Research in Model Theory: Generalized Amalgamation Properties

这个项目是在一阶理论和非初等类的框架下对与广义合并属性相关的对象进行系统研究。这些性质是在希拉的基本工作分类的一阶理论，以及结果优秀的类。初步结果允许推测，失败的广义合并是见证了某些范畴理论的对象。对这些对象的研究是本研究的中心议题。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的理论广义合并代数闭集稳定的一阶理论，延长工作Hrushovski。这样的理论似乎是丰富的，即使是完全范畴理论，可能是独立的兴趣。本研究的一个新的特点是短语广义合并的概念函子嵌入模型的单纯复形（一阶理论或那些原子类）。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（术语“一阶理论”和“非初等类”指的是使用的某些类型的公理）。模型理论的一个子领域，称为分类理论，试图识别一类结构上的条件，这些条件提供了关于该类元素整体行为的实质信息。几何模型理论的更精细的分析试图将结构的组合几何学（用模型理论术语定义）与来自经典已知结构的几何学联系起来。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在不同数学领域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的努力解决一个长期存在的（分类理论）猜想的希拉，并可能有助于发展几何方法的模型理论的非小学类。

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.