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CAREER: Model theory, measures and combinatorics

职业：模型理论、测量和组合学

基本信息

批准号：
1651321
负责人：
Artem Chernikov
金额：
$ 45万
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：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1651321&HistoricalAwards=false
关键词：
CAREER Model theory measures combinatorics

项目摘要

Model theory is an area of mathematical logic studying families of (typically) infinite structures and their properties definable in a formal language. While this method originates in foundational questions, over the years profound applications to the study of many central objects of classical mathematics and computer science were discovered. This project will investigate an emerging connection to the area of graph theory. Graphs are the basic discrete mathematical objects used to model networks and related systems. In combinatorics, one often investigates the asymptotic behavior of various quantitative properties (such as the density of the edges, or the size of a certain regular pattern) for large finite graphs. Model theory provides a method of converting asymptotic quantitative questions about properties of a family of graphs into qualitative questions about the shape, volume or dimension of a certain limiting infinite object (via the so-called ultraproduct construction). Infinitary model-theoretic machinery to study such limit objects will be used to address questions in finite graph combinatorics, and conversely rich body of results in combinatorics will be used to attack open questions in model theory.As demonstrated in the previous work of the principal investigator and other researchers, recent advances on pseudo-randomness and Ramsey-type phenomena for restricted families of graphs (e.g. semialgebraic regularity lemma by Fox et al., Tao's algebraic regularity lemma over finite fields, regularity lemma for graphs of finite Vapnik-Chervonenkis dimension by Lovasz-Szegedy, etc.) admit a uniform model-theoretic treatment well-aligned with the methods and ideas in Shelah's classification theory. This project initiates a systematic program of investigating these connections, centered around the study of Keisler measures and various model-theoretic notions of dimension (e.g. coming from forking independence or pseudofinite counting) in tame classes of first-order structures (stable, distal, dependent, simple, NTP2, etc.), as well as the finer questions on generalized "incidence bounds" and their relation to Zilber's trichotomy phenomena.

模型论是数学逻辑的一个领域，研究（典型的）无限结构族及其可用形式语言定义的性质。虽然这种方法起源于基础问题，但多年来在经典数学和计算机科学的许多中心对象的研究中发现了深刻的应用。这个项目将研究图论领域的一个新兴联系。图是用于网络和相关系统建模的基本离散数学对象。在组合学中，人们经常研究大型有限图的各种定量性质（如边的密度，或某个规则模式的大小）的渐近行为。模型理论提供了一种方法，将关于图族性质的渐近定量问题转化为关于某个极限无限对象的形状、体积或尺寸的定性问题（通过所谓的超积构造）。研究这些极限对象的无限模型论机制将用于解决有限图组合学中的问题，相反，组合学中丰富的结果将用于解决模型论中的开放问题。在本课题的前期工作中，作者和其他研究人员对受限族图的伪随机性和ramsey型现象的最新研究进展(如Fox等人的半代数正则引理，Tao有限域上的代数正则引理，Lovasz-Szegedy有限Vapnik-Chervonenkis维数图的正则引理，等)承认一种统一的模型论处理方法与希拉分类理论的方法和思想是一致的。该项目启动了一个系统的计划来研究这些联系，围绕着Keisler测度和各种模型理论的维度概念（例如来自分叉独立性或伪有限计数）在一阶结构（稳定、远端、依赖、简单、NTP2等）的研究，以及关于广义“关联界”的更精细的问题及其与Zilber三分法现象的关系。

项目成果

期刊论文数量（18）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Zarankiewicz’s problem for semilinear hypergraphs

半线性超图的 Zarankiewicz 问题

DOI：
10.1017/fms.2021.52
发表时间：
2021
期刊：
Sigma
影响因子：
0
作者：
Basit, Abdul;Chernikov, Artem;Starchenko, Sergei;Tao, Terence;Tran, Chieu-Minh
通讯作者：
Tran, Chieu-Minh

Model theory, Keisler measures, and groups - Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay, Groups, measures, and the NIP. Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596. - Ehud Hrushovski and Anand Pillay, On NIP an

模型理论、凯斯勒测度和群 - Ehud Hrushovski、Yaacov Peterzil 和 Anand Pillay，群、测度和 NIP。