Structure Theorems in Model Theory

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。Laskowski正在继续他的模型理论研究，这是数理逻辑的一个分支。他的大部分研究都围绕着在三个不同的背景下获得更好的结构定理这一主题展开。他将继续研究可分类理论模型的精细结构，着眼于对不可数语言的扩展，寻找适用于更多结构类别的谢拉卓越概念的推广，并寻找更多稳定性理论假设自动限制公式量词复杂性的实例。此外，Laskowski将尝试回答关于依赖公式、阿基米德有序阿贝尔群的初等扩展、理论的Schroeder-Bernstein性质以及符号动力学中的一个长期存在的可判定性问题的具体问题。模型理论关注的是理论之间的相互作用，即非常形式化语言中的句子集，以及满足这些句子的代数结构(模型)的类。有一个完善的理论分类，它基于某些元素配置在理论模型中的可嵌入性或不可嵌入性。Laskowski已经确定了与数据压缩和计算学习理论的局限性有直接联系的某些理论类别，并将继续他对这些理论的检验。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Laskowski is continuing his research in model theory, which is a branch of mathematical logic. The bulk of his research is organized around the theme of obtaining better structure theorems in three distinct settings. He will continue his investigations of the fine structure of models of classifiable theories with an eye on extensions to uncountable languages, look for a generalization of Shelah's notion of excellence that is applicable to more classes of structures, and seek more instances where stability theoretic assumptions automatically bound the quantifier complexity of the formulas. Additionally, Laskowski will attempt to answer specific questions about dependent formulas, elementary extensions of expansions of Archimedean ordered abelian groups, the Schroeder-Bernstein property of a theory, and a long standing decidability question from symbolic dynamics.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. There is a well established taxonomy of theories, which is based on the embeddability or non-embeddability of certain configurations of elements into models of the theory. Laskowski has identified certain classes of theories that have direct connections to limitations on data compression and computational learning theory, and will continue his examination of these theories.