Topics in Model Theory

模型理论主题

• 批准号: 0071746
• 负责人: Michael Chris Laskowski
• 金额: $ 7.98万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071746&HistoricalAwards=false
• 关键词: Topics; Model; Theory

ABSTRACTLaskowski is continuing his research on three projects within the realm of model theory. First, in collaboration with Hart and Hrushovski, the investigator is seeking to determine the `fine structure' of models of classifiable theories. This amounts to determining the amount of flexibility one has in choosing a decomposition tree for a model of such a theory. Recently, Laskowski has discovered that any shallow, strictly stable theory that does not have the dimensional order property interprets an infinite group. In collaboration with Shelah, Laskowski is seeking to understand the extent to which this group determines the dependence relation of dividing. Finally, in collaboration with Dolich, Laskowski is seeking to identify a class of theories that contains the simple theories, yet also a satisfactory notion of independence. Model theory is concerned with the interplay between theories (i.e., sets of sentences in a very formal language) and the classes of algebraic structures (models) that satisfy these sentences. Most of Laskowski's research to date concerns broad classes of theories, as opposed to concentrating on a specific theory or structure. Because of the breadth of the context, applications have been found in other areas of mathematics and in computer science, specifically in the analysis of structured data bases. These classes of theories are described in terms of the embeddability or non-embeddability of certain configurations of elements into models of the theory. Laskowski is continuing to analyze three classes of theories. In all three cases, the combinatorial assumptions on the class of theories imply that the models of such theories contain some algebraic content. The investigator is studying the extent to which this algebraic component determines the behavior of a model of the theory.

拉斯科夫斯基正在继续他在模型理论领域内的三个项目的研究。首先，研究人员与哈特和赫鲁晓夫斯基合作，试图确定可分类理论模型的“精细结构”。这相当于决定了为这种理论的模型选择分解树时所具有的灵活性。最近，Laskowski发现，任何不具有维序性质的浅的、严格稳定的理论都可以解释无限群。在与谢拉的合作中，拉斯科夫斯基试图了解这个群体在多大程度上决定了分裂的依赖关系。最后，与多利奇合作，拉斯科夫斯基正在寻求确定一类既包含简单理论，又包含令人满意的独立概念的理论。模型论关注的是理论(即一种非常正式的语言中的句子集合)和满足这些句子的代数结构(模型)类别之间的相互作用。到目前为止，Laskowski的大部分研究都涉及广泛的理论类别，而不是专注于特定的理论或结构。由于上下文的广泛性，已经在数学和计算机科学的其他领域中找到了应用，特别是在结构化数据库的分析中。这些理论类别是根据某些元素配置在理论模型中的可嵌入性或不可嵌入性来描述的。Laskowski正在继续分析三类理论。在所有这三种情况下，关于理论类的组合假设意味着这类理论的模型包含一些代数内容。研究人员正在研究这种代数成分在多大程度上决定了理论模型的行为。