Research in Model Theory: Generalized Amalgamation Properties

本课题是在一阶理论和非初等类的框架下，系统地研究与广义合并性质有关的对象。这些性质是Shelah关于一阶理论分类的基础工作的核心，也是优秀类的结果。初步结果可以推测，某些范畴论对象见证了广义合并的失效。对这些物体的研究是本研究的中心主题。Kolesnikov的方法允许将一阶理论和非初等类中的广义合并性质视为常见现象的特殊情况。本研究的出发点是发展稳定一阶理论中代数闭集的广义合并理论，扩展了赫鲁晓夫斯基的工作。这样的理论即使对于完全直言的理论来说也显得很丰富，而且可能具有独立的兴趣。本研究的一个新特点是将简单复合体的泛函嵌入到模型（一阶理论或原子类的模型）中来表达广义合并的想法。用同调代数对广义合并的预测表征将导致新思想涌入一阶理论和非初等类的模型理论。模型论是数学逻辑的一个分支，旨在分析以某种方式公理化的数学结构类（术语“一阶理论”和“非初等类”指的是所使用的某些公理）。模型理论的一个子领域叫做分类理论，它试图确定一类结构的条件，这些条件提供了有关该类元素整体行为的大量信息。几何模型理论的精细分析试图将结构的组合几何（用模型理论术语定义）与来自经典已知结构的几何联系起来。广义合并性质起源于分类理论。人们期望科列斯尼科夫的研究将把性质的失效与另一个数学领域所研究的某些对象的存在联系起来。这项研究将有助于更广泛地解决Shelah长期存在的（分类理论）猜想，并可能有助于发展非初等类模型理论中的几何方法。

This project is a systematic study of objects related to generalized amalgamation properties in the frameworks of first-order theories and non-elementary classes. The properties are at the heart of Shelah's fundamental work on classification of first-order theories, as well as the results on excellent classes. Preliminary results allow to conjecture that failure of generalized amalgamation is witnessed by certain category-theoretic objects. Study of such objects is the central topic of this research. Kolesnikov's approach allows to treat generalized amalgamation properties in first-order theories and non-elementary classes as particular cases of a common phenomenon. The starting point of this research is the development of a theory of generalized amalgamation for algebraically closed sets in stable first-order theories, extending the work of Hrushovski. Such a theory appears to be rich even for totally categorical theories and could be of independent interest.A novel feature of this research is the idea to phrase generalized amalgamation in terms of functorial embeddings of simplicial complexes into models (of a first-order theory or those of an atomic class). The projected characterization of generalized amalgamation in terms of homological algebra would lead to influx of new ideas into model theory of both first-order theories and non-elementary classes.Model theory is a branch of mathematical logic that aims to analyze classes of mathematical structures that are axiomatized in some way (the terms "first-order theory" and "non-elementary class" refer to certain kinds of axioms that are used). One of the subfields of model theory, called classification theory, attempts to identify conditions on a class of structures that provide substantial information about the overall behavior of the elements of the class. The finer analysis of geometric model theory attempts to connect the combinatorial geometry of structures, defined in model-theoretic terms, with geometries coming from classically known structures. Generalized amalgamation properties originated in classification theory. The expectation is that Kolesnikov's research will connect the failure of properties to the existence of certain objects studied in a different area of mathematics. This research will contribute to the broader efforts on solving a long-standing (classification-theoretic) conjecture of Shelah and, potentially, will help in developing geometric methods in model theory of non-elementary classes.