Mathematical Sciences: Research in Model Theory: Strongly Minimal Fusions and Other Homogeneous-Universal Models

9510377 Holland Holland的研究领域是数学逻辑的一个分支——模型论。近年来，许多模型理论家感兴趣的新结构被构造为具有维数概念的有限模型类的齐次泛型模型。在这些新结构中，有赫鲁晓夫斯基采用聚变方法在现有结构基础上建造的新的强最小结构。霍兰德已经开始从几何角度对任意融合进行全面、系统的研究。由此获得的新视角使荷兰能够在特定情况下制作比赫鲁晓夫斯基更简单，更透明的建筑。她打算全面推广这种简化，并继续系统地研究融合，从几何的角度来看，看看从强最小结构中产生的几何分类提供了什么新信息，从模型理论的角度来看，看看强最小结构的变化是否可能提供模型理论兴趣的进一步例子。与此同时，Holland打算建立一个统一的框架来看待所有关于同质普遍模型的新研究。Holland打算利用规划期来完善她的研究项目，形成具体的、可信的推测，这些推测应该构成一份可信的NSF提案。＊＊＊

9510377 Holland Holland's research is in model theory, a branch of mathematical logic. In recent years, many new structures of interest to model theorists have been constructed as homogeneous-universal models for classes of finite models with a notion of dimension. Among these new structures are the new strongly minimal structures built from existing ones by the method of fusion introduced by Hrushovski. Holland has begun a general, systematic study of arbitrary fusions from a geometric point of view. The new perspective gained therefrom enabled Holland to produce a simple, more transparent construction than that of Hrushovski in a particular case. She intends fully to generalize the simplification and to continue the systematic study of fusions both from a geometric point of view, to see what new information the provide on the classification of geometries that arise from strongly minimal structures, and from a model theoretic point of view, to see whether variations on the strongly minimal construction might provide further examples of model theoretic interest. In parallel, Holland intends to work toward a unified framework in which to view all of the new work on homogeneous-universal models. Holland intends to use the planning period to refine her research project, formulating the specific, plausible conjectures that should constitute a credible NSF proposal. ***