dimensional' cases; this thesis is devoted to a number of partial results in the higher dimensional cases. In particular, in ACF_0 and over o-minimal fields, we prove that all higher dimensional strongly minimal structures whose definable sets satisfy certain geometric restrictions are locally modular. We also make progress toward verifying these geometric restrictions in any counterexample. Finally, in the last chapter we give a full proof of local modularity for strongly minimal expansions of higher dimensional groups in ACF_0.