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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.

模型论研究数学对象的类可以用某种受限的形式语言来定义的方式，以及这些可定义性假设所隐含的结构属性。这种研究方法起源于围绕数学哲学和基础的问题，但近年来它在经典数学和计算机科学的一些中心对象的研究中发现了惊人的应用。该项目进一步研究这些联系，特别是在图（描述网络和相关系统的数学方法）和群作用（描述空间对称性集合的数学方法）的背景下。这项研究有可能为强大的无穷模型论机器在有限图组合学中的应用开辟一条道路，反过来也为组合学中的深层结果在模型论中的应用开辟一条道路。受莫利关于一阶理论的不可数模型的可能个数的猜想的启发，希拉分离出几个重要的“驯服”理论类别，并发展出一套丰富的分析模型和可定义集合的机器，特别是对稳定理论。后来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数学其他领域的重要现象。本项目将研究两个这样的联系。1)Fox等人和Tao等人分别给出了半代数图和大有限域上代数图的改进Ramsey型界和强正则引理，并在相应的领域中得到了广泛的应用。这些结果可以自然地被看作是关于图的结果，这些图可以用适合于分类图的某些结构来定义。2)可定义群作用的研究与拓扑动力学中的某些问题密切相关，特别是围绕弱概周期动力系统和驯服系统的研究。研究人员将致力于发展广义稳定性的进一步方法，并将其应用于各种“驯服”结构（稳定，O-最小，远端，依赖）中可定义的图形组合学问题，并研究这些结构中可定义的群作用的动力学性质。