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

模型论是数理逻辑的一个分支，研究数学结构的一般性质。 模型论的工作经常回答其他数学领域的问题。近年来，模型论在组合学、分析学和代数学中的应用取得了令人兴奋的新进展。本项目主要研究可定义拓扑群及其在紧致流形上的作用和等分布相关问题。该项目还解决了组合问题的背景下，不独立的属性（NIP）relationships.This project built upon the investigator's previous work on definable group action and combinational properties of theories without independence properties.本文系统地研究了拓扑群作用和极值组合学在非独立性条件下的轨道闭合问题。更具体地说，该项目研究NIP理论中的极值组合学，例如边缘关系具有NIP的可定义图的密度。该项目还将研究可在o-最小和强最小理论中定义的超图的Elekes-Ronayai问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.