Definable groups in d-minimal and related structures

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This project lies at the nexus of model theory (mathematical logic) and group theory. The main objects of study are groups definable in various structures, which can be of topological/geometric nature, such as o-minimal structures and tame expansions of them, such as d-minimal structures.In the o-minimal setting, definable groups have been fairly understood with perhaps the most notable result being the solution of Pillay's conjecture, which draws an explicit connection between those groups and real Lie groups. Extensions of Pillay's conjecture in one of many possible expansions of o-minimal structures will be investigated in this project. Concrete examples include(a) the expansion of the real field by a predicate for the set of all integer powers of 2,(b) the expansion of the real ordered group by a predicate for the set of integer numbersThose settings have recently seen the development of model-theoretic tools, which will be used in this project.Potential applications include:- The solution of Compact Domination Conjecture in the above settings- Affine embeddings for definable groups in the above settings.