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

模型理论在极值组合学和G空间紧化中的应用

• 批准号: 1500671
• 负责人: Sergei Starchenko
• 金额: $ 27万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500671&HistoricalAwards=false
• 关键词: Applications; Model; Theory; Extremal; Combinatorics

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.

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，模型论在组合数学和分析中的应用取得了令人兴奋的新发展。 该项目推进了可定义拓扑群及其作用的研究，以及在远端理论背景下的组合问题。这个研究项目建立在研究者以前关于可定义的群作用和没有独立性质的理论的组合性质的工作之上。本计画系系统研究群作用之可定义紧化，以及非独立性设定下之极值组合学。 更具体地说，该项目研究远端理论中的极值组合学和NIP理论中可定义的图的Erdos-Hajnal猜想。该项目还将研究o-极小和其他驯服理论中Ramsey函数的增长率，并将为连续群作用的紧化，特别是黎曼对称空间，开发一个模型理论框架。