Model Theory and Combinatorial Geometry, Algebraic and O-Minimal Flows.

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Model theory, a branch of mathematical logic, studies general properties of mathematical structures. Work in model theory often answers questions in other areas of mathematics. In the recent years there have been exciting new developments in applications of model theory to combinatorics, analysis and algebra. This project advances research on definable topological groups, their actions on compact manifolds and equidistribution related problems. The project also addresses combinatorial questions in the context of not-independence-property (NIP) relations.This project builds upon the investigator's previous work on definable group actions and combinatorial properties of theories without independence properties. The research undertakes a systematic study of orbit closures for topological groups actions and extremal combinatorics in the not-independence-property (NIP) setting. More specifically, the project studies extremal combinatorics in NIP theories, e.g. densities of graphs definable whose edge relation has NIP. The project will also investigate Elekes-Ronayai problem for hypergraphs definable in o-minimal and strongly minimal theories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.