Model-Theoretic Classification, Graph Combinatorics, and Topological Dynamics

模型理论分类、图组合学和拓扑动力学

基本信息

批准号：
1600796
负责人：
Artem Chernikov
金额：
$ 18万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600796&HistoricalAwards=false
关键词：
Model Theoretic Classification Graph Combinatorics

项目摘要

Model theory studies the ways in which classes of mathematical objects can be defined in some restricted formal language, and what structural properties are implied by these definability assumptions. This method of study originated in questions around the philosophy and foundations of mathematics, but in recent years it has found striking applications in the study of some central objects of classical mathematics and computer science. This project investigates further these connections, in particular in the context of graphs (mathematical ways of describing networks and related systems) and group actions (mathematical ways of describing collections of symmetries of a space). This study has the potential to open up a route for applications of the powerful infinitary model-theoretic machinery to open questions in finite graph combinatorics, and conversely for applications of deep results in combinatorics to open questions in model theory.Motivated by Morley's conjecture on the possible number of uncountable models of first-order theories, Shelah isolated several important classes of "tame" theories and developed a rich machinery for analyzing models and definable sets for some of those classes, particularly for stable theories. Later work by many researchers demonstrated that notions and methods of generalized stability reflect important phenomena in other areas of mathematics. This project will investigate two such connections. 1) Improved Ramsey-type bounds and strong regularity lemmas were obtained for semi-algebraic graphs by Fox et al. and for algebraic graphs in large finite fields by Tao, with numerous applications in the corresponding areas. These results can naturally be viewed as results about graphs definable in certain structures fitting into the classification picture. 2) Study of definable group actions turns out to be closely related to certain questions in topological dynamics, especially around weakly almost periodic dynamical systems and tame systems studied by Glasner and others. The investigator will work on developing further methods of generalized stability and applying them to questions in combinatorics of graphs definable in various "tame" structures (stable, o-minimal, distal, dependent) and to study dynamical properties of definable group actions in those structures.

模型理论研究了在某些受限制的形式语言中可以定义数学对象类别的方式，以及这些可确定性假设所暗示的结构属性。这种研究方法起源于围绕数学的哲学和基础的问题，但是近年来，它在研究古典数学和计算机科学的一些中心对象中发现了惊人的应用。该项目进一步研究了这些连接，特别是在图表的上下文（描述网络和相关系统的数学方式）和组动作（描述空间对称集合的数学方式）的情况下。这项研究有可能为在有限的图表组合学中开放问题的强大无限模型理论机械的应用开辟一条途径，相反，与莫利（Morley）在模型理论中开放问题的深度结果的应用。其中一些课程，特别是对于稳定的理论。后来的许多研究人员的工作表明，广义稳定性的概念和方法反映了其他数学领域的重要现象。该项目将调查两个这样的连接。 1）Fox等人获得了改进的Ramsey型边界和强有力的规律性引理。对于Tao的大有限磁场中的代数图，在相应的区域中有许多应用。这些结果自然可以看作是针对分类图片中的某些结构中可定义的图表的结果。 2）对可确定的群体行为的研究与拓扑动态中的某些问题密切相关，尤其是在格拉斯纳和其他人研究的弱周期性动态系统和驯服系统周围。该研究者将致力于开发广义稳定性的进一步方法，并将其应用于可在各种“驯服”结构（稳定，o-inimal，远端，依赖性）中定义的图表组合中的问题，并研究这些结构中可定义的组动作的动力学特性。