Unstable Model Theory

本研究将运用模型理论的方法研究不稳定理论模型中有限组合对象族之间的一般相互作用。这些问题自然承认并受益于图论和有限组合学的思想和方法。更准确地说，Malliaris将致力于进一步发展特征序列理论，特征序列是在一阶公式的参数空间上定义的超图的可数序列。Malliaris已经证明了图论和组合学的一系列技术，包括Szemeredi正则，可以通过特征序列应用于模型论结构。Malliaris建议进一步探索这些有前途的联系，这些联系与不稳定理论中的许多结构问题有关，包括有序和独立的精细结构以及具有独立性的理论的分类，以及在此背景下理解边缘密度和超图正则性等图论现象的模型论意义。此外，这些研究揭示了类型族在规则超功率中实现和省略的方式，因此与长期存在的不稳定理论（可数理论的预序，粗略地说，衡量产生饱和规则超功率的难度）上的Keisler序结构的开放问题有关。模型理论是数学的一个分支，它研究某一类数学对象的基本结构，即给定理论的模型。在给定的类内（或不同类的可比较模型之间）发生的结构变化揭示了理论本身固有的简单性或复杂性。这项提议的工作源于新的迹象，即某些理论可能具有深刻的、以前未被发现的结构相似性。这些相似性与可能的极限行为有关，也就是说，与理论模型中有限组合对象族产生的无限构型的复杂性有关。这项工作的一个广泛目标是开发更精细的工具来检测理论的某些不变量中的这种行为（例如，图在某些T公式的特征序列中的持久性和分布）。这些想法将为正在进行的对不稳定理论进行分类的程序提供新的杠杆作用。在过去的四十年里，模型理论分类理论已经开发出一个信息量丰富的工具箱，用于分析一阶理论的复杂性，并分离出许多有用的复杂性指标，但仍有许多工作要做。特别是，这项工作的第二个潜在目标是提供一种语言，用于精确地询问和探索这些指标的分布和密度的大部分开放问题：它们如何和在哪里聚集，以及它们如何与模型中的其他对象交互，例如类型的基础集。

The proposed research will apply methods from model theory to study the generic interactions between families of finite combinatorial objects in models of unstable theories. These questions naturally admit, and benefit from, ideas and methods from graph theory and finite combinatorics. More precisely, Malliaris will work to further develop the theory of characteristic sequences, which are countable sequences of hypergraphs defined on the parameter space of a first-order formula. Malliaris has shown that a deep collection of techniques from graph theory and combinatorics, including Szemeredi regularity, can be brought to bear on model-theoretic structure via the characteristic sequence. Malliaris proposes to further explore these promising connections, which are relevant to many structural questions in unstable theories, including the fine structure of order and independence and the classification of theories with the independence property, as well as to understanding the model-theoretic significance of graph-theoretic phenomena such as edge density and hypergraph regularity in this context. Moreover, these investigations shed light on the ways in which families of types are realized and omitted in regular ultrapowers, and are therefore relevant to the longstanding open problem of the structure of Keisler's order on unstable theories (a preorder on countable theories which, roughly speaking, measures the difficulty of producing saturated regular ultrapowers).Model theory is a branch of mathematics which studies the fundamental structure of certain classes of mathematical objects, the models of a given theory. The structural variation which occurs within a given class (or between comparable models of different classes) sheds light on the inherent simplicity, or complexity, of the theory itself. The work in this proposal arises from new indications that certain theories may have deep and previously undetected structural similarities. These similarities have to do with possible limit behavior, that is, with the complexity of infinite configurations which arise from families of finite combinatorial objects within models of the theory. One broad aim of this work is the development of finer tools to detect this behavior in certain invariants of the theory (e.g. the persistence and distribution of graphs in the characteristic sequence of certain formulas of T). These ideas would give new leverage in the ongoing program of classifying unstable theories. Model-theoretic classification theory has, over the last forty years, developed a richly informative toolbox for analyzing the complexity of first-order theories and isolated many useful indicators of complexity, but much remains to be done. In particular, a second underlying goal of this work is to give a language in which to precisely ask, and to explore, the largely open questions of distribution and density of these indicators: how and where they cluster and how they interact with other objects in the model, e.g. base sets for types.