The primary emphasis of this research is the study structure theorems in model theory, which is a branch of mathematical logic. In many instances, a theory (i.e., a set of sentences in a formal language) is strong enough to allow for a structure theorem on the class of its models (i.e, the algebraic structures satisfying every sentence of the theory). It is known that if a stable theory forbids the encoding of second-order information, then there is a strong structure theorem for the class of sufficiently saturated models of the theory. Extending such a structure theorem to classes of models with less saturation requires understanding combinatorial phenomena, frequently in the presence of a definable group. In many instances, techniques from Descriptive Set Theory are beneficial.
This research will contribute to the taxonomy of algebraic structures. Information present in structure theorems investigated here yield bounds on the complexity of certain neural networks.
