Classification of Unstable Theories

提出的研究重点是不稳定理论，主要是具有独立性的理论。本工作的一个方向是从渐近（超功率）的角度研究不稳定理论。具体来说，Malliaris计划研究Keisler秩序的结构，这是一个长期存在的比较理论复杂性的程序，其结构在不稳定理论中自1967年以来一直开放。Keisler的顺序直接比较了独立于语言的理论，人们很早就认识到，这种顺序可能会提供重要的模型理论信息。这项工作的补充方向包括进一步发展模型论和图论之间的联系。例如，Malliaris计划进一步发展超图的特征序列理论（与公式相关），这个框架将图论和有限组合学（如Szemeredi正则）的思想和技术引入不稳定理论中的类型分析，以及不稳定理论中独立性和顺序之间的各种权衡。非正式地说，与图论和超超能力的联系提供了一个视角，在这个视角中，局部噪声被平滑，伪有限结构复杂性的显著跳跃的本质可以更清楚地看到。这是研究独立性的理想环境，在这种环境中，看到由某些随机结构引起的“噪声”或不一致性之间的细微区别以及更多的全局兼容性是至关重要的。从广义上讲，本研究计划旨在为不稳定理论建立一个更统一的结构理论，并旨在发现和进一步发展不稳定理论固有的复杂性的生产性概念。提出的工作是在模型理论，特别是不稳定理论的分类。模型是由底层集合的数据给出的结构，以及在固定的背景语言L中对所有函数、关系和常量符号的解释；理论是一阶逻辑的l句的完全一致集合；而初等类恰恰是某些理论的模型的类，例如特征为0的代数闭域。模型理论的一个主要兴趣是通过模型的基本类来研究理论，也就是说，通过对与其相关的基本类中的可能变化进行分类来研究手边的理论。模型理论分析最丰富的背景之一是稳定性理论，即所谓的稳定或“驯服”理论的研究，由希拉在20世纪70年代发展起来。虽然从稳定性的角度来看，更大的一类非稳定或“不稳定”理论相当复杂，但它们具有重要的意义。许多数学结构是不稳定的，因此提供了理论动机和例子；不稳定理论存在着许多猜想和问题，自然吸引着人们的研究。提出的工作旨在建立工具和视角，从更统一的角度识别和分析不稳定理论中模型理论复杂性的跳跃，以及与有限和渐近组合学发展某些富有成效的类比和相互作用。与此同时，这些工具将通过Keisler命令的大规模结构猜想进行部分测试。

The proposed research focuses on unstable theories, primarily theories with the independence property. One direction of this work involves studying unstable theories from an asymptotic (ultrapower) point of view. Specifically, Malliaris plans to work on the structure of Keisler's order, a long standing program of comparing the complexity of theories, whose structure on the unstable theories has remained open since 1967. Keisler's order directly compares theories independent of language, and it was recognized early on that this order would likely give significant model-theoretic information. A complementary direction of this work involves developing further connections between model theory and graph theory. For instance, Malliaris plans to further develop the theory of characteristic sequences of hypergraphs (associated to formulas), a framework which brings ideas and techniques from graph theory and finite combinatorics such as Szemeredi regularity to bear on the analysis of types in unstable theories, and on various tradeoffs between independence and order in unstable theories. Informally, both the connections to graph theory and to ultrapowers give a perspective in which local noise is smoothed out and the nature of the significant jumps in the complexity of pseudofinite structure can be more clearly seen. This is an ideal context for studying the independence property, in which it is crucial to see finer distinctions between 'noise' or inconsistency arising from certain random structure, and more global compatibility. Broadly speaking, the proposed research program builds toward a more unified structure theory for the unstable theories, and aims to discover and further develop productive notions of complexity inherent to unstable theories.The proposed work is in model theory, specifically the classification of unstable theories. A model is a structure given by the data of an underlying set along with an interpretation for all function, relation, and constant symbols in a fixed background language L; a theory is a complete consistent set of L-sentences of first-order logic; and an elementary class is precisely the class of models of some theory, e.g. algebraically closed fields of characteristic 0. A major interest of model theory is studying theories via elementary classes of models, that is, studying a theory at hand by classifying the possible variations within its associated elementary class. One of the most fertile contexts for model theoretic analysis has been stability theory, the study of so-called stable or 'tame' theories, developed by Shelah in the 1970s. While the much larger class of non-stable or 'unstable' theories are quite complex from the point of view of stability, they are of significant interest. Many mathematical structures are unstable, so provide both theoretical motivation and examples; and there are many conjectures and problems about unstable theories which naturally attract research. The proposed work aims to build tools and a perspective from which to identify and analyze jumps in model-theoretic complexity among the unstable theories from a more uniform point of view, as well as to develop certain productive analogies and interactions with finite and asymptotic combinatorics. At the same time, these tools will be tested, in part, via the large scale structural conjecture of Keisler's order.