Applications of Model Theory to Extremal Combinatorics and Compactifications of G-spaces

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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 and analysis. This project advances research on definable topological groups and their actions, and also on combinatorial questions in the context of distal theories. This research project builds upon the investigator's previous work on definable group actions and combinatorial properties of theories without independence properties. This project undertakes a systematic study of definable compactifications of group actions, and extremal combinatorics in the not-independence-property (NIP) setting. More specifically, the project studies extremal combinatorics in distal theories and the Erdos-Hajnal conjecture for graphs definable in NIP theories. The project will also investigate growth rates of Ramsey functions in o-minimal and other tame theories, and will develop a model theoretic framework for compactifications of continuous group actions, in particular Riemannian symmetric spaces.