Grant: Discovery Grants Program - Individual

The proposal is in the area of model theory, a branch of mathematical logic studying the descriptions of mathematical structures in a suitable formal (first order) language. During the last two decades there have been some major advances in the development of "geometric" model theory, with applications in number theory, algebraic geometry and other branches of mathematics. The main objects of study in this field are structures having a certain invariant independence relation and an associated notion of dimension, e.g. vector spaces or algebraically closed fields. The subject has been constantly evolving by extending its scope to wider classes of structures (stable, simple, o-minimal etc). I plan to continue my previous research on expansions of first order structures, both in simple and non-simple settings. Expansion is the result of adding new relations to the language. Besides being one of the major sources of new examples in model theory, expansions can also serve as an effective tool for the study of model theoretic properties of the underlying structures. While the greater expressive power of the expanded language allows it to capture the features of a structure that were not definable in the original language, one may lose certain "nice" model theoretic properties (e.g. stability or simplicity) when passing to the expansion. Thus one of the major questions is the existence of expansions that are still analyzable from the model theoretic point of view. Once such an expansion is found, one tries to characterize various properties of the original structure in terms of the expansion. Often this helps to discover new model theoretic conditions, or to generalize certain notions to a wider class of structures. I plan to focus on two specific kinds of expansions: expansions with a distinguished elementary substructure (elementary pairs) and expansions with "externally definable" subsets. The goal is to further investigate pairs of simple structures and their applications, to use pairs to clarify and refine the hierarchy of definability conditions within the class of simple structures, to extend the pairs technique to non-simple settings , and to explore "generic" expansions with externally definable sets.