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。模型理论的工作经常回答其他数学领域的问题。近年来，模型理论在组合学、分析学和代数中的应用取得了令人兴奋的新进展。本课题主要研究可定义拓扑群及其在紧流形上的作用和相关的等分布问题。该项目还解决了非独立属性（NIP）关系背景下的组合问题。这个项目建立在研究者之前关于可定义的群体行为和没有独立性质的理论的组合性质的工作之上。本文系统地研究了非独立性（NIP）条件下拓扑群作用和极值组合的轨道闭包问题。更具体地说，该项目研究了NIP理论中的极值组合，例如边关系具有NIP的可定义图的密度。该项目还将研究在0极小和强极小理论中可定义的超图的Elekes-Ronayai问题。该奖项反映了美国国家科学基金会的法定使命，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.