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。鲍德温将继续研究关于强子模型概念的齐次通用模型的构造。这种方法的一个应用将稳定性理论与有限模型上的概率联系起来，不仅提供了证明0-1定律的技术，而且提供了一种获得几乎必然理论的模型理论性质的技术。几乎必然理论的这些模型论性质可以应用于有限模型论中的问题。鲍德温一直是将稳定性理论应用于有限变量逻辑发展的领导者。这项工作首次在Cherlin、Harrington、Lachlan和Zilber关于均匀结构的深入工作和有限模型理论的主流之间提供了严肃的联系。Baldwin和Baizhanov正在研究稳定结构通过任意谓词的扩展；这是受到Benedikt早期在嵌入式有限模型理论方面的工作的启发。鲍德温与霍兰德的早期工作包括在代数领域的应用，特别是群论，以及复数通过任意子集展开的应用。这个项目将试图找到一些特定的新结构，通过乘法群的子群来扩展复数。鲍德温的大部分工作都是在数理逻辑的一个分支--模型理论中进行的。这项工作的总体目标是在更高的抽象水平上理解“普通数学”。这种抽象允许在从概率论到组合学再到代数几何的广泛不同的数学领域中发现共同的特征。这类工作在更好地理解代数结构和为数据库理论的研究提供背景方面都是卓有成效的。尤其值得一提的是，鲍德温与朗讯科技的合著者迈克尔·贝内迪克特在《嵌入式有限模型理论》方面的研究发现，数据库查询的可表现性受到了限制。鲍德温将继续他的教育工作-主要专注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NSF赞助的另外两个项目有关。

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.