CAREER: Model theory, measures and combinatorics

模型论是数理逻辑的一个领域，研究（典型的）无限结构的族及其在形式语言中可定义的性质。虽然这种方法起源于基础问题，但多年来，它在经典数学和计算机科学的许多中心对象的研究中得到了深刻的应用。这个项目将调查一个新兴的连接到图论领域。图是用于对网络和相关系统进行建模的基本离散数学对象。在组合数学中，人们经常研究大型有限图的各种定量性质（如边的密度或某种规则模式的大小）的渐近行为。模型论提供了一种方法，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、体积或维数的定性问题（通过所谓的超积构造）。研究这种极限对象的无穷模型理论机器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攻击模型理论中的开放问题。关于图的受限族的伪随机性和Ramsey型现象的最新进展（例如Fox等人的半代数正则性引理，Tao的有限域上的代数正则性引理，Lovasz-Szegedy的有限Vapnik-Chervonenkis维数图的正则性引理等。承认一个统一的模型理论的治疗以及对齐的方法和思想在希拉的分类理论。该项目启动了一个系统的研究这些连接的计划，围绕着Keisler措施和各种模型理论的维度概念（例如，来自分叉独立或伪有限计数）在驯服类的一阶结构（稳定，远端，依赖，简单，NTP 2等）的研究，以及关于广义“关联界”及其与Zilber分裂现象的关系的更精细的问题。

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.