Investigations in Model Theory

模型理论研究

• 批准号: 0100594
• 负责人: John Baldwin
• 金额: $ 9万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100594&HistoricalAwards=false
• 关键词: Investigations; Model; Theory

Baldwin proposes investigations in stability theory, the model theory of algebra and finite model theory. Baldwin will continue to investigate the the construction of homogeneous universal models with respect to a notion of strong submodel. One application of this method links stability theory with probability on finite models by providing a technique for not only proving 0-1 laws but obtaining model theoretic properties of the almost sure theory. These model theoretic properties of the almost sure theory can be applied to problems in finite model theory. Baldwin has been a leader in the application of stability theory to the development of logic with finitely many variables. This work has, for the first time, provided serious links between the deep work of Cherlin, Harrington, Lachlan, and Zilber on homogeneous structures and the mainstream of finite model theory. Baldwin and Baizhanov are studying expansions of stable structures by arbitrary predicates; this is motivated by the earlier work with Benedikt on embedded finite model theory. Baldwin's earlier work with Holland has included applications in the area of algebra, specifically group theory, and in expansions of the complex numbers by an arbitrary subset. The present project will try to find some specific new structures which expand the complex numbers by subgroups of the multiplicative group.Most of Baldwin's work has been in model theory, a branch of mathematical logic. The general aim of this work is to understand `ordinary mathematics' at a higher level of abstraction. This abstraction allows the discovery of common features in widely different areas of mathematics, ranging from probability theory to combinatorics to algebraic geometry. This kind of work has been fruitful both in a better understanding of algebraic structures and in providing a background for investigations in database theory. In particular, Baldwin's work with his co-author Michael Benedikt of Lucent Technologies on `embedded finite model theory' has found limits on the expressibility of database queries. Baldwin will continue his educational work - primarily focusing on developing innovative and effective ways to prepare mathematics teachers. The educational work is connected with two other NSF sponsored programs.

鲍德温提出调查稳定性理论，模型理论的代数和有限模型理论。Baldwin将继续研究关于强子模型概念的同质通用模型的构造。这种方法的一个应用链接的稳定性理论与概率有限模型提供了一种技术，不仅证明0-1法律，但获得模型理论的性质几乎肯定的理论。 几乎处处理论的这些模型论性质可以应用于有限模型理论中的问题。Baldwin是将稳定性理论应用于多变量逻辑发展的领导者。这项工作，第一次，提供了严重的联系之间的深入工作的切尔林，哈灵顿，拉克兰，齐伯均匀结构和主流的有限模型理论。Baldwin和Baizhanov正在研究任意谓词的稳定结构的扩展;这是由早期与贝内迪克特在嵌入有限模型理论上的工作所激发的。鲍德温的早期工作与荷兰已包括应用领域的代数，特别是群论，并在扩大复杂的数字由任意子集。本项目将试图找到一些具体的新结构，扩大复杂的数字由子群的乘法group.Most鲍德温的工作一直在模型论，一个分支的数理逻辑。 这项工作的总目标是理解“普通数学”在更高的抽象层次。 这种抽象允许发现广泛不同的数学领域的共同特征，从概率论到组合数学到代数几何。 这种工作已经卓有成效，在更好地理解代数结构，并在数据库理论的调查提供了一个背景。特别是，Baldwin与他的合著者Lucent Technologies的Michael贝内迪克特在“嵌入式有限模型理论”方面的工作发现了数据库查询的可表达性的限制。鲍德温将继续他的教育工作-主要集中在开发创新和有效的方法来准备数学教师。 教育工作与其他两个NSF赞助的计划。