Investigations in Model Theory

鲍德温提出了稳定性理论、代数模型理论和有限模型理论的研究。Baldwin将继续研究关于强子模型概念的齐次普遍模型的构造。该方法的一个应用是将有限模型上的稳定性理论与概率联系起来，不仅提供了一种证明0-1定律的技术，而且获得了几乎确定理论的模型理论性质。几乎确定理论的这些模型理论性质可以应用于有限模型理论中的问题。鲍德温是将稳定性理论应用于有限多变量逻辑发展的先驱。这项工作第一次将Cherlin、Harrington、Lachlan和Zilber关于同质结构的深入研究与有限模型理论的主流联系起来。Baldwin和Baizhanov正在研究用任意谓词展开稳定结构；这是由早期与Benedikt在嵌入式有限模型理论方面的工作所激发的。Baldwin与Holland的早期工作包括在代数领域的应用，特别是群论，以及用任意子集展开复数。本课题将尝试寻找一些特定的新结构，通过乘群的子群来展开复数。鲍德温的大部分工作都是在模型论，一个数理逻辑的分支。这项工作的总体目标是在更高的抽象层次上理解“普通数学”。这种抽象允许在广泛不同的数学领域中发现共同的特征，从概率论到组合到代数几何。这类工作在更好地理解代数结构和为数据库理论研究提供背景方面都卓有成效。特别是，Baldwin与朗讯科技公司（Lucent Technologies）的合著者Michael Benedikt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，主要致力于开发创新和有效的方法来培养数学教师。这项教育工作与另外两个国家科学基金会资助的项目有关。

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.